/sci/ - Science & Math

>>10229303
Based Lurie

what can I expect from the Math GRE?

Chad Algebra
https://www.amazon.com/Hot-Algebra-Exposed-Danica-McKellar/dp/0452297192/

Virgin Algebra
https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X/

So, you have the long line, which is comprised of an uncountable amount of unit intervals. Does it have a first interval?

>>10230082
Lots of calculus. Which is worse than it sounds, because they're all backasswards trick problems and you only have basically a minute or two per problem to figure out what to do. The majority of your time spent preparing should be learning calculus tricks from a prep book.

The stuff that isn't calculus on the test (abstract algebra/topology/probability and combinatorics/etc.) is fairly basic. If you've even taken a course on those topics you should get them all without too much trouble.

Also, I'd recommend taking a week or so to go back and refresh high school geometry. A handful of geometry problems always show up and it's rough going to try and improvise when you haven't touched a geometry problem since 10th grade.

Hi i want to start apostol calculus and to make sure i dont procrastinate i want to chat with someone daily to make sure i put in work that day, where can i find a place to find people like this?
Also because its /mg/ i can pursue 'computational' maths as a masters with either CS or math bachelors, since i can do it with a CS bachelors what exactly would i be behind/lack knowledge of if i decided to go with the CS bachelor? Will i need any of the pure maths? Or what?

>>10230976
Start up a discord study group, you'll always find newfags here wanting to learn the basics.

I know this guy was joking and all, but I ask the teachers of /mg/; is this actually doable?

>>10231011
that just sounds like it'll attract a bunch of cancer, also
>discord

>>10231040
>I know this guy was joking and all, but I ask the teachers of /mg/; is this actually doable?
We prefer this.

>>10231154
>discord
I'm not caught up with /g/ memes, which one is fresh there now?
I'd say you can add me and have me nag you into studying, but I'm not in the mood for that.
>>10231040
>is it doable
No. Normal people are normal people.
>>10231160
If you move everything one year forward, that is actually possible.
It's just really stupid.

>>10231163
hi im >>10230976
im frazzles#4215 at discord, im fine if you want something other than discord

>>10231217
Fuck it, whatever, invite sent.
Can't promise to remind you every day, but I'll try.

>http://pi.math.cornell.edu/~hatcher/TN/TNpage.html
>Topology of Numbers
>The plan is for this to be an introductory textbook on elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. The title "Topology of Numbers" is intended to convey this idea of a more geometric slant, where we are using the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning of a set with certain specified subsets called open sets. A fair portion of the book is devoted to studying Conway's topographs associated to quadratic forms in two variables, so perhaps the title could have been "Topography of Numbers" instead.

wtf I love number theory now!

>>10231040
>real analysis and abstract algebra in 9th grade
Soon to be teacher here, no. I had to take those two classes in one semester with the other 2 classes being joke classes, and it was still bloody. We can't even get a lot of teenagers to past Algebra 1 to graduate high school. If /sci/ had their way we'd be teaching kindergarteners about Noetherian rings and Lebesgue integration. I can see Calculus 1 in 9th grade being common place if we revamped everything, but anything else is a pipe dream.

>>10231478
11th grade*, can't read when under caffeinated, still holds true though.

>>10230881
No, read the construction again.

>>10231478
I don't understand why most everyone can manage to learn basic chemistry or biology but so many struggle with basic algebra. Is it because algebra is purely abstract?

>>10231503
Different, topologically-illiterate anon here, couldn't you consider 0x[0, 1) as the first segment?

>>10231510
wait nevermind that's the long ray not the long line

>>10231359
>Hatcher
Who allows this hack continue to write terrible books?

>>10231518
is this a meme? was about to start hatcher's AT book

>>10231570
>was about to start hatcher's AT book
don't

>he gets nervous/anxious when doing math

>>10231589
Tell is what you saw, anon.

>>10231577
why. give me a concrete reason

>>10231589
does anyone outside of exams or frantically trying to find results to justify your continued existence to your advisor?

>>10231040
Certified teacher. No. These people have obviously never been in front of a class

Is there a non-trivial field on which matrices commute?

>>10231728
no

>>10231730
Thanks.

>>10231730
wikipedia claims this generalizes to matrices on any ring but I can't find a proof anywhere

>>10231764
for a non-unital ring*. A ring with unity is ez

>>10231768
...which they assume in the intro. kill me

>>10231764
if you have a multiplicative identity just take the generators of so(3) I suppose.

>>10231506
Don't ask me, I had a really hard time in brainlet general chemistry 1 but always got near perfect scores on brainlet algebra and at least got a B with a terribad professor in abstract algebra. I had a shit work ethic when I took chemistry as an underclassman though.

>>10231764
Hard to do that without a proper definition of "trivial ring".
The trivial field is just the 0 field.
Hit me up with the link.

