/sci/ - Science & Math

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/mg/ Anonymous Sat Nov 11 21:02:26 2017 No.9290579 [Reply] [Original]

Anonymous Sat Nov 11 21:05:07 2017 No.9290585

>>9290579
What's the fastest route towards mathematical maturity? Inhaling as much rigorous mathematical text and resources that your mind allows, and exhaling nothing but solutions to difficult exercises?

Cuz that's my current plan, I'm tired of sucking eggs and want to 'git gud'.

>>	Anonymous Sat Nov 11 21:09:54 2017 No.9290599 >>9290585 >rigorous If you have to explicitly say this, you still have a long road ahead of you.

>>	Anonymous Sat Nov 11 21:11:00 2017 No.9290601 >>9290585 I've never seem amyone becoming good in more than one subject of math without going to university. You hardly even see anyone finishing a book.

>>	Anonymous Sat Nov 11 21:12:10 2017 No.9290604 >>9290599 I know I do, but you're comment is unhelpful regardless.

>>	Anonymous Sat Nov 11 21:12:50 2017 No.9290608 File: 3.10 MB, 776x5164, A guide to mathematics (scaled).png [View same] [iqdb] [saucenao] [google] >>9290585 There's no fastest route to it - you just follow the route at a faster pace or not

>>	Anonymous Sat Nov 11 21:13:35 2017 No.9290609 >>9290601 I'm in uni but we use shit tier books imo, I imagine in the upper division classes that changes a bit though.

>>	Anonymous Sat Nov 11 21:15:06 2017 No.9290614 >>9290608 Certainly there's more and less efficient routes, idiot. The guide you posted is a prime example.

>>	Anonymous Sat Nov 11 21:15:45 2017 No.9290617 >>9290585 There Is No Royal Road.

>>	Anonymous Sat Nov 11 21:18:11 2017 No.9290624 >>9290617 I was not asking what the royal road was, just opinions on practices that lead there fastest.

>>	Anonymous Sat Nov 11 21:23:58 2017 No.9290634 File: 89 KB, 1011x94, ff0424df34c9ff8e8bf16eeabe077b92.png [View same] [iqdb] [saucenao] [google] >>9290614 I'd be surprised if anyone here can actually go to Gelfand's Algebra and solve every problem unaided

>>	Anonymous Sat Nov 11 21:26:17 2017 No.9290640 >>9290609 >we use shit tier books imo That's to be expected. Is there anything stopping you from using good books?

>>	Anonymous Sat Nov 11 21:32:24 2017 No.9290650 >>9290634 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

Anonymous Sat Nov 11 21:42:37 2017 No.9290674

>>9290640
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 anon please answer this question, >>9290585 ,
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.

Anonymous Sat Nov 11 21:49:25 2017 No.9290692

>>9290674
>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.

>>	Anonymous Sat Nov 11 21:53:41 2017 No.9290704 >>9290674 >learning calculus If you're learning it at all you have already failed. >reading comprehension You might want to check out >>>/lit/ for that. This is a mathematics thread.

>>	Anonymous Sat Nov 11 22:20:25 2017 No.9290754 >>9290704 >if you're learning it at all you have already failed Lol what? >/lit/ massive top kek, r u seriously implying math has nothing to do with reading comprehension?

>>	Anonymous Sat Nov 11 22:21:34 2017 No.9290757 >>9290692 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 >>9290754 Are you meaning to say that one should rather be learning analysis, not calculus?

>>	Anonymous Sat Nov 11 22:23:37 2017 No.9290759 >>9290757 wrong anon, meant for >>9290704

>>	Anonymous Sat Nov 11 22:25:04 2017 No.9290765 >/mg/ is actually anything other than about magnesium

Anonymous Sat Nov 11 22:25:32 2017 No.9290767

>>9290757
>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.

>>	Anonymous Sat Nov 11 22:27:59 2017 No.9290770 >>9290757 >learning analysis If you're learning it at all you have already failed.

Anonymous Sat Nov 11 22:47:17 2017 No.9290795

>>9290585
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.

Anonymous Sat Nov 11 23:04:54 2017 No.9290821

>>9290767
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.

>>9290770
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.

>>9290795
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.

Anonymous Sat Nov 11 23:11:06 2017 No.9290834

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

Anonymous Sat Nov 11 23:12:49 2017 No.9290840

>>9290834
>"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".

>>	Anonymous Sat Nov 11 23:13:09 2017 No.9290843 >>9290821 How far through apostol and spivak are you out of curiosity

Anonymous Sat Nov 11 23:29:54 2017 No.9290872

>>9290843
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.

Anonymous Sat Nov 11 23:31:51 2017 No.9290879

>>9290840
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?

Anonymous Sat Nov 11 23:43:38 2017 No.9290903

>>9290872
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 /sci/? Professors or classmates?

Anonymous Sun Nov 12 00:08:17 2017 No.9290934

>>9290903
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.

Anonymous Sun Nov 12 00:11:08 2017 No.9290941

>>9290934
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.

>>	Anonymous Sun Nov 12 00:48:55 2017 No.9290986 This thread needs more anime

>>	Anonymous Sun Nov 12 00:56:53 2017 No.9290990 File: 68 KB, 552x706, 1509682394055.jpg [View same] [iqdb] [saucenao] [google] >>9290986 Why don't you post some then?

>>	Anonymous Sun Nov 12 00:57:03 2017 No.9290991 >>9290986 k

Anonymous Sun Nov 12 00:59:43 2017 No.9290995

>>9290941
> 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.

>>	Anonymous Sun Nov 12 01:09:15 2017 No.9291001 File: 479 KB, 1280x720, [Coalgirls]_Magical_Girl_Madoka_Magica_the_Movie_II_-_Eternal_(1280x720_Blu-ray_FLAC)_[4A5E05FA].mkv_snapshot_00.33.53_[2016.03.07_23.38.51].png [View same] [iqdb] [saucenao] [google] >>9290986

Anonymous Sun Nov 12 01:14:14 2017 No.9291006

>>9290995
>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.

>>	Anonymous Sun Nov 12 01:20:22 2017 No.9291013 >>9291006 Pls respond though anon, 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.

>>	Anonymous Sun Nov 12 02:31:29 2017 No.9291080 What's a good classical mechanics textbook?

Anonymous Sun Nov 12 02:36:09 2017 No.9291084

>>9291013
Sorry anon 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.

>>	Anonymous Sun Nov 12 02:38:28 2017 No.9291085 >>9291013 >>9291084 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).

Anonymous Sun Nov 12 03:07:16 2017 No.9291103

>>9291080
Kleppner - Introduction to Mechanics.

>>9291084
>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 anon.

>>	Anonymous Sun Nov 12 06:20:37 2017 No.9291284 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?

>>	Anonymous Sun Nov 12 10:39:24 2017 No.9291532 >>9290650 As in the inequality for n terms

Anonymous Sun Nov 12 11:54:10 2017 No.9291654
File: 208 KB, 2560x1440, 1507425119513.png [View same] [iqdb] [saucenao] [google]

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?

>>	Anonymous Sun Nov 12 12:16:37 2017 No.9291692 >>9291654 >Am I missing something here? Yes. The system being consistent means that it has Some solution. Not that All (x1,x2,x3) are solutions.

Anonymous Sun Nov 12 12:43:20 2017 No.9291744

>>9291692
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?

Anonymous Sun Nov 12 13:42:00 2017 No.9291849

>>9291744
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.
http://www.wolframalpha.com/input/?i=row+reduce+%7B%7B1,4,-3%7D,%7B-2,-5,3%7D,%7B5,8,-3%7D%7D
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.

>>	Anonymous Sun Nov 12 14:57:57 2017 No.9291994 A non-zero element in a prime ideal contains a prime factor. Help.

>>	Anonymous Sun Nov 12 15:45:57 2017 No.9292116 >>9291994 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

>>	Anonymous Sun Nov 12 17:12:13 2017 No.9292288 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?

>>	Anonymous Sun Nov 12 19:29:46 2017 No.9292632 >>9290579 > 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%?

>>	Anonymous Sun Nov 12 20:25:54 2017 No.9292737 >>9292288 [eqn]f(x,y) = \begin{cases} e^{-1} & \text{ if } x^2 + y^2 \leq 1\\ e^{-(x^2+y^2)} & \text{ if } x^2 + y^2 > 1 \end{cases} [/eqn]

Anonymous Sun Nov 12 21:17:49 2017 No.9292804 [DELETED]

File: 248 KB, 1000x1200, 1492618231265.png [View same] [iqdb] [saucenao] [google]

>>9292288
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].

