/mg/

I'd be surprised if anyone here can actually go to Gelfand's Algebra and solve every problem unaided

>we use shit tier books imo
That's to be expected. Is there anything stopping you from using good books?

wlog let b>a
0 < (b-a)^2
0 < b^2 + a^2 - 2ab
2ab < a^2 + b^2
2ab/4 < a^2/4 + b^2/4
ab < a^2/4 + b^2/4 + 2ab/4
ab < ((a+b)/2)^2
sqrt(ab) < (a+b)/2

QED

I'm going through good books right now, because I was growing tired of the happy medium my school aims for (trying to please everyone with the same classes, resulting in me learning calculus like an engineer).

Could one single user please answer this question, ,
and not simply throw shadows and projections of all colors my way? I'm beginning to think if everyone's reading comprehension is such shit I shouldn't be listening to any of you anyway.

>What's the fastest route towards mathematical maturity?
That's an open problem.
>Inhaling as much rigorous mathematical text and resources that your mind allows
That's certainly a part of it.
>exhaling nothing but solutions to difficult exercises
This is worthless if you don't actually understand the solutions.

>learning calculus
If you're learning it at all you have already failed.
>reading comprehension
You might want to check out for that. This is a mathematics thread.

>if you're learning it at all you have already failed
Lol what?

>Veeky Forums
massive top kek, r u seriously implying math has nothing to do with reading comprehension?

I know this, but this is still a useless reply. Certainly there's opinions, and I'm inviting you to share yours, and critique mine

Are you meaning to say that one should rather be learning analysis, not calculus?

wrong user, meant for

>/mg/ is actually anything other than about magnesium

>but this is still a useless reply.
The first sentence of that reply implies this is currently the only kind of reply you can possibly get.
>I'm inviting you to share yours
I believe I did. "solving" problems without really understanding them isn't really that valuable, I think it's too obvious to even be stated.

>learning analysis
If you're learning it at all you have already failed.

Well for one, your goal is stupid. Get a real goal. But before you do that, you need to develop taste. That's acquired by doing things on your own. Kind of like that wizard in the picture did, but you don't need to be that extreme obviously.

The more you read books and less you try to figure things out on your own, the more dependent you get on reading what other people produce other than producing things for yourself.

Right now your goal is just "I wanna be smart," when really your mindset is all wrong, because you don't even seem to really appreciate math at all.

That's fine, if you want to be an applied mathematician of course. Math should be a means to an end. Just make sure you understand what you're actually looking for. If your goal is really so imprecise as this, then you really have no possible way to find a fast route cause you don't even really have a goal.

Obviously you would ideally intend to understand the problems. I, too, thought such things where too obvious to be stated, yet here I am. And yes, I know my first sentence admitted to such a question lacking a definite answer, but any non-autist would identify that I'm looking for an individuals take on said question, not facts that are to be objectified.

dummy detected, I don't even know what you're trying to communicate anymore other than vague attempts at perturbing my collection of cast iron jimmies.

Achieving mathematical maturity is certainly a very real goal. And I am doing plenty of things on my own.

My goal is not "wanting to be smart". At least not the one I'm stating. My goal is to be able to transition from texts like Stewart's "Calculus" to Rudin's "Principles of Mathematical Analysis" efficiently, and right now I'm banging my head against a wall 6 hours a day trying to plow through Spivak and Apostol's work on Calculus, with Landau's on Analysis in an attempt to get there. I'm looking for critiques of my current approach, and suggestions as to where it might be improved.

>You don't even seem to appreciate math at all
LOL, again, nice projections. You would think my departure and dissatisfaction from my schools Stewart and friends (TM) curriculum says, if anything, that I do appreciate mathematics. This demonstrates, again, the shit projections I'm getting, rather than constructive and coherent advice.

I posted this at the end of the last thread but then it died, so:

Is the Evan Chen Napkin a good way to get a crash course/"just a taste" of the major topics in mathematics? I studied pleb CS and applied math at university but I've been enjoying working through Dummit+Foote when I'm off work

>"just a taste"
Princeton's Companion to Mathematics (or whatever it's called), but I don't know what you're talking about. If you want a more thorough taste and oversight of mathematics, perhaps try John Stillwell's "Elements of Mathematics".