>>10231040
I've taught modulo arithmetic to 4th graders with clocks. That list itself probably isn't doable but things can certainly be sped up compared to the standard curriculum.

>>10231570
I wasnkind of memeing, but I honestly am not a big fan of Hatcher.
Hatcher writes in a long-winded style and tends to hide his arguments and definition between piles of text. He often argues by appealing to geometric intuition, which, while interesting and a good way to think about the material, completely lacks the rigor you should have for your first foray into AT if it's your main text. Something like Spanier is better in my opinion, and Hatcher as a supplement would work well. The big boy book is tom Dieck. Hatcher does have good problems though, so I would look at those.

I also hate Hatcher's near lack of margins. Dear lord, is it ugly.

>>10229303
Given a finite field extension [math]F(\alpha_1,...,\alpha_n)/F[/math] is there an efficient algorithm for finding a primitive element [math]\beta[/math] (i.e. so that [math]F(\beta)=F(\alpha_1,...,\alpha_n)[/math])?

Every proof of the Primitive Element Theorem I've read proves the case for [math]n=2[/math] which proves the full theorem by induction. But I'm looking for something that constructs [math]\beta[/math] directly from the [math]\alpha_1,...,\alpha_n[/math] instead of constructing [math]n-1[/math] primitive elements to get to [math]\beta[/math].

>>10231768
>>10231771
Every ring has a multiplicative identity, brainlet.

>>10231771
We write down the general format of two 2x2 matrices.
(a b)(e f)
|c d| |g h|
The first element commutes iff ae+bg=ae+fc
By annulation, bg=fc for any duo of elements in the ring.
Since we can naturally multiplicate by the sum's identity, we summarize the condition as ab=0 for any two elements of the ring.

>>10231921
My bad, that's only for commutative rings.
ae+bg=ea+fc
We can make the matrix a=0, b=0 so you have the nice condition back, multiplication of any two elements has to be zero, and that solves us back into commutativity.
Of course, a bunch of rings satisfy ab=0 for any a and b.
But if it does satisfy that, then it obviously commutes.

Lads.. I don't get it. Why is [math]\text{Gal}(K/F)[/math] the identity. Can we not have some automorphism that permutes the extra roots? I would have thought if anything, it would be [math]S_n[/math]. Note that the minimal polynomial of [math]a_i\in F[/math] is denoted [math]\text{min}(F,a_i)[/math].

Can an automorphism fixing the ground field of an algebraic extension only permute the elements that belong to the same minimal polynomial?

>>10231967
minimal polynomial of [math]a_i\in K[/math] over [math]F[/math] ****

>>10231967
Never mind... after some digging around on google, I found that that is actually true, yet there was NO MENTION of it in the book... This is the second result that he uses without mentioning...

>>10231891
Dear God, please excuse my LaTeX vomit...

>>10231999
This is an elementary fact that's used pretty commonly in Galois theory without explicit mention. It's more or less obvious too, think about what a homomorphism does to your minimal polynomial.

>>10232054
yeah, but it's an intro to Galois theory book, so you'd expect some mention.. and yeah, in the end, it is obvious to prove, but so is the first isomorphism theorem

>>10231967
One of the things that people don't usually learn in Galois theory is that it isn't the big, unreadable fifth degree polynomials that stop you from solving fifth degree polynomials by radicals.
It's the really shitty, easy to solve ones. Like, x^6+1 has S_6 as its permutation group, because its roots are so shitty you can permute them however the fuck you want without it mattering. Polynomials with 5 different positive irrational roots that you wouldn't guess in a million years all have that fucking identity only Galois group.

>>10232283
>Like, x^6+1 has S_6 as its permutation group
It doesn't though. x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1)

>>10232305
Right, my bad. I suck at examples.

>>10232283
>x^6+1 has S_6 as its permutation group, because its roots are so shitty you can permute them however the fuck you want
Lolwut? It's not even irreducible over Q.

how would you guys go about proving that there exists infinite solutions for 3 co-prime squares x^2,y^2,z^2 in arithmetic progression? anything except rational solutions on the circle.

>>10232522
Infinite solutions to what?

>>10232559
(x,y,z) = (1,5,7) are such solutions since their squares are co-prime, and
1*1 = 1
5*5 = 25
7*7 = 49
they are in arithmetic progression. prove there are infinite triplets of solutions.

>>10232522
Assume it has a largest solution, probably.

>>10232522
(z^2-y^2)=(z-y)(z+y)=(y^2-x^2)=(y-x)(y+x).
Therefore, you just need to show that
(z-y)(z+y)=(y-x)(y+x)
Has infinite solutions.

>>10231908
wrong. any interesting ring does, but a general ring need not be unital.