>>	Anonymous Sun Nov 12 23:02:41 2017 No.9292958 >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

>>	Anonymous Mon Nov 13 01:10:42 2017 No.9293136 wtf you stupid mod stop deleting every post

>>	Anonymous Mon Nov 13 10:19:50 2017 No.9293643 >>9292288 [eqn]f(x,y) = \frac{2+\cos(x)}{1+x^2+y^2} [/eqn]

Anonymous Mon Nov 13 11:13:56 2017 No.9293743
File: 2.32 MB, 2242x3269, 1376268869485.jpg [View same] [iqdb] [saucenao] [google]

>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

>>	Anonymous Mon Nov 13 12:06:25 2017 No.9293834 >>9293743 Yeah, that part of stats is actually math.

>>	Anonymous Mon Nov 13 12:08:48 2017 No.9293842 >>9292958 download more iq

Anonymous Mon Nov 13 12:35:01 2017 No.9293885

>>9293743
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]

>>	Anonymous Mon Nov 13 12:36:06 2017 No.9293891 didn't mean to quote, sorry.

Anonymous Mon Nov 13 15:42:28 2017 No.9294318 [DELETED]

Take any [math]1[/math]-form [math]\omega[/math] on [math]\mathbb{R}^n[/math] and Hodge decompose it into [math]\omega = d\alpha +
\delta \beta + \gamma[/math] where [math]\alpha[/math] is a [math]0[/math]-form,
[math]\beta[/math] is a [math]2[/math]-form, and [math]\gamma[/math] is a harmonic [math]1[/math]-form where [math]\nabla \gamma = 0[/math]. Let [math]C \in H_1(\mathbb{R}^n)[/math] be a [math]1[/math]-cycle, then
[eqn]
\langle \omega , C\rangle = \int_C \omega = \int_C (d\alpha + \delta \beta + \gamma)
[/eqn]
The first term vanishes by Stokes's theorem. Suppose [math]\langle C, \omega\rangle =
\omega|_0[/math], it then suffices to show that [math]\langle C, \omega\rangle[/math] does not depend on [math]\beta[/math]. To do this note that [math]\ast: C^1(\mathbb{R}^n) \tilde{\rightarrow} C^{n-1}(\mathbb{R}^n)[/math] is a vector space isomorphism, and by Poincare duality [math]H^k(\mathbb{R}^n) \cong H_{n-k}(\mathbb{R}^n) \cong (H_{n-k}(\mathbb{R}^n))^*[/math], there exists [math]\tilde{C} \in H^1(\mathbb{R}^n)[/math] and [math]\tilde\beta \in (\operatorname{Im}\ast)^*[/math] such that [math]\langle C, \delta \beta \rangle = \langle \tilde \beta, d \ast \tilde{C}\rangle = \langle \partial \tilde\beta , \ast \tilde{C} \rangle = 0[/math], thus [math]\langle C,\omega \rangle
= \langle C, \gamma \rangle[/math], and since [math]C\in H_1(\mathbb{R}^n)[/math] is arbitrary [math]\langle C, \omega \rangle =
\gamma|_0[/math].
This assumes the smoothness of [math]\omega[/math] and [math]C \in \operatorname{Ker}\partial / \operatorname{Im}\partial[/math].

>>	Anonymous Mon Nov 13 22:45:29 2017 No.9295248 This is probably a retarded question but what use does the Cauchy-Schwarz inequality actually have?

Acbn !7CUUUUUUUU Mon Nov 13 23:06:23 2017 No.9295285

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

Anonymous Mon Nov 13 23:08:11 2017 No.9295289

>>9292958
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.

>>	Anonymous Mon Nov 13 23:12:11 2017 No.9295295 >>9295248 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.

>>	Anonymous Mon Nov 13 23:13:34 2017 No.9295297 >>9290579 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.

Anonymous Mon Nov 13 23:19:14 2017 No.9295306

>>9292632
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.

Anonymous Tue Nov 14 00:31:22 2017 No.9295397

>>9295285
I don't know about courses but...

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

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

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

>>	Anonymous Tue Nov 14 06:36:00 2017 No.9295759 >>9295248 I used it in my riemannian geometry class just the other day

>>	Anonymous Tue Nov 14 06:37:02 2017 No.9295760 >>9295297 Gauss lemma

>>	Anonymous Tue Nov 14 06:43:16 2017 No.9295762 >>9295760 thx

>>	Anonymous Tue Nov 14 09:11:49 2017 No.9295898 >>9295297 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.

>>	Anonymous Tue Nov 14 13:29:43 2017 No.9296260 What subject does \mathbb{F}_4 ^{3} refer to? I know it has something to do with vectors

>>	Anonymous Tue Nov 14 14:03:03 2017 No.9296316 >>9296260 It's the F_4 is probably the (unique) field of 4 elements. https://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.

>>	Anonymous Tue Nov 14 14:53:03 2017 No.9296397 Is there an explicit definition of an (∞,n)-category for arbitrary n? Like there is for an (∞,1)-category in terms of simplicial sets.

Anonymous Tue Nov 14 16:20:25 2017 No.9296563

>>9295248
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

666Aryan88Warrior88of88Lucifer(...) !DSmwcsMTP6 Tue Nov 14 17:07:10 2017 No.9296649 [DELETED]

File: 119 KB, 1280x720, yamete anime.jpg [View same] [iqdb] [saucenao] [google]

>>9296397
>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.

666Aryan88Warrior88of88Lucifer(...) !DSmwcsMTP6 Tue Nov 14 17:08:14 2017 No.9296651
File: 186 KB, 555x313, 9c3cb899.png [View same] [iqdb] [saucenao] [google]

>>9296397
>>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.

Anonymous Tue Nov 14 17:16:05 2017 No.9296661

Hey /sci/ 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?

>>	Anonymous Tue Nov 14 17:43:21 2017 No.9296706 >>9296651 Why does higher category theory have to be such a bitch?

>>	Anonymous Tue Nov 14 18:04:33 2017 No.9296751 >>9296661 Do whatever the fuck you want. It almost literally couldn't matter less.

>>	Anonymous Tue Nov 14 18:45:38 2017 No.9296832 >>9296751 one would certainly lead to more reduced options though, mad anon

>>	Anonymous Tue Nov 14 18:52:18 2017 No.9296847 >>9296661 How is your retarded question appropriate for this thread? Fuck off.

>>	Anonymous Tue Nov 14 18:57:45 2017 No.9296856 How to really learn mathematics?

>>	Anonymous Tue Nov 14 19:02:21 2017 No.9296859 >>9290585 Whatever

>>	Anonymous Tue Nov 14 19:09:40 2017 No.9296873 >>9296856 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.

>>	Anonymous Tue Nov 14 19:37:54 2017 No.9296915 >>9296832 It really won't.

>>	Anonymous Tue Nov 14 19:42:11 2017 No.9296922 >>9296873 Thanks

>>	Anonymous Tue Nov 14 20:34:48 2017 No.9297003 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

>>	Anonymous Tue Nov 14 20:43:46 2017 No.9297031 >>9297003 Nevermind, it's n^m. Remember not to pick example sets with 2 elements each.

>>	Anonymous Tue Nov 14 21:44:22 2017 No.9297128 >>9296706 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.

>>	Anonymous Tue Nov 14 21:53:21 2017 No.9297144 File: 65 KB, 951x625, Capture.png [View same] [iqdb] [saucenao] [google] TURN ON CNN A proof of the Jacobian conjecture https://arxiv.org/pdf/1711.04967.pdf >https://arxiv.org/pdf/1711.04967.pdf https://arxiv.org/pdf/1711.04967.pdf

>>	Anonymous Wed Nov 15 00:09:26 2017 No.9297360 >>9295898 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?

Anonymous Wed Nov 15 05:23:07 2017 No.9297711

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.

>>	Anonymous Wed Nov 15 05:24:20 2017 No.9297712 >>9297711 pretty poor treatment*

>>	Anonymous Wed Nov 15 05:31:13 2017 No.9297721 >>9297711 direction vector even

>>	Anonymous Wed Nov 15 06:08:51 2017 No.9297751 File: 89 KB, 739x909, Axler.jpg [View same] [iqdb] [saucenao] [google] >>9297711 See pic. If you don't understand something, ask.

Anonymous Wed Nov 15 06:22:38 2017 No.9297762

>>9297711
As >>9297751 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.

>>	Anonymous Wed Nov 15 09:10:13 2017 No.9297925 >>9297360 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.

>>	Anonymous Wed Nov 15 10:45:32 2017 No.9298058 >>9293743 Probability theory is not statistics.

Anonymous Wed Nov 15 12:59:43 2017 No.9298251

>>9290585
>Math maturity
Most universities would call it completion up to Calc 3. However, I'd say it's just a drive to do more than the textbook demands. Being curious enough to generalize a form and prove it for yourself or at least strive for that. It's fine if you never achieve it but doing the work required for it and busting your ass is it. There's no fast route.