How far through apostol and spivak are you out of curiosity

You're gonna roast me but I'm not a third of the way through either. I ask you only to withhold any assumptions and projections in your reply, and I'd bet money that I'll be done with both before Christmas. I recently stumbled upon a wealth of free time, and I'm looking to take advantage of it.

So basically I mean a pretty reasonably-sized introduction to a broad range of concepts with a few sample problems but not a crazy amount of rigor (the rigor would come with subsequent explorations of more rigorous textbooks of the specific pieces that I find most interesting. Again, I'm now a run-of-the-mill pleb software engineer, not a research mathematician.)

I've got a copy of the Princeton Companion to Mathematics but it generally seems like it's better as a quick reference, not a "read through this and then you'll have learned a bunch" since - correct me if I'm wrong - the layout of the book isn't really pedagogical?

you should withhold your assumptions that i would assume anything or make projections at all!

From the sounds of your situation you should try focusing on Apostol. Spivak can be pretty balls to the walls at times, although people will probably disagree with me.

Do you have much discussion with people about mathematics outside of Veeky Forums? Professors or classmates?

Forgive me, both for my prior assumptions and for this upcoming post (I’m on mobile).

>convserations
I frequent #math on freenode, /mg/ and /sqt/ on here and a couple other math forums like stack exchange when I’m really stuck (in order of frequency and duration). When I’m enrolled, I very rarely have conversations with classmates or professors.

>apostol before spivak
Thank you for saying this, because this is what my personal experience has been suggesting, yet everywhere else I’ve read to proceed in the opposite order.

Actually, expanding on that last bit about communication, I’ve even been considering lately paying a PhD or so out of pocket to proof read my answers to some of the text I’m self studying, and since this curious situation has been nagging at me, if you have alternative suggestions I’d be absolutely open to them. I’m just trying to minimize the hurdles an autodidact faces, while simultaneously reaping the benefits of human communication in learning.

This thread needs more anime

Why don't you post some then?

k

> I’ve even been considering lately paying a PhD or so out of pocket to proof read my answers to some of the text I’m self studying
a very good idea, however
>When I’m enrolled, I very rarely have conversations with classmates or professors.

The benefit of being at university is that you are already paying these people for that purpose! Don't hesitate to ever visit your professors in office hours to talk about stuff outside of the current course. You will find that people who are passionate about math usually love to talk shit to other people about it, especially students who come in with questions! Just make sure you know exactly what it is you're asking so you can be to the point.

Honestly I have made very good social connections with professors just by routinely going in and starting conversations about problems i'm struggling with. It is absolutely fine to be an autodidact but don't ignore how enlightening a good talk with another human being can be.

...

>know exactly what it is you're asking
I agree completely with asking the people I'm paying for such a service, but I've never had a question an internet search or post hasn't answered in time. This puts me in this weird phase I'm at right now, where I feel I'd learn faster and cheaper at home, but I know there's no way to do math professionally like that, afaik (I'm even LaTeXing my notes and solutions for a future github post to buff out my resume, need be).

If one could get by as an autodidact mathematician and bypass undergrad I'd put all my eggs in that basket, I know I could and would do it, but unfortunately it doesnt seem to be an option and I'll probably be paying to continue to be a certified autodidact.

Pls respond though user, I've never understood the 'ask questions, talk to professor and classmate' memes. I did when i was in high school, but now I'm much more adept at simply using the internet.

What's a good classical mechanics textbook?

Sorry user I was away from the computer~
How can I put this... I'd like to say that you won't understand the meme til you give it a go but that isn't very constructive.
I feel much the same way as you about being an autodidact and learning using the internet, I rarely attend classes in person. Math lends itself to this kind of behavior.
But I guess I'm old fashioned because sometimes I've forced myself to not look up questions on google, saving it to go make a visit. In my experience using the internet or forums can definitely get you answers and progress, but face to face conversation gives you exactly that; conversation, discussion. While you sound like a smart individual these professors are usually years of invaluable experience ahead of you. They can provide insights into conclusions you would have never reached yourself. Also you get more than just an answer, when you build relationships with these people they can give you personalized recommendations for further study etc etc. also as an autodidact sometimes you get sick of learning all these really cool things and having nobody to share it with.
But yeah the internet can surely provide all of those things too, maybe face to face conversation is a meme?