>>10232283
you really shouldn't make posts explaining galois theory when your grasp of galois theory is pants-on-head retarded

>>10233033
>but a general ring need not be unital.
You mean a rng.

>>10233237
>>10233033
>The conventions used by my book/professor are unique
autism

>>10233237
that's not even a word

i dont get it

>>10234044
>that's not even a word
https://en.wikipedia.org/wiki/Rng_(algebra)

This is the entrance exam for the best uni in my country. It's not in English but just glancing at it would you say it's brainlet tier or decent difficulty?

http://prijemni.etf.bg.ac.rs/resenja/2018/matematika.pdf

>>10234582
Seems really easy, but I might be mistaken, since I can't read that shit.
You've got it in the bag.

>>10234613
Yeah even with my brainlet math skills I can tell it's most everything just simplifying things. Got like 6 months to study that but gonna have to ace it due to poor grades. Though I think it'll take much less than that

How many ways to choose 3 elements from the set {x in N: 1 <= x <= 40} such that their sum is an odd number?

>>10235084
>How many ways to choose 3 elements from the set {x in N: 1 <= x <= 40} such that their sum is an odd number?
What have you tried?

>>10235084
You have to pick either 3 odd numbers or 1 odd number and 2 even numbers so it's
[eqn]{20 \choose 3}{20 \choose 0} + {20 \choose 1}{20 \choose 2} = 4940[/eqn]

>>10235101
This is definitely the worst pasta on the board.
>>10235109
This.

>>10234582
It just looks like a lot of busy work and none of it is exactly relevant when you get to non-brainlet math. Entrance exams are retarded.

>>10235122
Well to get into a uni here you have 40 points from high school (sum of the 4 year's GPAs * 2) and the entrance exam which is different for every uni and gives a maximum of 60 points. That's it. No bullshit AP classes, extracurriculars, letters of recommendation. If you perform the best, you know you're getting in 100%.

is this a good book for solving geometric problems? what about solid geometry?

>>10236010
Why don't you read it and find out?

If the number set isn't specified where it should be, is it assumed that it's the real numbers?

>>10236186
If the base set isn't specified it is literally always completely obvious from the context which set it is. Obviously R is not _always_ the de facto choice: if you're reading a number theory book you would generally assume any theorems you read are about N, or in algebraic geometry the field is generally assumed to be C when it's unspecified.

>>10235109
Thanks, I've been reading up on combinatorics and it's very interesting.

Here's another one:
Make a 5 member group of 2 mathematicians and 8 physicists, such that it consists of at least one mathematician.

[eqn]{10 \choose 5} - {10 \choose 2} + {10 \choose 1} = 197[/eqn]

The solution is 196 though.
It's probably obvious but I can't tell why there is an extra one.

Also, can you recommend a rigorous introductory combinatorics book?

>>10236245
Take all possible combinations of 4 physicists, and multiply by two, because you could top it with either mathematician. Then take all combinations of 3 physicists.

>>10236254
I see. Can you explain what the fault in my reasoning was? Is it just a coincidence that what I got was off by 1 from the solution?

>>10236010
try coxeter geometry revisited

>>10236186
write the sentence from what book/class and ill tell you

>>10236358
Right, so from what I'm seeing, you took the whole set of combinations, removed two of them (because those had to go to mathematicians) and added one option back in (between the mathematicians).
The problem is that the mathematicians are a fixed two of the ten people. Even if you'll unpick two of the picked 5 to replace the mathematicians, you might be unpicking exactly those mathematicians.
Or maybe I'm completely misunderstanding your intuition.
The answer was close by coincidence.

>>10236836

To be honest looking at this now, I can't even tell what I was going for here. What I WANTED to do is subtract all the lists that don't fit the criteria from the total combinations. So take out all the lists that don't have even one mathematician:

[eqn]{10 \choose 5} - {8 \choose 5} = 252 - 56 = 196[/eqn]

Though I can't tell if this is more intuitive to me than what you suggested. Wonder if there's another way to find the solution.

>>10236892
Right.
Imagine we had 5 mathematicians to pick from. So we could do some trickery and reduce the problem to one simple calculation, but that often enough doesn't come to us, so we do this:
We set i as the number of mathematicians picked, and we set up the number of possible combinations of mathematicians and physicists based on this i. Then, we put a big Sigma on the beginning, and tell it to sum as i goes from 2 to 5.
What I'm trying to say is the main trick of combinatorics is breaking down the problem into different combinations of problems, and that's the method you'll usually want to go for, since it works a solid 90% of the time.
Of course, then stuff like superpermutations just look at you like you're retarded, but that's life.
Finally, that's a very nice solution.

>reading a book on algebraic geometry
>the book keeps using X for anonymous sets and k[X] for polynomials
Every uni should teach students the greek and hebrew alphabets on the first semester to avoid this type of stuff.