>>	Anonymous Wed Nov 15 13:29:41 2017 No.9298292 >>9296847 cuz theres math kiddos in here with answers

>>	Anonymous Wed Nov 15 16:35:09 2017 No.9298645 any pointers on showing that x^4-17=0 has a solution modulo all odd primes?

>>	Anonymous Wed Nov 15 17:05:56 2017 No.9298698 >>9298645 Quadratic reciprocity. Always quadratic reciprocity.

Anonymous Wed Nov 15 17:24:50 2017 No.9298729

>>9298645
[math] x^4 = 17 \mod 3 \iff x^4 = 2 \mod 3 [/math]

???????????????????????????????????????

To be quite honest I'm pissed because I was trying to solve your problem and I was going crazy because nothing worked. I even thought I was a brainlet for a moment until I decided to just test it out with small primes and realized you were the brainlet all along.

>>	Anonymous Wed Nov 15 18:26:55 2017 No.9298838 >>9298645 >odd primes are there any non-odd primes other than 2 or what?

>>	Anonymous Wed Nov 15 18:49:16 2017 No.9298869 >>9298838 -2 is prime too.

>>	Anonymous Wed Nov 15 19:49:20 2017 No.9298934 >>9290579 Help me to understand what tensor is and where are they used.

>>	Anonymous Wed Nov 15 22:06:00 2017 No.9299138 >>9298838 It's mainly a condensed way to say all primes other than 2.

>>	Anonymous Wed Nov 15 22:25:18 2017 No.9299164 >>9298934 "tensor" just means tensor product of vector spaces. in differential geometry, just like you can have vector fields, you can have tensor fields...

>>	Anonymous Wed Nov 15 22:35:17 2017 No.9299178 >>9299164 >tensor product of vector spaces brainlet.png

Anonymous Wed Nov 15 22:48:03 2017 No.9299206

Is there just no full rigorous proof of greens, stokes and gauss theorems without using meme differential forms? The generalized "stokes" theorem is beautiful and elegant, but I first need the proof for the real world cases as a sanity check, but I can't find anything.

Anonymous Wed Nov 15 22:53:53 2017 No.9299213

>>9299206
doing it on specific cases would probably be just as hard as doing the general differential forms proof. it's actually a really simple proof once you've defined differential forms and proved their elementary properties. do carmo's "differential forms and applications" has a good treatment

>>	Anonymous Wed Nov 15 22:56:59 2017 No.9299215 >>9299206 >>9299213 also, think about what "without differential forms" implies. what the hell is a line / surface integral if not the integral of a differential form? good luck defining it any other way

>>	Anonymous Wed Nov 15 23:01:56 2017 No.9299221 >>9299206 Yes of course there is. I'm sure any analysis book contains it.

Anonymous Wed Nov 15 23:07:49 2017 No.9299226

>>9299215
What? I'm not talking about the theorem in some abstract manifold, but baby's theorem for vector valued functions in R^3. You can define those integrals without using differential forms.
>>9299221
Not really, most just talk about differential forms. I'm referring to an elemental proof.

>>	Anonymous Wed Nov 15 23:09:15 2017 No.9299228 >>9299226 >You can define those integrals without using differential forms I mean, you are just not calling them differential forms, but otherwise doing all the work...

>>	Anonymous Wed Nov 15 23:15:54 2017 No.9299237 >>9299226 A proof without then would just be a confusing slog through the exact same ideas. Differential forms came around to clean up the messy arguments you seem to seek.

Anonymous Wed Nov 15 23:29:36 2017 No.9299254

>>9299228
>>9299237
Yes I understand that, but you are not getting the point. I don't care about elegance, but about how elementary the proof is. Let's say the first thing you saw in calculus is differential forms and said "everything is a particular of stokes theorem and tada why do we even learn calc XD if a fag". You will obviously call that shit bullshit as it was specifically DEFINED in that sort of way to make things easier, but with a loss in the abstraction. I'm not saying it's bullshit, but I think it's important to have knowledge in the lower level stuff to then apply all this nice abstractions. Hell, it's like starting a course in differential geometry without talking first about curves or surfaces. It's more of a sanity check than a real concern of the validity of the logic being used, but I'm just amazed that I can't find a long but rigorous proof of stokes theorem that doesn't use differential forms. And thid doesn't sound as such an amazing claim considering I have a proof of the change of variable theorem in R^n in the most general sense (for a Riemann integral) that is 46 pages long, but exists.

Anonymous Wed Nov 15 23:33:52 2017 No.9299258

>>9299254
I found one in 20 seconds though https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-c-line-integrals-and-stokes-theorem/session-92-proof-of-stokes-theorem/MIT18_02SC_MNotes_v13.3.pdf

It's a reduction to green's theorem, which uses the fact that you can locally project your surface into one of the three canonical axis planes

here's a sketch of green's
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-c-greens-theorem/session-67-proof-of-greens-theorem/MIT18_02SC_notes_67.pdf
it's annoying

>>	Anonymous Wed Nov 15 23:37:17 2017 No.9299261 >>9299254 It's just literally the same thing written differently, though. It's not a simplification like the other things you mention.

>>	Anonymous Wed Nov 15 23:49:49 2017 No.9299278 >>9299258 I've seen those, but are quite lacking to what I want. Thankd though.

Anonymous Wed Nov 15 23:52:48 2017 No.9299281

>>9299278
lacking how? it's all there. is there any part you don't know how to do?

e.g. you can always locally write a surface in R^3 as a function of two of x,y,z because of the implicit function theorem, you can approximate regions in R^2 with rectangles because C^1 functions have bounded variation, etc etc

>>	Anonymous Thu Nov 16 00:09:12 2017 No.9299295 File: 49 KB, 451x461, 1484154623418.jpg [View same] [iqdb] [saucenao] [google] >>9290579 >I wish it were as easy to banish hunger by rubbing my belly

>>	Anonymous Thu Nov 16 00:09:21 2017 No.9299296 >>9299206 >real world cases as a sanity check Fuck off to some engineering thread.

Anonymous Thu Nov 16 04:26:48 2017 No.9299536

>>9299206
Look up the proof of Cauchy's theorem in complex analysis from Goursat's lemma. You can do something similar for Green's theorem (which is no coincidence since Cauchy's theorem follows from Green's theorem): Prove Green's theorem for triangle following the same strategy, then all polygones, then the general case.

Anonymous Thu Nov 16 04:37:16 2017 No.9299546

>>9299536
Wot, greens theorem requires C^1. Well I suppose it depends how you constructed the notion of holomorphic function, but the way my book follows it doesn't use the fact that that they are infinitely diferentiable, but just that the complex derivative exists in the region.
>>9299296
>He thinks math is just symbol manipulation
The state of nu-math is depressing

>>	Anonymous Thu Nov 16 04:50:26 2017 No.9299559 >>9299546 >>He thinks math is just symbol manipulation Who are you quoting? I never claimed such a thing. You needing a "sanity check" even when convinced of an argument being valid means you belong in an engineering thread.

Anonymous Thu Nov 16 05:09:12 2017 No.9299577

>>9299559
No you retard, is just a way to see if you can prove things with lower level stuff. You can prove all of classical geometry using only analytical geometry/linear algebra, but that doesn't mean you shouldn't learn classical methods that give you a different kind of intuition and motivates different aspects of a more abstract theory. If I ask for a proof of Lebegues criterion, it's really easy using measure theory, but It's nice to see it can be done with basically your run of the mill calc definition to set foot in something more tangible. A perfect example is Jordan curve theorem. There exists many elegent proofs, but because it's something thag appears so evident, it's perfectly reasonable to ask if there is an elementary proof, which there are plenty but they are pretty long and tedious, but it gives you a different perspective. You fag.

>>	Anonymous Thu Nov 16 05:13:22 2017 No.9299579 >>9299577 Why the homophobia though

>>	Anonymous Thu Nov 16 05:24:54 2017 No.9299584 >>9299579 >>>/r/taiwan/

Anonymous Thu Nov 16 05:27:40 2017 No.9299588

>>9299577
>methods that give you a different kind of intuition
>it gives you a different perspective
Anyone who could possibly benefit from this would be able to construct a long and elementary proof of the thing they are interested in. Since you clearly lack the brain power to do so, I doubt such a perspective would be useful to you.
Try understanding the usual proof first, that might actually help.