I could be mean and point out that you're really only just starting as a mathematician, you might as well give the meme a try before deciding this way or that.

Maybe what I'm trying to say is that Math really is meant to be a collaborative thing, it's certainly a lot funner that way (imo).

Kleppner - Introduction to Mechanics.

>saving it for a visit
Interesting idea! I'm always adverse to this, because I can't help but feel before using the professors time, that I should be trying to figure it out for myself. But as you go on to mention:

>provide insights, build relationships, personalized recommendations, having somebody to share it with
These were all really solid reasons, maybe I'll save a couple questions next time that I feel might be better answered in person - or perhaps I'll just intentionally and routinely be a bit lazy and force myself to ask for clarification! I think I'm also a tad dissuaded because I'm at a community college, so it's not like any of these people I meet will be real 'connections' for research, internships or what have you - but I have passed up a handful of really cool professors I wish I had gotten to know better, I'm sure there's so much untapped insights I could've gotten out of them (I don't come from an educated crowd, so it's hard to find conversations like that irl).

Anyhow thank you, kind user.

I am looking for a way to determine the number of points one can select from a 2x2x2x2 terrasect (4-cube) without forming an isosceles triangle or a line of 3 points. Anyone got any ideas?

As in the inequality for n terms

I don't get it.
If we choose b3 = b2 = 1, and b1 = 2, then this system clearly isn't consistent because if we take the top row and we (for example) let x1 = x2 = x3 = 1, then we have 1 - 2 + 5 = 4 =/= b1, hence the system isn't consistent.
Am I missing something here?

>Am I missing something here?
Yes. The system being consistent means that it has Some solution. Not that All (x1,x2,x3) are solutions.

Oh woops, that makes sense.
But another thing I don't get, is why did he take the bottom row and say that the system is always consistent when 0 = b3 + b2 - b1? Sure, I can see that in the matrix for row 3, but how does he know that the other two rows are consistent too if row 3 is consistent?

If the system is consistent, then 0 = b3 + b2 - b1 must hold.
But,you are right, the solution doesn't prove "If 0 = b3 + b2 - b1 holds, then the system is consistent".

Anyway, it's better to think this stuff in terms of linear maps.
Name the first matrix in the pic: A.
Consider the column vector x=(x1,x2,x3)^T
Consider the linear map x|-->Ax.
The vectors that lie on the image of that map are, by definition, precisely the vectors (b1,b2,b3)^T that make the system consistent.
A(x1,x2,x3) = x1 (first column of A)+ x2(second column of A) + x3 (third column of A)
Therefore the image is composed precisely by all the linear combinations of the columns of A.
So, just take A and column reduce (the span stays the same). Or equivalently take A^T and row reduce.
wolframalpha.com/input/?i=row reduce {{1,4,-3},{-2,-5,3},{5,8,-3}}
You get image of A = span( (1,0,1) , (0,1,-1) ) = (c1,c2,c1-c2); c1, c2 in R
If (b1,b2,b3) is in the image, then b1 can be anything, b2 can by anything and b3=b1-b2 which can be rewritten as b3+b2-b1=0.

A non-zero element in a prime ideal contains a prime factor.

Help.

consider any element x in the prime ideal P. Then write x as a product of irreducibles. Since P is prime, the at least one of these irreducibles y is in P, so x can be written as x=yz, with prime factor y

What is an example of a continuous function [math]f:\mathbf{R}^2\to(0,\infty)[/math], [math]\lim_{\|x\|\to\infty}f(x)=0[/math] such that it has infinitely many local maxima but has no local minima?

> find a mistake in lecture notes
> decide to correct it
> fall into a rabbit hole
> eventually prove everything
> feel better
Why do I feel better even if I know that I will forget 90%?