>>10236975
>the book keeps using X for anonymous sets and k[X] for polynomials
define "anonymous sets"

>>10236983
Random sets.

>>10236986
>Random sets.
Random in what sense?

>>10237159
In the sense of "Let X be a set".

>>10237168
>In the sense of "Let X be a set".
What's random about that?

>>10237179
Ah, I understand. You also don't have anything to do.
Watched any nice animes recently?

>>10236975
yeah, i hate that k[X] can be both the coordinate ring of X or the ring with elements of X as variables, and even worse, when X itself is a variable

umm.. help?

>>10237983
17^2 = (10 + 7)^2 =100 + 140 + 49 = 289
In general
(a + b)^2 = a^2 + 2ab + b^2

>>10238003
>>10237983
i think it has something to do with a categorical generalisation but im not sure

>>10237983
What's the context?

>>10238020
im not sure

>>10238023
>tfw remember I never learned what an ultrafilter was and that I need to do it sometime
Anyhow, 18x16=290, because (a-1)(a+1)=a^2-1.
That should do it.

>>10238033
My bad, 288

>>10238033
>>10238035
I don't think this argument works, because we're using a different type of distributivity here

>>10238040
>the amazing and natural applications of category theory

>>10238049
it's more concrete than you'd think

redpill me on stacks

>>10238521

Can someone explain the proof of Rouché's theorem to me? My book basically says "well for each loop γ_k, f(γ_k) and (f+g)(γ_k) are homotopous by this homotopy, so they have the same winding number around 0, so f and f+g have the same number of zeroes", but i don't see how it uses that |f|<|g| on γ.

>>10239164
i mean * |g|<|f|

>>10239164
It might loop de loop if you don't need |g|<|f|.
For example, if there's a point where f=-g then you done fucked up.

>>10239204
thank you and merry christmas

what linear algebra is needed for basic functional analysis? just need a few things to know what direction to prep in

>>10239386
Merry christmas!
>>10239483
The entire linear algebra. With eigenvectors and algebras and stuff. In particular eigenvectors.
You'll also want to know real and complex analysis, basic point set topology, and some PDEs and ODEs so you don't feel like everything you're reading is pointless.
Measure theory is nice, but not a requirement. I would still advise reading it before tho, because it's a nice subject.

>>10239483
functional analysis IS linear algebra with extra structure. so, everything really.

>>10239684
Hm... without the Lebesgue integral and function spaces, it's pretty hard to talk about weak solutions, traces, etc. Plus, measure theory gives you an appreciation for pathological functions. Maybe the abstract measure theory can be postponed, but even that is useful for when you switch between measures, which you end up doing sometimes in statistics, probability, and certain energy estimation techniques for PDE.

What books can I look at for set theory, computability theory, proof theory, infinite combinatorics/graph theory, large cardinal properties, inner model theory, and all that jazz?

What about stuff on categorical logic?

>>10239723
>categorical logic
Goldblatt's book on topoi

>>10239723
>set theory, computability theory, proof theory, infinite combinatorics/graph theory, large cardinal properties, inner model theory, and all that jazz
>categorical logic
not math

>>10239723

> infinite combinatorics

she just left me bros. thats it, no more bitches till the fields medal

What is this sum supposed to mean?

it is from a paper about boolean functions (https://eprint.iacr.org/2007/290.pdf))

My best interpretation:
> The inner sum isn't really there
> They include it to show they are treating the binary representation of i as a vector
> because deg g <= deg f, the terms in the sum with nonzero coefficients will only have d or less set bits

>>10239813
Everything is correct there. Just look at a specific example like n=3 then
[eqn]g(x_1,x_2,x_3) = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3 + a_{12} x_1 x_2 + a_{13} x_1 x_3 + a_{23} x_2 x_3 + a_{123} x_1 x_2 x_3 [/eqn]

>>10239717
It is very important to the subject, but it's still not a strict requirement.
And you mostly use the absolute basics of the basics: the definition of a measure, the definition of an integral, Radon-Nykodim and Fubini. But that's all essential Real Analysis too.

>>10239780
>hurr durr
https://en.wikipedia.org/wiki/Infinitary_combinatorics

>>10239751
>math
please define

>>10239751
what's with all of these scifags bashing on set theory and category theory?

Merry Christmas, /mg/!

>>10240427
christ wasn't born on the 25th
roman pagans were giving gifts on solstice since the dawn of time
materialism has corrupted all sense of 'good will' and has forced gift giving as an obligation
therefore we celebrate nothing but our degeneracy.

>>10240427
>Merry Christmas
Happy holidays.

>>10240527
Merry Christmas, /his/!
>>10240531
Merry Christmas, /biz/!

>>10240547
the only thing that's merry is the bottom of a bottle. god i'm going to fucking kill myself

>>10240549
Merry Christmas, /r9k/!