Anonymous Thu Nov 16 05:47:40 2017 No.9299607

>>9299588
Understanding the actual proof only gives you insight if the proof is sufficiently constructive. You can prove the fundamental theorem of algebra in many different eleganr ways, but that doesn't mean you get a better understanding of the theorem from the proof. And no, it's not always so easy to construct a long elementary proof as its the case with greens theorem and jordan curve theorem. Understanding the usual proof only means you can follow the logical steps and constructions used to arrive at the theorem, but it doesn't .ean that shows you "how the theorem works". Is it so fucking difficult for nu mathematicians to understand that mathematics isn't just formalism?

Anonymous Thu Nov 16 06:04:34 2017 No.9299626

>>9299607
>if the proof is sufficiently constructive.
Just double-negation translate the proof into constructive logic if that bothers you so much.
> it's not always so easy to construct a long elementary proof
It is actually pretty easy provided you understand the usual proof of the theorems. Since you don't, it might actually be non-trivial for you.
You need more mathematical maturity before attempting to prove results you don't even fully grasp using just elementary methods.
>n* mathematicians
You are one yourself though, basically by your own admission.
>understand that mathematics isn't just formalism
You aren't talking to a formalist. It seems like you have reading comprehension problems, not just problems with mathematics.

>>	Anonymous Thu Nov 16 06:14:10 2017 No.9299641 >>9298934 Kind of the equivalent of a n dimensional matrix. Why restrict yourself to transformations that depend only on two numbers when they could depend on n numbers?

>>	Anonymous Thu Nov 16 09:21:38 2017 No.9299837 >>9299559 Every mathematician does this.

Anonymous Thu Nov 16 09:24:58 2017 No.9299841

>>9299626
>>9299588
you have a huge stick up your ass, and you're unwilling to admit your super formalist, immature stance on math

mathematicians use sanity checks heavily, informal pictures and intuition that gets reinforced when proving things in different ways, different contexts, using different language, etc

Anonymous Thu Nov 16 09:38:13 2017 No.9299847

>>9299841
>formalist
I am pretty much as anti-formalist as it gets since I'm a platonist. You seem to be retarded since you can't read and comprehend a thing which has been explicitly stated two times at this point.
>sanity checks
Understanding a logical argument and confirming its validity is enough for anyone with a basic level of mathematical maturity to not need a "sanity check" or "real world" examples. Undergrads with engineering tendencies such as yourself might not fully realize it (or even be able to realize it) though.
>informal pictures and intuition that gets reinforced when proving things in different ways, different contexts, using different language, etc
Yes, that's used. However, it is not related to the garbage you are trying to push here.

Anonymous Thu Nov 16 09:41:45 2017 No.9299852

>>9299847
>Understanding a logical argument and confirming its validity is enough for anyone with a basic level of mathematical maturity to not need a "sanity check"
no. no one's talking about "real world" examples either you asshole. after you prove an obtuse theorem, it does wonders to grab corollaries in known settings in order to see it working and understand what it "really" means.

>>	Anonymous Thu Nov 16 09:48:01 2017 No.9299855 >>9299852 >real world cases see >>9299206 Confirmed for being a retard with no reading comprehension skills. >after you prove an obtuse theorem Don't respond to posts you haven't even read.

>>	Anonymous Thu Nov 16 09:52:40 2017 No.9299862 >>9299855 >you can't disagree with my idiotic rant because I started by replying to something objectionable! fucking idiot, might as well just say "sorry I'm being so retarded, please agree with me, the other guy is wrong!"

>>	Anonymous Thu Nov 16 09:54:58 2017 No.9299868 >>9299862 You didn't bother to get the full context, why even respond? Your retarded reply makes no sense, since it depends on us talking about proving an obtuse theorem which the person doesn't really understand yet.

>>	Anonymous Thu Nov 16 11:15:20 2017 No.9299974 Serious question. What level of math do I need to know to understand Neumann's Theory of Games and Economic Behavior?

>>	Anonymous Thu Nov 16 11:48:33 2017 No.9300018 Why is there a meaningful correspondence between so many different kinds of mathematical objects and polynomials. By "meaningful" I mean that properties of the former can be derived from properties of the latter e.g. its roots.

>>	Anonymous Thu Nov 16 11:56:41 2017 No.9300029 File: 877 KB, 1080x1920, c3a25a82fafe7efb743027e67afe7d7e.jpg [View same] [iqdb] [saucenao] [google] >lecture 3 in stats >maximum-likelyhood >stats is not math

>>	Anonymous Thu Nov 16 12:10:13 2017 No.9300059 >>9299868 Lel Im the original guy and you didn't bothered to read everything. "Real wolrd" was only making reference that the proof was in R^3, but I haven't see application where tye region of integration isn't a simple one.

>>	Anonymous Thu Nov 16 12:28:17 2017 No.9300078 >>9300029 >>stats is not math You shouldn't believe everything you hear on nu-/pol/ aka /sci/

>>	Anonymous Thu Nov 16 13:03:50 2017 No.9300148 I'm trying to into topology and am working through Flegg "From Geometry to Topology" and it seems like its last chapter covers the same material as the first chapter of Munkres. I'm understanding everything in Flegg well. Is this a good path to follow?

>>	Anonymous Thu Nov 16 14:20:39 2017 No.9300272 >>9300059 >Lel >>9300078 >nu Get the fuck back to >>>/r/eddit/

>>	Anonymous Thu Nov 16 14:25:31 2017 No.9300276 >>9300272 Your link does not work, newfriend

Anonymous Thu Nov 16 19:06:05 2017 No.9300768

>>9300148
munkres is technical for a first reading, i.e. you will easily lose yourself if you don't know what to skip (but it is a good reference)
i'm not familiar with english introductions to topology, maybe you can try 'an illustrated introduction to topology and homotopy' by kalajdzievski (it covers usual point set topology and then goes to first bits of homotopy, i.e. fundamental group, which is an algebraic object who measures some kind of holes, and covering spaces, who gives you a simple way to compute some fundamental groups and a galois theory for topological spaces; this is the shit you would see in a first course of topology in uni, it is not totally trivial)

>>	Anonymous Thu Nov 16 19:18:13 2017 No.9300798 >>9300768 Munkres isn't technical at all. It's a very straightforward, easy to follow book.

Anonymous Thu Nov 16 19:47:10 2017 No.9300853
File: 294 KB, 1200x1920, Screenshot_20171117-013031.png [View same] [iqdb] [saucenao] [google]

>>9300798
it is and it is very cute, but it is not for newfags, who will get bored before reaching the end of page 1(sorry for the hyperbole)
at least he would have to avoid all that middle stuff oriented toward analysis (and probably more)

>>	Anonymous Thu Nov 16 19:49:39 2017 No.9300858 >>9300853 it is straightforward*, forgot a word

>>	Anonymous Thu Nov 16 21:44:36 2017 No.9301082 Since I started learning calculus I realized my knowledge of trigonometry is rather lacking, can someone suggest a good text on the subject? Preferably proof based so I can practice doing those

>>	Anonymous Thu Nov 16 21:51:13 2017 No.9301092 >>9301082 What exactly do you lack? You shouldn't be nervous if you don't remember every single identity.

>>	Anonymous Thu Nov 16 22:33:45 2017 No.9301152 >>9301092 I feel like I should be able to derive any of the identities on demand, which I'm not able to do at the moment. And have some more intuition about how it all fits together

>>	Anonymous Thu Nov 16 22:48:21 2017 No.9301175 >>9295248 triangle inequality is bretty good for probability and analysis proofs

Anonymous Thu Nov 16 23:18:18 2017 No.9301218

am i doing this right?

joint density fxn f(x,y) = 8xy where 0<y<x<1, except less than or equal to, f(x,y) = 0 elsewhere

to find the marginal density, f(x), i do this right..

indefinite integral (8xy)dy = 4xy^2

then for f(y) it's

indefinite integral (8xy)dx = 4yx^2

except i don't know how to account for the bounds here..

do i need to adjust the boudns for f(y) to be from 0 to x?

Anonymous Thu Nov 16 23:21:32 2017 No.9301221
File: 22 KB, 540x404, 1488099882979.jpg [View same] [iqdb] [saucenao] [google]

>>9295248
>what use does the Cauchy-Schwarz inequality actually have
https://en.wikipedia.org/wiki/Uncertainty_principle#Robertson.E2.80.93Schr.C3.B6dinger_uncertainty_relations

>>	Anonymous Thu Nov 16 23:39:19 2017 No.9301239 >>9301218 >do i need to adjust the boudns for f(y) to be from 0 to x? Yes. You always integrate over the domain when getting a marginal density. Your goal in the end is a function of x, so it's not a problem if one of your integral bounds depends on x.

>>	Anonymous Thu Nov 16 23:48:43 2017 No.9301249 File: 524 KB, 663x1000, Anime_Taylor.png [View same] [iqdb] [saucenao] [google] Threadly reminder to work with physicists

>>	Anonymous Fri Nov 17 00:15:22 2017 No.9301282 >>9290608 I know it says either proof book is fine, but which would be better to use?