Let [math]M[/math] be a closed 2-dimensional manifold and [math]U_\alpha[/math] be an open cover with [math]\phi_\alpha:U_\alpha \rightarrow\mathbb{R}^2[/math] homeomorphisms. Let [math]f:\mathbb{R}^2 \rightarrow (0,\infty)[/math] and put [math]f_\alpha = f \circ \phi_\alpha[/math], then by the gluing lemma one can patch together [math]f_\alpha[/math] with the coordinate transition functions to form a continuous function [math]g:M \rightarrow (0,\infty)[/math]. Now put [math]\mathfrak{f} = g|_{[0,1]}[/math], then [math]\mathfrak{f}[/math] is a Morse function and hence its degree is proportional to its Euler chaaracter, and by the Gauss-Bonnet theorem [eqn]\chi(M) \propto \int_{\mathbb{R}^2} d^2x \sum_i \delta(x-x_i),[/eqn] where [math]x_i[/math] are the regular singular points of the pullback [math]\mathfrak{f}^*[/math] along [math]\phi_\alpha[/math]'s onto [math]\mathbb{R}^2[/math]. This means that you can construct, for [math]M[/math] such that [math]\chi(M) = 1[/math], a function [math]g[/math] such that [math]\mathcal{f}[/math] has one local max in the interval [math][0,1][/math], and you can be sure that they cannot have a local min, since that'd make the Euler characteristic of [math]M[/math] less than 1. By picking infinitely many [math]M[/math] with this property, you can construct infinitely many such [math]g_i[/math]'s from [math]M_i\rightarrow [0,1][/math].
The tough part is the issue of gluing these [math]g[/math]'s together such that [math]g_i[/math] maps to [math][i,i+1][/math]. If you can do it then you can just pick patches [math]U_{\alpha,i}[/math] to restrict to and then precompose it with [math]\phi_{\alpha,i}[/math] to obtain the desired map [math]f:\mathbb{R}^2 \rightarrow [0,\infty)[/math].

>get stuck on problem for an hour
>look at solution
>it's so fucking easy and I feel like a retard.

How do i solve this

wtf you stupid mod stop deleting every post

[eqn]f(x,y) = \frac{2+\cos(x)}{1+x^2+y^2} [/eqn]

>stats is not math
i'm on the second lecture and everything is math, proving estimators converge, proving that the quadratic error converges to zero, checking which estimator has least variance etc

Yeah, that part of stats is actually math.

download more iq

an AR(p) process [eqn]y_n = c + \sum\limits_{i=1}^{p}\alpha_i y_{n-i} + \epsilon_n,
\ \epsilon_n \sim \mathcal{N}(0, \sigma)[/eqn] is said to be stationary if the roots of the polynomial [eqn]x^p - \sum\limits_{i=1}^{p}\alpha_i x^{p-i}[/eqn] all lie within the unit circle.

how do i sample uniformly from the set [math]\left\{(\alpha_1, \cdots, \alpha_p) \in \mathbb{R}^p \ \vert \ \vec{\mathbf{\alpha}}\ \text{makes for a stationary}\ AR(p)\ \text{process} \right\}[/math]

didn't mean to quote, sorry.

This is probably a retarded question but what use does the Cauchy-Schwarz inequality actually have?

Okay, so I have a background in set theory and some proofs along with predicate logic, but I want to get more into category theory for practical usage as a software developer,as many concepts appear frequently

Any good intro courses?

Coming from a CS/CE background

It's called Egg of Columbus (you have no idea how hard it was for me to find that term)

The only general way to get past this is by doing more practice. You could also try looking up whatever it is you're learning and seeing if there's any other insight that helps your understanding of whatever subject you're on.

It's integral for probability, as covariance depends on it. It's useful in linear algebra for the triangle equality as well. You see it used in some proofs in linear algebra at the beginning every so often because of this.

What possible proofs are there of an integer-coefficient polynomial being irreducible over the rationals? I'm aware of Cohn's criterion, Klein's criterion, and Eistenstein's criterion.

When you prove everything up to a point you get a solid fundamental understanding of the subject up to that point. In no way is it useless; there's plenty of times where doing this not only helps you later on in further courses, but also allows you to teach or tutor others on the subject as well.

I don't know about courses but...

bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/

youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_

youtube.com/watch?v=3XTQSx1A3x8&list=PLbgaMIhjbmElia1eCEZNvsVscFef9m0dm

I used it in my riemannian geometry class just the other day

Gauss lemma

thx

My favorite is checking that it's irreducible mod some prime. Apologies if that's one of the first two you listed; I don't recognize the names.