Does being happy with antidepressants make you better at math than sad without them?

>>10240427
Merry Christmas

>>10240527
Merry Yuletide(?)

>>10240371
It's geofags kvetching about any math they can't relate to Euclidean n-space

>>10240845
Atiyah is entirely right though

>>10240596
significant cognitive impairment is an almost ubiquitous symptom of moderate to severe depression
if you're depressed enough to need medication for it taking them will absolutely improve your academic performance

>>10240976
>implying that geometric intuition is the only form of intuition

>>10240994
im too beta to go to a doctor or whatever, what do?

>>10241673
kill yourself

>>10240976
Case in point

>>10240845
>>10240976
>>10241151 is not only correct, but it's also the single worst intuition people have.
I'd post a bait Euclidean problem but I don't keep folders with that stuff, so I'll just say "Weierstrass function".

The only thing I like is math, but I can no longer dent the fact that jobs in academia are basically impossible. Is it time to bite the bullet and accept the fact that I’ll probably be a software engineer?

>>10242565
*deny

is this the most intimate position?

>>10242565
I'm going to fall for the machine learning meme myself I think

>/g/ tries to figure out conditional probability
>>>/g/69069311

>>10242408
i recently forced myself to stop appealing to my geometric intuitions so much because they often lead to a dead end.

I spent a lot of time last summer doing math

turns out I basically rediscovered this shit https://en.wikipedia.org/wiki/Algebraic_normal_form#Recursively_deriving_multiargument_Boolean_functions

Let's say we have some [math] u \in H^1(\Omega) [/math] with
[eqn]\Delta u = 0 [/eqn]
and
[eqn] tr(u) = f [/eqn]
in the weak sense for some [math] f \in C^\infty [/math], where [math] tr: H^1(\Omega) \to L^2(\partial\Omega) [/math] is the [math] H^1 [/math] trace operator.

Is [math] u \in C^\infty (\Omega) [/math] (maybe with some condition on [math]\partial \Omega [/math] like lipschitz boundary or even piecewise lipschitz)?

>>10243856
I mean, firstly, shouldn't u be in H2 so its weak derivatives even exist?

>Δu=0
This is just weak harmonicity, right?

If u isn't just H2, but also L1 and Ω is open (and either bounded or u locally L^1) this directly implies harmonicity and smoothness, see Weyl's lemma.

>>10243876
I was referring to the variational problem associated to the laplace equation, i.e.
[eqn]
\int_\Omega \nabla u \cdot \nabla v = 0~~~\forall v\in H^1_0(\Omega)
[/eqn]
thus only requiring [math] u\in H^1[/math].

Weyl's Lemma kind of looks like it's what I'm looking for, even though I'm surprised it doesn't require some regularity for [math] \Omega [/math]

>>10242408
That's only because people have shit intuition, that's all. You won't go anywhere in, say, riemannian geometry or geometric group theory, or dynamics without some physical and geometric/topological intuition.

>>10243894
Weyl's lemma really has very few assumptions about Ω, your Integral seems something very close to weak harmonicity (that would be the case in which v is in C^infinity with compact support), which would directly imply the very weak harmonicity required for the Weyl lemma.
The proof of Weyl's lemma can be done via mollification, which kinda explains why it really doesn't matter much what exactly happens at the boundary of Ω.

I've already seen fifteen different definitions of regular and rational maps for specific cases.
What sort of notation hell is this.

What is a dynamical system, /mg/? And why are some research groups obsessed with it? Explain to me like I'm 5yo.

>>10245132
Imagine you have a function that describes the state of something at some point in continuous or discrete time.
That's a dynamical system. The position of a point in time is an example, the temperature in a room, the value of stocks, whatever.

>>10237983
kek is this from Mathematics Made Difficult? a true classic indeed

>>10245298
yeah

>>10239813
they're writing g as a sum over all possible monomials of degree <=d. the first sum is over all degree 1 monomials, the next (not shown explicitly) is over all degree 2 monomials, and continuing all the way up to degree d

So, we are given the function [math]f:\mathbb{R}^{n}\rightarrow{\mathbb{R}}[/math] defined by [math]f(x) = \frac{1}{2}x^{T}Ax - x^{T}b[/math], where [math]b\in{\mathbb{R}^{n}}[/math] is given and [math]A\in{\mathbb{R}^{nxn}}[/math] is given and assumed to be symmetric positive definite.

How do we know that [math]x=A^{-1}b[/math] is the unique minimizer of [math]f[/math]? Further, how does one compute:
[math]\nabla{f(x)} = Ax - b[/math]
and [math]\nabla^{2}{f(x)} = A[/math]?

>>10245142
Ah, it's not as crazy as I first thought.

>>10245335
Man, your pic depicts my performance in this semester's finals so well.