>>	Anonymous Fri Nov 17 00:32:02 2017 No.9301312 >>9301249 Why would I work with subhumans?

>>	Anonymous Fri Nov 17 01:56:37 2017 No.9301404 File: 657 KB, 700x995, Anime_Landau_Purple.png [View same] [iqdb] [saucenao] [google] >>9301312 To make them happy

>>	Anonymous Fri Nov 17 02:02:13 2017 No.9301412 >>9301282 it makes literally no difference grab a book and start reading instead of jerking off over what the "best" source is

>>	Anonymous Fri Nov 17 02:02:31 2017 No.9301413 >>9301404 Animals exist to make people happy, not the other way around.

>>	Anonymous Fri Nov 17 02:04:17 2017 No.9301415 Threadly remainder most mathematicians can not solve the most basic physics problems because they believe their fake ivory tower of abstraction makes them superior.

>>	Anonymous Fri Nov 17 02:12:31 2017 No.9301422 >>9301415 >physics problems Discuss your garbage somewhere else, this is a mathematics thread.

>>	Anonymous Fri Nov 17 02:29:53 2017 No.9301441 File: 1.68 MB, 1537x1962, Anime_Gravitation.png [View same] [iqdb] [saucenao] [google] >>9301413 You're a meanie :[

>>	Anonymous Fri Nov 17 03:07:45 2017 No.9301482 >>9299974 BUMP. This is a serious question please. I just started reading it and can't understand shit.

Anonymous Fri Nov 17 04:15:28 2017 No.9301541

Is there any text that has a rigorous approach to the calculus of variations? In the sense that I don't think I can tackle functional analysis just yet, but I want to understand variational problems rigorously. I suppose I mean like a spivack approach to calculus to gain skills. I just don't want a physicists text.

>>	Anonymous Fri Nov 17 06:48:30 2017 No.9301719 Bamp

>>	Anonymous Fri Nov 17 07:26:33 2017 No.9301746 >>9301441 jesus christ how small is the font for that book

>>	Anonymous Fri Nov 17 07:31:30 2017 No.9301752 >>9301746 one pico pica

>>	Anonymous Fri Nov 17 08:47:43 2017 No.9301812 >>9300018 Polynomials are one of the simplest and most natural examples of an algebraic object, so it seems pretty normal that they happen to appear everywhere

>>	Anonymous Fri Nov 17 08:50:59 2017 No.9301815 >>9301152 Literally any of the identities you might have to prove are basically converting tans to sin/cos and sums of angles to single angles. So the only ones you should really need are tan=sin/cos, pythagorean, and sums of angles for sin and cos

>>	Anonymous Fri Nov 17 09:03:37 2017 No.9301830 >>9301152 >I feel like I should be able to derive any of the identities on demand [math] e^{ia}=\cos(a)+i\sin(a) [/math] [math] e^{i(a+b)}=e^{ia}e^{ib} [/math] I wish I knew that in high-school. It would save a lot of frustration.

>>	Anonymous Fri Nov 17 09:24:42 2017 No.9301845 >>9301282 People here put way too much thought into books. Just use both.

>>	Anonymous Fri Nov 17 11:15:14 2017 No.9302009 What's a rigorous definition of infinity?

>>	Anonymous Fri Nov 17 11:21:39 2017 No.9302016 https://proofwiki.org/wiki/Definition:Infinity

>>	Anonymous Fri Nov 17 12:31:41 2017 No.9302124 >>9301541 Anyone?

>>	Anonymous Fri Nov 17 15:08:38 2017 No.9302483 >>9299974 BUMP

>>	Anonymous Fri Nov 17 15:59:11 2017 No.9302573 >>9297144 This seems almost too simple, but I couldn't spot any easy mistakes. It might just be a coming together of very powerful theorems. Anybody know if it's been confirmed legit?

>>	Anonymous Fri Nov 17 19:27:18 2017 No.9303028 >>9299584 What does Taiwan have to do with homophobia?

>>	Screenfetch thread Fri Nov 17 19:34:51 2017 No.9303043 >>9301282 Book of Proof is entirely free. You can finish the book with every exercices in 2-3 weeks if you have 1 hour per day.

Anonymous Fri Nov 17 19:44:03 2017 No.9303066

>>9297144
>>9302573
>Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.
every time there is a "proof" of the jacobian conjecture (aka 3 times a year) I almost get a heart attack
everything's wrong with this shit problem, even citing other people on partial progress is dangerous

Anonymous Fri Nov 17 20:59:18 2017 No.9303195

>>9302573
There's a very simple mistake, ie. the statement of Ax-Grothendieck theorem: every injective polynomial mapping from C^n to itself is also surjective, but the inverse need not be polynomial (that's the crux of the Jacobian problem) and he assumes that all along, so he basically didn't do anything

Anonymous Fri Nov 17 22:39:11 2017 No.9303424 [DELETED]

Is this right?
[math]A $n$-sided polygon represented by an $(n+1)\times3$ matrix, $X$, of the $x,y$ coordinates of its vertices, $<x_0,y_0,1>...<x_n,y_n,1>$ is congruent to another $n$-sided polygon $Y$, also represented by a $(n+1)\times3$ matrix, in the Euclidean plane if and only if there is some orthogonal $(n+1)\times3$ matrix $M$ such that $X \cdot M=Y$[/math]

Anonymous Fri Nov 17 22:41:46 2017 No.9303427

Is this right?

A [math]n[/math]-sided polygon represented by an [math](n+1)\times3[/math] matrix, [math]X[/math], of the [math]x,y[/math] coordinates of its vertices, [math]<x_0,y_0,1>...<x_n,y_n,1>[/math] is congruent to another [math]n[/math]-sided polygon [math]Y[/math], also represented by a [math](n+1)\times3[/math] matrix, in the Euclidean plane if and only if there is some orthogonal [math](n+1)\times3[/math] matrix [math]M[/math] such that [math]X \cdot M=Y[/math].

>>	Anonymous Fri Nov 17 22:45:57 2017 No.9303433 >>9299641 tensors are not merely multidimensional arrays, they maintain invariant quantities.

Anonymous Fri Nov 17 22:52:04 2017 No.9303444

>>9303427
yes. this is just an obnoxious way of saying you can always find a rotation/translation/scaling transformation to move one shape into the other. Orthogonal matrices only ever give rotations and reflections, the translation (since it's not properly a linear transformation) has to be shoehorned in using homogeneous coordinates which is why you have that extra 1 at the end of the vectors, since you're technically rotating and projected to look like a translation.

>>	Anonymous Fri Nov 17 23:23:37 2017 No.9303497 >>9303444 Nice trips. Obnoxious yes, but it's complete and rigorous, right? I mean, it's high school geometry in one line.

>>	Anonymous Sat Nov 18 00:26:24 2017 No.9303640 File: 13 KB, 840x501, oc_brainlet.png [View same] [iqdb] [saucenao] [google] >Using the axiom of choice >proof by contradiction

>>	Anonymous Sat Nov 18 00:34:11 2017 No.9303654 >>9303640 like it or not, LEM and AC fit our intuitions on what things should be like. math relies on them heavily.

>>	Anonymous Sat Nov 18 00:36:25 2017 No.9303657 >>9303654 https://plato.stanford.edu/entries/mathematics-constructive/ sweetie...

>>	Anonymous Sat Nov 18 00:39:03 2017 No.9303660 >>9303657 >constructive mathematics exists and is unable to prove basic results yes.......

>>	Anonymous Sat Nov 18 00:42:35 2017 No.9303662 >>9303657 >sweetie... >>>/tumblr/ Please fuck off

>>	Anonymous Sat Nov 18 01:06:25 2017 No.9303688 >>9303654 >"intuitions" You're lowering yourself to the level of engineers. Sure, they are used pretty heavily, but that doesn't mean you shouldn't try avoiding unnecessary hypotheses whenever possible.

>>	Anonymous Sat Nov 18 01:36:13 2017 No.9303714 >>9301441 tfw no anime girl physicist gf

>>	Anonymous Sat Nov 18 01:46:49 2017 No.9303725 File: 165 KB, 303x311, int2.png [View same] [iqdb] [saucenao] [google] >>9303640 >constructivists

Anonymous Sat Nov 18 05:51:18 2017 No.9303984

Can someone please point me to a link explaining what Tensors are? I keep seeing them mentioned, but I never understood what the fuck they are.
All I find after google searching are engineering-tier bullshit. I just want to see a mathematical and intuitive approach.