What subject does \mathbb{F}_4 ^{3} refer to? I know it has something to do with vectors

It's the F_4 is probably the (unique) field of 4 elements.
en.wikipedia.org/wiki/Finite_field

(F_4)^3 probably means considering the triplets of elements of F_4 as a vector space over F_4.

Is there an explicit definition of an (∞,n)-category for arbitrary n?

Like there is for an (∞,1)-category in terms of simplicial sets.

Surprisingly many, see "A Cauchy Schwarz master class" (you can find it on google). It's one of the basic tools for proving inequalities (which is what analysis is all about), very easy to prove, and, when used right, occasionally really sharp

>>Higher-dimensional categories are like a vast mountain that many people are trying to conquer. Some intrepid explorers have made the ascent, each taking a different route and each encountering different hazards. Each has made a map of his route, but do we know how all these maps fit together? Do we know that they fit together at all? In fact, are we even climbing the same mountain?
From the book by Cheng and Lauda.

Hey Veeky Forums check my career reasoning plox:

>don't know what I want to do
>just vaguely know it'd be comfy if it'd involved math
>am undergrad, have to declare major

So,
>stats
>math
Pick one.

AFAIK, statistics is really in demand in many fields, especially if you know how to code. I don't want to close any doors, and to the best of my knowledge, it's best to keep undergrad more general, and grad school more specialized, so I'm thinking I'll do my undergrad in math. That should leave industry jobs open for stats/machine learning positions open right after uni right? Or do i really HAVE to major in stats to get that? And this would be keeping /comfy/ grad school options open for either pure, applied or stats?

Why does higher category theory have to be such a bitch?

Do whatever the fuck you want. It almost literally couldn't matter less.

one would certainly lead to more reduced options though, mad user

How is your retarded question appropriate for this thread? Fuck off.

How to really learn mathematics?

Whatever

Look for an unsolved problem and keep working on it until you find a solution. Then go to the next problem. That's how all famous mathematicians got their skills.

It really won't.

Thanks

Consider a set A with m elements and a set B with n elements. How many relations are there from A -> B such that the relations are also functions?

I want to say m*n relations

Nevermind, it's n^m. Remember not to pick example sets with 2 elements each.

Learning higher category theory is a bit like eating a far too big pizza. It's hard to stomach at first, but in the end it's just a pizza, just like "normal" category theory. Also, it's a pineapple-pizza.

TURN ON CNN

A proof of the Jacobian conjecture
arxiv.org/pdf/1711.04967.pdf
>arxiv.org/pdf/1711.04967.pdf
arxiv.org/pdf/1711.04967.pdf

Thanks! Is there some general routine for doing so? Like if you have a 100 degree polynomial with coefficients which don't all eliminate modulo that prime, is there some way of algrebraically proving that, or do you need to check manually?

I feel like I'm missing something really obvious here, but could someone please give me some intuition as to why you'd want to take the projection of a vector connecting two points onto a distance vector or normal vector, when you're trying to do stuff like calculate distance between a point and a line or a point and a plane? I understand how to do it but not really why. My course material focuses a lot on computation and seems to have a pretty treatment of geometric properties of dot and cross products in general.

pretty poor treatment*

direction vector even

See pic.
If you don't understand something, ask.

As mentioned, the projection is used in problems of minimizing distances. For example, if you have a vector line and a point outside that line, you find the distance between the point and the line by projecting the point onto the line.

But the projection also has other important properties. For example, projections give rise to what is known as Fourier coefficients which are very important for many reasons, one being that it is possible to prove that with Fourier coefficients one can find the "best" approximation of a vector under certain conditions.

Another important example is that other "geometric" transformations can be written in terms of the projection. For example, the reflection transformation. If you have a vector line and you want to reflect a point relative to that line then what do you do? You take your vector and project it onto the line. Then you move the vector in that same distance and direction so that it ends up on the other "side" of the line. And that is your reflection, in terms of the projection. And what is the reflection? One the first isometries that one can study in a vector space. And then the study of isometries can lead to elegant proofs of various inequalities, for example.

The great thing is that for a given prime, checking whether a polynomial is irreducible mod that prime is a finite check. I'd usually just have magma do it.

Probability theory is not statistics.