>>10245335
I don't get the question, you've posted the answer. Apply gradient to f, set grad f = 0 and solve for x. Apply another gradient to (grad f) to get the last one.

>>10245417
Well, I suppose I would then ask how one would determine that the convex, so you knew that it was a *minimum* instead of a maximum. Also, how do you compute these gradients? I checked the wikipedia page and realized that they are the gradients, but how were they computed?

>>10245637
>how one would determine that the convex
Prove it by definition. There is also a characterization using the second differential, which is A and thus positive definite and therefore f is convex.
>Also how do you compute these gradients?
Try doing it yourself, I'm not going to do the rest of your homework.

>>10229303
Is this a good book for a math brainlet to help with improving mental math skills?

>>10246143
No

>>10246240
why not

>>10246258
The best way to get better at mental maths is do mental maths. It doesn't have much to do with memory anyway.

>>10246285
Oh well.
I just thought it would be a good book to supplement studying and provide some useful learning techniques is all

>>10246294
No harm in checking it out.

I'M PROCRASTINATING AGAIN /mg/ HOW DO I STOP IT?

>>10246536
I DON'T KNOW, BUT IF YOU FIGURE IT OUT PLEASE TELL ME

Howdy.
I've been trying to find a text on pointless topology but libgen.io doesn't return anything for any of the alternative names wikipedia lists (namely Pointfree topology, Locale theory, or variations with spaces, etc).
Any tips? Is it basically lattices and not worth actually studying or something?

>>10246660
it's basically lattices, but from a thorough categorical standpoint. the idea is like rings<=>affine schemes, you have an equivalence locales with enough points<=>sober spaces. And like with schemes, this equivalence extends to a more general adjunction. But there's a lot in the locale literature that you won't find in most of the order theory literature; things like tensor products of frames become important, you do sheaf theory on them, and you want to understand the behavior of "topological" conditions on locales (hopefully stated in terms of formal open sets) under things like base change. So, while nominally the same subject as order theory, what you care about (and hence the theorems you prove) are quite different and more geometric in nature.

>>10247073
Thank you very much!
Any major requirements other than sheaf theory?

Is it true that [math]\forall S(x), P(x)[/math] is equivalent to [math]S(x) \rightarrow P(x)[/math]? If so, where can I find a formal explanation to this. The only book where I could find this is "A Transition to Advanced Mathematics", by Smith, Eggen, and Andre, wich is a book with no references. I searched on Enderton's book on logic and other similar books, but I found nothing.

>Galois

Excuse me, I need that for OC.
>>10247232
>a literal comma separating S(x) and P(x)

>>10247264
There we go.

>>10229303
Is "Calculus III" literally just multivariable and vector calc in one class?

>>10247455
I took "Calculus 3" at a community college that used the quarter system. We learned out of Stewart. The tiers looked like this:
>Calc 1
Limits, differentiation, related rates
>Calc 2
Riemman sums, integration, FTC, optimization
>Calc 3
Sequences and series, introduction to vectors and vector operations, brief intro to differential equations
>Calc 4
Multivariable calculus, parametric curves, etc

>>10245844
>Try doing it yourself, I'm not going to do the rest of your homework.
I understand your frustration, but I have only ever seen a gradient in terms of a function in the form f(x,y). So seeing a "gradient" in this context (and also referred to as a derivative, but just with the nabla symbol) is rather confusing. Wikipedia wasn't very helpful either. If you don't feel like spoon feeding me, what sort of book do you think I should check out to get up to speed on these topics? Is this just analysis with a bit of topology? Haven't taken a proper analysis course yet...

>>10247455
Calc III is vector calc, II is integral and I is differential, IV is Multi-variable then people typically take ODE’s and Lin alg and stat analysis.

>>10247520
>>10247520
https://en.wikipedia.org/wiki/Gradient#Definition
I dunno, the wikipedia gives a nice definition. But I'll write it down real pretty for you.
Let [math]f: \mathbb{R}^n \rightarrow \mathbb{R}[/math].
We define [math]\nabla f(x_1, x_2, ..., x_n)= \sum_i^n \frac{ \partial f}{\partial x_i}[/math].

>>10247556
Gee, thanks

>>10247556
Actually my bad, that's the other thingy.
The gradient is [math]\nabla f=(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, ..., \frac{\partial x_n}{\partial x_n})[/math]

>>10247573
Ugh, my finger slipped right at the end.
Fucking multivariable calculus.
The gradient is [math]\nabla f=(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, ..., \frac{\partial f}{\partial x_n})[/math]

>>10247088
not really. be comfortable with order-theoretic proofs and elementary category theory like limits and stuff. it's helpful if you know basic commutative algebra because many of the terms are borrowed from it and it will help ground algebraic intuition. you can even pick up the topology along the way tbqh but it mostly has to do with size stuff (mainly basics like be connected or compact and variations on those). it can be learned relatively in isolation, other than the sheaf theory, which is nice and fun. I also highly suggest using vickers to supplement it with the logic/computation side to show you some applications.