>>	Anonymous Sat Nov 18 05:52:55 2017 No.9303987 >>9303984 >Can someone please point me to a link explaining what Tensors are? https://en.wikipedia.org/wiki/Tensor

Anonymous Sat Nov 18 06:33:26 2017 No.9304022

Is the distributivity property basically just a glorified commutative property? F is a field. G = set of finite tuples of elements of F. Let X: G -> F be given by the sum of the finite tuple and let Y: G -> G be given by the product of the scalar a with the tuple. Then distributivity is just XY = YX.

Anonymous Sat Nov 18 06:34:01 2017 No.9304023

>>9290608
I've seen this picture posted a million times. Is it actually good for studying maths and setting a good foundation or is it a meme? I've done up to Calc II (will start Calc III soon) in school, but I was never taught math with proofs, I've been taught the traditional school approach. Is it worth it to start at the very beginning of that chart and redo all those basic math books if I already know up to Calc II? Do they contain some fundamental ideas/ways to do math that I've missed out on? Or should I finish up my Calc III and move on from there using the chart.

>>	Anonymous Sat Nov 18 06:44:09 2017 No.9304030 >>9304023 >Is it actually good for studying maths and setting a good foundation or is it a meme? meme

>>	Anonymous Sat Nov 18 06:54:31 2017 No.9304039 >>9303662 What's wrong with sweetie?

>>	Anonymous Sat Nov 18 07:04:15 2017 No.9304044 >>9304030 Why do you think so? Also what path would you recommend after Calc II, Linear Algebra and Diff Eq?

Anonymous Sat Nov 18 07:12:12 2017 No.9304053

>>9304044
>Why do you think so?
because it's a memelist like the other lists, any list with Lang on it should make you suspicious

>Also what path would you recommend after Calc II, Linear Algebra and Diff Eq?
depends on what you want to do, those 3 open up many different roads. calc 3 is essentially calc + linear algebra, can learn about PDEs after that. can learn more algebra (i.e. groups, rings, fields...) that doesn't immediately rely on linear algebra but that the experience with linear will help with, etc.

it's really up to you

Anonymous Sat Nov 18 07:12:42 2017 No.9304055
File: 514 KB, 1920x1200, Screenshot_20171118-131006.png [View same] [iqdb] [saucenao] [google]

>>9303984
your question is a bit vague, meaning tensor over a vector space, maybe with enough structure, are quite simple, see for example 'linear algebra via exterior products' by winitzki (this is a nice and free book, and gave me a good impression of k-vectors/forms/tensors)
the stuff gets a bit messy (as with everything) when you go to manifolds, where you have a vector space for each point and maybe you want enough structure to relate them, but then tensors behave quite well and are extensively used in diff geometry / physics(project all your memes here)
I'm not sure about a reference for the second case, maybe dodson's 'tensor geometry' (pic related, tldr its a physics book but tensors aren't)

>>	Anonymous Sat Nov 18 07:17:27 2017 No.9304061 >>9304053 Ah thank you, I appreciate the advice. Could you refer me to somewhere that will guide me that isn't a meme list? Like a uni site with a curriculum that lists textbooks or some other site?

>>	Anonymous Sat Nov 18 07:21:26 2017 No.9304069 >>9304061 >Could you refer me to somewhere that will guide me that isn't a meme list? guide you to what?

>>	Anonymous Sat Nov 18 07:21:46 2017 No.9304070 >>9290579 Anyone got recommendations for online accessible advanced number theory texts? Especially Galois theory, polynomials, generating functions, local fields, etc.

>>	Anonymous Sat Nov 18 07:33:11 2017 No.9304085 >>9304069 Some sort of list or listing of topics/textbooks to study to become a mathematician. I wish I could do it in a university but for the moment I'll just have to do with self-teaching.

Anonymous Sat Nov 18 07:36:00 2017 No.9304087

>>9304023
You haven't missed anything then - in fact, you're essentially on the "a taste" step. Finishing Calc III is quite important to progress into the deeper sections, but given that you're taking it an university already, you should be able to progress into the proof books quite naturally (assuming your precalc knowledge isn't absolute dogshit). Just skip the part on set theory though and go straight to (rigorous) linear algebra or analysis

>>	Anonymous Sat Nov 18 07:39:21 2017 No.9304092 >>9304070 >Anyone got recommendations for online accessible advanced number theory texts? Especially Galois theory, polynomials, generating functions, local fields, etc. Falko Lorenz - Algebra (I and II)

Anonymous Sat Nov 18 07:39:53 2017 No.9304094

>>9304085
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 Sat Nov 18 07:40:24 2017 No.9304096

>>9304094
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 [math] C [/math] 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 Sat Nov 18 07:40:56 2017 No.9304097

>>9304096
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 Sat Nov 18 07:41:26 2017 No.9304099

>>9304097
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 Sat Nov 18 07:41:57 2017 No.9304102

>>9304099
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 Sat Nov 18 07:42:28 2017 No.9304104

>>9304102
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 Sat Nov 18 07:42:59 2017 No.9304106

>>9304104
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 Sat Nov 18 07:43:30 2017 No.9304107

>>9304106
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 Sat Nov 18 07:51:50 2017 No.9304113

>>9304023
Also, the fact that the other anon told you to do PDEs straight after linear algebra and calc 3 literally has no clue what he's talking about. You can't do anything in PDEs without a strong background in analysis unless you want to learn meme "methods in PDEs" in which case you might as well be studying engineering

>>	Anonymous Sat Nov 18 07:54:08 2017 No.9304116 >>9304094 >>9304096 >>9304097 >>9304099 >>9304102 >>9304104 >>9304106 >>9304107 >someone actually took the time to type this

>>	Anonymous Sat Nov 18 07:54:27 2017 No.9304117 >>9304092 Thanks!

>>	Anonymous Sat Nov 18 07:54:28 2017 No.9304118 >>9304113 analysis is a joke, you can learn what you need for PDEs along the way, there's 0 need to take a class in it

>>	Anonymous Sat Nov 18 08:12:15 2017 No.9304138 >>9304118 If it's so easy then why don't you solve the Navier-Stokes equations in 3D?

>>	Anonymous Sat Nov 18 08:12:50 2017 No.9304141 >>9304138 >If it's so easy then why don't you solve the Navier-Stokes equations in 3D? did you misread my post?

>>	Anonymous Sat Nov 18 08:24:06 2017 No.9304162 >>9304118 t. undergrad

>>	Anonymous Sat Nov 18 08:24:46 2017 No.9304163 >>9304162 >t. undergrad wrong

>>	Anonymous Sat Nov 18 08:25:09 2017 No.9304164 >>9304163 neet then lmao

>>	Anonymous Sat Nov 18 08:25:31 2017 No.9304167 >>9304164 >neet then lmao wrong

Anonymous Sat Nov 18 09:05:58 2017 No.9304223

>>9304167
I'm actually on your side, taking an undergrad class in PDEs almost as much of a waste as taking a class on "vector calculus" and graduate level PDEs would only be useful if you want to become a mathematician and want some basic knowledge of the foundations for how current PDE research is done.

Anonymous Sat Nov 18 09:19:57 2017 No.9304240

I took an Ordinary Differential Equations class (mathematics department) and it was essentially all about memorizing a bunch of tricks. It was fucking horrible studying it. Felt like a history class or something.
There's also an ODE 2 class as well as a PDE class available. Are those classes still about memorizing shit?

>>	Anonymous Sat Nov 18 09:20:25 2017 No.9304241 >>9304240 >Are those classes still about memorizing shit? Ask the instructor teaching the class

>>	Anonymous Sat Nov 18 09:23:52 2017 No.9304245 >>9304116 Copy pasta some crazy Russian PhD Harvard math.

Anonymous Sat Nov 18 14:29:39 2017 No.9304739
File: 10 KB, 645x773, 33.png [View same] [iqdb] [saucenao] [google]

What's the hardest skills to learn in Algebra 1 and 2, /sci/?

Ages since I passed those and I thought it would be a good idea to relearn everything I forgot starting with the most difficult skills.