>>10247088
also, the elephant is actually a great book. its length is intimidating, but actually makes it more accessible than topos theory imo. Also, I suggest using borceux (I think its volume 3?) as a companion because it has a lot of the details spelled out.

>>10247088
looking over it again, C1 from the elephant is just so fantastic (and is definitely available on russian websites). knowing a bit of topos theory is also a great motivator, so maybe learn it in tandem (Lawvere-Tierney topologies and sublocales are closely related). It's just so beautifully clear in connecting it to logic, order theory, topos theory, and topology.

>>10247674
>Sketches of an elephant
>1100 pages
Alright. I'm terrified, but I can figure myself out.
Thank you once again for the advice.

>>10246660
>I've been trying to find a text on pointless topology but libgen.io doesn't return anything for any of the alternative names wikipedia lists (namely Pointfree topology, Locale theory, or variations with spaces, etc).
>Any tips?
Look up the ones mentioned in https://en.wikipedia.org/wiki/Pointless_topology#Bibliography

https://arxiv.org/pdf/1812.09999.pdf
>Potential automorphy over CM fields
>Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack A. Thorne
>(Submitted on 25 Dec 2018)

>Let [math]F[/math] be a CM number field. We prove modularity lifting theorems for regular [math]n[/math]-dimensional Galois representations over [math]F[/math] without any self-duality condition. We deduce that all elliptic curves [math]E[/math] over [math]F[/math] are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for [math]GL_2(\mathbb{A}_F)[/math].

>>10248028

>>10248049
You just know

>>10248096
Just know what? Lol

>>10248028
>>10248049
holy fuck this is baste

https://terrytao.wordpress.com/2018/12/24/elias-stein
>I was deeply saddened to learn that Elias Stein died yesterday, aged 87.
>My weekly meetings with Eli would tend to go something like this: I would report on all the many different things I had tried over the past week, without much success, to solve my current research problem; Eli would listen patiently to everything I said, concentrate for a moment, and then go over to his filing cabinet and fish out a preprint to hand to me, saying “I think the authors in this paper encountered similar problems and resolved it using Method X”. I would then go back to my office and read the preprint, and indeed they had faced something similar and I could often adapt the techniques there to resolve my immediate obstacles (only to encounter further ones for the next week, but that’s the way research tends to go, especially as a graduate student).

>>10236245
Combinatorics: A Guided Tour

>>10248252
You just know that Scholze did everything and put other people's names because they're his friends.

convince me to believe in sets

>>10249274
{}

How do you deal with burnout? I haven't enjoyed studying math in months. I was briefly motivated to learn more algebraic topology but I lost interest in that as well.

>>10229303
Hey guys I'd appreciate some advice

What would be a good program to graph in that is based off rule propagation, if then statements, rather than based of calculating, preferable for a grid which can I can program a color organizing schema around as well

>>10249388
I have long term plans that require learning different maths
So that additional fact that it's more than just fun helps

>>10249274
elaborate on your ontological objections to them

>>10249388
If you convince yourself math isn't a field, but instead a large group of fields connected by next to nothing, you can burn yourself out in one and move to another.
>>10249399
What?

>>10249490
I can try that I guess. I've been going ham on algebra for ages now, maybe I just need a change of pace.

Can one eventually do math for 18 hours a day for years?

>>10250197
Why would anyone do that?

>>10250207
math monk status

>>10250197
Yes, it's called grad school

>>10250327
need an expectation vs reality for this meme

>>10247564
imagine putting a baby in that tit monster.

>>10250432

Mochizuki and Yamashita doubling (tripling?) down on Scholze-Stix's 'id-version' of IUT with revisions and lecture notes:

http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf
>(2018-12-28) Slightly modified the wording of (LbEx4) and added (LbEx6)

http://www.kurims.kyoto-u.ac.jp/~motizuki/Yamashita%20---%20Keio2018Nov_id_version.pdf
>Brief excerpt on the "id-version" of IUTeich from the lecture notes for a series of lectures on IUTeich given by Go Yamashita in November 2018 at Keio University

>>10247232
it's kind of equivalent. when we write VSx(Px) what we really mean, in terms of classical formal logic, is Vx(Sx -> Px). in mathematics it would be equivalent, but in the field of logic it would not be equivalent, because VSx(Px) is not well-formed and neither is Sx -> Px in classical logic.