>>	Anonymous Sat Nov 18 14:40:33 2017 No.9304767 How do i solve something like ( (5)^1/2 + (7)^1/2 )^1/2 basically the square root of a sum of square roots

>>	Anonymous Sat Nov 18 14:43:15 2017 No.9304774 >>9304240 Shame, ODEs are a beautiful subject with interesting applications to pure math

Anonymous Sat Nov 18 16:00:07 2017 No.9304936
File: 1.54 MB, 2000x1000, do it for wildberger.png [View same] [iqdb] [saucenao] [google]

>Virgin Terry A. Davis
>wastes a decade to write useless ancient-looking OS, no one knows or cares
>schizo neet, does nothing but stream and shitpost all day
>internet stalks a coalburner, finally loses virginity to a crack whore with AIDS
>gets his life fucked up even more than usual by /g/, currently under arrest warrant for punching the only people to care about him into a coma

>CHAD NORMAN J. WILDBERGER
>reinvents the entire field of mathematics, leaves everyone baffled by his insights
>associate professor in reputable university, has enlightened thousands of students as well as viewers from around the world
>happily married with a daughter
>online shitposting has no effect on him, brushes away the cries of brainlets that don't understand his thinking with his smug visage and acerbic wit

Anonymous Sat Nov 18 16:00:45 2017 No.9304937

>>9304774
I mean, it wasn't all about learning tricks. There were linear systems, Picard-Lindeloff, etc. But still, some equations had certain forms which you had to remember certain ways to approach them. You couldn't just find that way by yourself in the duration of the exam; it wasn't always simple.

>>	Anonymous Sat Nov 18 16:29:11 2017 No.9304998 >>9304087 Thanks for the advice anon, so you think working off those textbooks like the chart says is fine?

>>	Anonymous Sat Nov 18 16:32:38 2017 No.9305009 >>9304998 >so you think working off those textbooks like the chart says is fine? no, it's a memelist

>>	Anonymous Sat Nov 18 16:36:28 2017 No.9305024 anyone know anything about p-adic galois stuff

>>	Anonymous Sat Nov 18 16:48:04 2017 No.9305039 >>9305024 >anyone know anything about p-adic galois stuff like?

>>	Anonymous Sat Nov 18 16:49:42 2017 No.9305044 >>9304936 >>happily married with a daughter very poor choice of words

>>	Anonymous Sat Nov 18 16:53:29 2017 No.9305055 >>9304767 t. brainlet

>>	Anonymous Sat Nov 18 16:59:50 2017 No.9305072 File: 154 KB, 315x499, file.png [View same] [iqdb] [saucenao] [google] >>9301541

>>	Anonymous Sat Nov 18 17:13:25 2017 No.9305097 >>9305055 Have a (you)

>>	Anonymous Sat Nov 18 17:19:46 2017 No.9305113 >>9305009 so....what do i go off? that giant ass list the other dude copy pasted or the meme chart? im lost.

>>	Anonymous Sat Nov 18 17:22:13 2017 No.9305121 >>9301541 http://4chan-science.wikia.com/wiki/Mathematics#Calculus_of_Variations

>>	Anonymous Sat Nov 18 17:24:34 2017 No.9305129 >>9305113 >that giant ass list the other dude copy pasted I'm not a "dude".

>>	Anonymous Sat Nov 18 17:29:55 2017 No.9305142 >>9305113 read what you want to read m8, why follow someone else's arbitrary list of books?

>>	Anonymous Sat Nov 18 17:40:49 2017 No.9305163 >>9304023 >>9305113 here's serious advice. grab Tao's Analysis I and Hoffman & Kunze's Linear Algebra. nothing more. work through both slowly, preferrably do the first chapters in Tao before anything else.

>>	Anonymous Sat Nov 18 17:41:44 2017 No.9305165 >>9305163 >Tao's Analysis I memebook

>>	Anonymous Sat Nov 18 17:43:12 2017 No.9305171 >>9305165 go on, explain. every person new to mathematics that started using it has been extremely satisfied, and I can't recommend it enough, especially for independent study.

>>	Anonymous Sat Nov 18 17:43:33 2017 No.9305173 >>9305171 >every person new to mathematics that started using it has been extremely satisfied [citation needed]

>>	Anonymous Sat Nov 18 17:44:31 2017 No.9305178 >>9305173 I'm clearly implying the people I have seen or known to use it, not everyone in the world you dense brainlet

>>	Anonymous Sat Nov 18 17:46:56 2017 No.9305181 >>9305163 >Tao's Analysis I If it didn't have Tao's name on it, it would be thrown in the trash where it belongs, never should have been published

>>	Anonymous Sat Nov 18 17:48:16 2017 No.9305185 >>9305181 you're still not elaborating buddy, did you have any difficulties with it? be specific

>>	Anonymous Sat Nov 18 17:50:41 2017 No.9305188 >>9305185 >did you have any difficulties with it? No, in fact I found that the book was rather trivial.

>>	Anonymous Sat Nov 18 17:51:36 2017 No.9305190 >>9305188 so your problem with the book is that you already knew the topics? I'm making a big effort to coax you into saying something meaningful here

>>	Anonymous Sat Nov 18 17:52:35 2017 No.9305194 >>9305190 >so your problem with the book is that you already knew the topics? What made you think that?

>>	Anonymous Sat Nov 18 17:54:47 2017 No.9305198 >>9305194 my fault for assuming you had anything to say, totally got me, sick bait, etc etc

>>	Anonymous Sat Nov 18 17:54:50 2017 No.9305199 >>9304245 uff, some time math looks kind of randomly generated I guess I'm a brainlet, a depressed one

Anonymous Sat Nov 18 17:57:13 2017 No.9305207

>>9305198
>my fault for assuming you had anything to say, totally got me, sick bait, etc etc
I don't quite understand your thought process, nowhere did I imply that I already knew the topics, especially since that wasn't the case when I read the book. Can you be more clear about what made you think that? I'm making a big effort to coax you into saying something meaningful here.

>>	Anonymous Sat Nov 18 17:57:43 2017 No.9305211 >>9305163 Should I use Tao's over Apostle's book?

>>	Anonymous Sat Nov 18 17:58:16 2017 No.9305213 >>9305207 >look mom I'm baiting him so hard lmao xd

>>	Anonymous Sat Nov 18 17:58:56 2017 No.9305215 >>9305213 >>look mom I'm baiting him so hard lmao xd Who are you quoting?

>>	Anonymous Sat Nov 18 17:59:26 2017 No.9305218 >>9304094 i can guarantee this person is a troll and no highschool in the world teaches that much material lmao

>>	Anonymous Sat Nov 18 18:00:21 2017 No.9305220 >>9305211 >Should I use Tao's over Apostle's book? nah, tao is a memebook, maybe grab something more worthwhile like barry simon's five volume 'comprehensive course on analysis'

Anonymous Sat Nov 18 18:00:33 2017 No.9305221

>>9305211
Apostol is calculus, Tao is Analysis. This means Tao will do proper analysis, introducing e.g. the topology of R properly. Tao's book also has a rather nice introduction to set theory and the construction of numbers. If you want to do real math later on, I'd use Tao, unless you don't know calculus yet, in which case ocw.mit.edu supplemented with apostol should be good.

>>	Anonymous Sat Nov 18 18:01:36 2017 No.9305224 >>9305220 >t. barry simon

>>	Anonymous Sat Nov 18 18:01:49 2017 No.9305226 >>9305218 >i can guarantee this person is a troll and no highschool in the world teaches that much material lmao t. american

>>	Anonymous Sat Nov 18 18:04:12 2017 No.9305229 >>9305220 hahahaha nice meme >>9305211 yeah you should read EGA for high school algebra too xd

Anonymous Sat Nov 18 18:05:55 2017 No.9305235

>>9305221
I know Calculus but its Calculus from the Stewart book (which from what i understand isnt too intuitive and lacks proofs). Ive learned the problem oriented approach fo Calculus rather than the pure math with proofs and solid foundations and stuff

>>	Anonymous Sat Nov 18 18:07:37 2017 No.9305238 >>9305235 >I know Calculus but its Calculus from the Stewart book (which from what i understand isnt too intuitive and lacks proofs) Are there really theorems in Stewart without proof? I have trouble believing this

Anonymous Sat Nov 18 18:07:41 2017 No.9305239

>>9305235
that's fine, that's what calculus mostly is. if you're comfortable with manipulating the integral and the derivative (integration by part, chain rule, l'hopital's rule, and so on) that's enough calculus and if you want to go further into math I'd go for analysis

Anonymous Sat Nov 18 18:14:47 2017 No.9305254

>>9305239
Thank you, I appreciate your advice, i got nowhere else to go to math advice so i actually am very grateful for guidance. I'm going to read that Book of Proof, then Linear Algebra Done Right (never was used to matrices, I think i should learn these now), and then i'll read Tao's Analysis books and move on from there. I'll also take a look at Apostle's calc book to see if its worth redoing calc with it. thx again

Anonymous Sat Nov 18 18:21:35 2017 No.9305273

>>9305238
I was going to reply with some examples, but after skimming the book I realize it's fairly complete and rigorous in its treatment. that's surprising. the only (understandable) shakiness is in the change of variable theorem for integrals (not proven at all) and stoke's theorem (only done in a special case)

>>	Anonymous Sat Nov 18 18:21:49 2017 No.9305276 >>9305163 Should I work on both simultaneously or one first and the other later and if so which one.

>>	Anonymous Sat Nov 18 18:26:07 2017 No.9305292 >>9305276 preferrably do the first chapters in Tao first, then both or either

>>	Anonymous Sat Nov 18 18:27:08 2017 No.9305295 >>9305254 that's a reasonable plan, good luck. if you run into any trouble post around in /sqt/ or make a thread if it's anything more interesting

Anonymous Sat Nov 18 18:52:06 2017 No.9305348

>>9305295
BTW when I was talking about Apostol's analysis i was talking about this https://www.amazon.com/Mathematical-Analysis-Second-Tom-Apostol/dp/0201002884/ref=sr_1_3?s=books&ie=UTF8&qid=1511048948&sr=1-3&keywords=Tom+Apostol.. I'm assuming when you talk about his calc books youre referring to this https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_3?s=books&ie=UTF8&qid=1511049070&sr=1-3&keywords=Tom+Apostol

Anonymous Sat Nov 18 19:00:20 2017 No.9305359

>>9305348
you're right, I assumed wrong.

by skimming through Apostol's analysis, it seems to include a lot of extra topics. it looks like way too much material for a first course, it's going to take ages to get to the crucial implicit and inverse function theorems. it looks great as a reference though or as a go-to for topics in analysis for someone with some experience. Tao's Analysis I is especially light to read and welcoming for newcomers, even more in comparison with this book, but is generally trash as a reference for later on.

>>	Anonymous Sat Nov 18 19:10:05 2017 No.9305372 >>9305359 Oh ok that makes sense, i did try to read through Apostle's Analysis before but I saw that I didn;t understand shit. How about his calc books, what do you think of those? worth going over?

>>	Anonymous Sat Nov 18 19:26:34 2017 No.9305393 >>9305372 to learn calculus, yeah sure. I kinda liked his book, I read it for like a day to review calculus. i'd just go for analysis tho

Anonymous Sat Nov 18 19:29:43 2017 No.9305397

>>9305393
Alright then, wont bother you anymore about this. Im just gonna finish multivariable in stewarts book, ill redo calc with apostles calculus vol 1&2, then read the how to proof books, then move on to analysis by tao and that linear algebra book. thx again for the help, ill remember u

Anonymous Sat Nov 18 19:32:28 2017 No.9305399

Anyone here favor the abacus? I tried to bring mine to my college math class, but they practically held a gun up to my head and forced me onto the calculator, which I have never learned to use. What could I do in this situation? I'm not comfortable with calculators and anything that can be done on the calculator can be done with an abacus.

>>	Anonymous Sat Nov 18 19:34:45 2017 No.9305402 >>9305399 get back in ur time machine

Anonymous Sat Nov 18 19:42:51 2017 No.9305409

Is the Squeeze Theorem and the Intermediate Value Theorem the reason why Calculus actually works?

In other words, assuming you don't take the infinitesimal approach, you have a dy/dx because it has to be on that point because it's the only value in between the last two points on other side, so it's squeezed in there and you know it has to exist because it's the only value that keeps in continuous?

>>	Anonymous Sat Nov 18 20:20:21 2017 No.9305489 >>9305254 Linear Algebra Done Right is known for eschewing matrices in favor of a more abstract approach, just FYI.

>>	Anonymous Sat Nov 18 20:25:58 2017 No.9305500 >>9305489 >Linear Algebra Done Right is known for eschewing matrices uh?

>>	Anonymous Sat Nov 18 20:40:47 2017 No.9305526 >>9305500 It completely downplays the matrix aspect. For instance, determinants don't show up until the last few pages.

Anonymous Sat Nov 18 20:44:28 2017 No.9305537

>>9305526
>It completely downplays the matrix aspect.
Have you even read the book? How does it downplay matrices? These are all in the table of contents

>Representing a Linear Map by a Matrix
>Addition and Scalar Multiplication of Matrices
>Matrix Multiplication
>Linear Maps Thought of as Matrix Multiplication
>The Matrix of the Dual of a Linear Map
>The Rank of a Matrix
>Eigenvectors and Upper-Triangular Matrices
>Eigenspaces and Diagonal Matrices
>Trace: A Connection Between Operators and Matrices
>Determinant of a Matrix

>>	Anonymous Sat Nov 18 20:57:36 2017 No.9305559 What do you think of "Mathematics. Content, methods and meaning." By a Russian guy starting with K?

>>	Anonymous Sun Nov 19 02:57:32 2017 No.9305951 >>9305044 t. esl subhuman it's "married to wife", "married with child"

>>	Anonymous Sun Nov 19 03:06:02 2017 No.9305956 File: 103 KB, 1219x250, tits or gtfo.jpg [View same] [iqdb] [saucenao] [google] >>9305129

>>	Anonymous Sun Nov 19 03:16:36 2017 No.9305963 File: 42 KB, 752x489, 1501239839470.png [View same] [iqdb] [saucenao] [google] >Spent a semester studying group theory and analysis >Completely forget linear algebra and differential equations tfw small brain capacity

>>	Anonymous Sun Nov 19 04:29:16 2017 No.9306020 >>9305956 please do not feed the trannyposters

>>	Anonymous Sun Nov 19 04:36:02 2017 No.9306026 File: 64 KB, 1440x1080, maxresdefault.jpg [View same] [iqdb] [saucenao] [google] >>9306020 >please do not feed the trannyposters >anyone that isn't a "dude" is a "tranny"

>>	Anonymous Sun Nov 19 04:48:21 2017 No.9306040 >>9306026 >there are women on 4chan

>>	Anonymous Sun Nov 19 04:51:48 2017 No.9306041 >>9306040 >>there are women on 4chan Uh... yeah, actually

>>	Anonymous Sun Nov 19 05:29:35 2017 No.9306064 any interesting recent applied math papers?

>>	Anonymous Sun Nov 19 05:35:23 2017 No.9306068 >>9306064 How do I get into applied mathematics?

>>	Anonymous Sun Nov 19 08:36:38 2017 No.9306259 >>9305399 decent meme, I like itq

>>	Anonymous Sun Nov 19 08:55:29 2017 No.9306281 >>9305963 You're not going to last very long if you don't shore up that linear algebra.

>>	Anonymous Sun Nov 19 09:06:22 2017 No.9306291 >>9305963 What the other guy said. Learn Linear Algebra more than anything else. It is THE most important subject in mathematics.

>>	Anonymous Sun Nov 19 09:31:48 2017 No.9306315 Threadly reminder that Rudin is a meme.

>>	Anonymous Sun Nov 19 10:25:14 2017 No.9306392 >>9306315 It's too late for a reminder now

Anonymous Sun Nov 19 10:44:15 2017 No.9306422

>tfw self studying optimization since the lectures for it was at the same time as the mathematical statistics lectures
It was a mistake desu, but was quite fun when I discovered that one of the phd students that helps with the stats class is also taking the same optimization class

>>	Anonymous Sun Nov 19 14:15:26 2017 No.9306924 What are some good book about the history of math?

>>	Anonymous Sun Nov 19 14:38:13 2017 No.9306977 >>9306291 >It is THE most important subject in "mathematics". Perhaps for engineer scum.

>>	Anonymous Sun Nov 19 14:43:35 2017 No.9306983 >>9306924 john stillwell - mathematics and its history it's what wildberger uses

>>	Anonymous Sun Nov 19 14:52:46 2017 No.9307005 >>9306041 What part of "tits or gtfo" is not clear?

>>	Anonymous Sun Nov 19 15:27:52 2017 No.9307093 >>9306977 it is though

>>	Anonymous Sun Nov 19 15:33:02 2017 No.9307108 >>9307093 Yeah, for engineer scum.

>>	Anonymous Sun Nov 19 16:01:42 2017 No.9307175 >>9306977 no, it just is for mathematics as well

>>	Anonymous Sun Nov 19 19:06:04 2017 No.9307642 >>9306977 Fuck off.

>>	Anonymous Mon Nov 20 00:15:03 2017 No.9308130 What's even the point of doing a pure maths major

Anonymous Mon Nov 20 01:01:23 2017 No.9308201

>>9301082

sit down with a pencil paper and compass. draw. see how much you can describe what you have drawn algebraically, as equations. then write those equations in statement form. voila, now you have just derived your own theorems about trigonometry. now look up identities and see what you missed. wait a day. write down everything you did the previous day. see what you missed. keep doing this until you have written an entire book on trigonometry or have discovered theorems not yet known to man.

>>	Anonymous Mon Nov 20 01:07:20 2017 No.9308206 >>9308130 when mommy and daddy are footing the bill or you are naive enough to think you are going to contribute something or you have nothing to live for except deduction by hand

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