What is going on here?

http://www.ams.org/journals/tran/1975-209-00/S0002-9947-1975-0409255-0/S0002-9947-1975-0409255-0.pdf

>>10229303
Thanks op, based find

>>10251209
>http://www.kurims.kyoto-u.ac.jp/~motizuki/Yamashita%20---%20Keio2018Nov_id_version.pdf
>that huge No!
lmao

>>10251209
sometimes I wish I could understand what was going on here but then I remember it's apparently completely useless outside proving the meme conjecture

>>10254387

A good reason to learn it would be to see if you can adapt it to something else

Sitting down and actually understanding all that shit would probably give you a bunch of ideas

>>10247455
if your school doesn't have a calc IV, then yes, calc III is multi and vector

Hit me up with the most autistic definition you know. Bonus points it's a simple concept.
>a ring is a duo of a group and a homomorphism onto its monoid of endomorphisms under pointwise addition

>read a paper in geometry
>see big jump in reasoning without sufficient explanations, don't understand it
>ask author for proof
>publishes 50 pages of explanations of the leap.
>lots of leaps in the 50p explanations of the leaps
>publishes 150 pages of explanations of the leaps in the explanations of the leaps in the original paper
>mfw im doing mathematics in 2070

>>10254693
If Scholze thinks its useless I doubt the depressed slacker I am will find something interesting in it

>>10254714
I still love defining a monad as a monoid in the category of endofunctions, especially when talking to CS babies

>>10254714
im sorry, but what the fuck is a duo

>>10254738
https://dictionary.cambridge.org/us/dictionary/english/duo

FUCK FUCK FUCK FUCK

MAKE IT STOP

>>10254823
The first one is literally "A field has an identity element lmao", the second is the exact fucking same, and the final one is fucking obvious.

>>10248680
Sounds like what my advisor is doing right now...

>>10254856
uhhh, yeah they're all obvious, but i dont see where the "identity element" is coming from

tfw realizing I'll never be a professor and I may as well get a masters in memechine learning and become a wagie instead of doing a PhD

>>10255508
[math]KL={kl|k belongs to K, and L belongs to L}[/math].
Since the multiplicative identity belongs to K, every element of L belongs to KL, and similarly for L. If every element of KL is separable, or inseparable, then the first two follow because literal subsets.
The third is left as an exercise to the reader.

>>10255527
I always forget the spaces.
[math]KL={kl~|~k~belongs~to~K,~l~belongs~to~L}[/math]

>>10255532
Use the TeX preview in the top left corner of the reply box before posting shit that won't compile right

>>10255525
>never be a professor
Why not? I honestly dont think it will be that hard. Just be willing to take on post docs and assistant professorships... You will get there

>>10255574
hahahahahahahahahahahahaha.
please talk to younger members of your department about career prospects in modern academia

>>10255532
>KL=kl | k belongs to K, l belongs to L

>>10255525
same. how the fuck am i supposed to cope guys?

>>10255527
>Since the multiplicative identity belongs to K, every element of L belongs to KL, and similarly for L. If every element of KL is separable, or inseparable, then the first two follow because literal subsets.
0/10 if you handed this as a homework solution

>>10255624
Same here, I'm going to legit kill myself

>>10248680

>>10255525
>>10255624
>>10255712
Wtf, in my country one can be a professor for the undergrad scum with only a masters degree. Isn't it enough for us/europe as well? I'm sure you guys could do it while working on your PhD. Or perhaps you guys don't want to teach the undergrad?

>>10255723
It's almost impossible in France if you're not a foreigner (and those foreigner are *much* less qualified than the french ones, but you know, affirmative action, thanks to the jews)

>>10255727
>France
Oh, everything makes sense now. I heard the unis there are vicious when it comes to math. Isn't moving to another country an option?

>>10255723
What country? And we don't just want to teach, the nice part of being a professor is being able to do research (or else we would just be high school teachers lol).
>>10255727
Seriously? Maybe I should learn French. I thought the French are insanely good at math though, why would they want foreigners?

>>10255752
France does not realize or care that other countries don't adopt this strategy of screwing over their own hard working citizens. The result is that foreigners get jobs both in France and their home country while French people are not prioritized either in France or abroad. How easy is it to find such a job in the US or in a country that's not third world? Also I need to speak the local language fluently, so this has to be French or English. Learning a new language as an adult to fluency is no joke

>>10255768
How easy as a foreigner I meant. Despite affirmative action I'm pretty sure they prioritize their own citizens. I'm also a white male. Tough luck for me I guess

>>10255764
They want foreigners for virtue signalling and pretending there is some equity when it's clear there fucking isn't. Positive discrimination is still discrimination. Almost all of my incompetent profs who couldn't solve a single exercise correctly from the tutorials or copied the solution online without understanding it (and leading to errors) were foreign

>>10255764
>What country?
Brazil, and research is somewhat encouraged as well here.
>>10255768
Man, I'm not sure how it works in US, but if you're a good you could always try getting in IMPA in Rio. Language is not a problem, you can give lectures in English. It's the third world, but you won't have the third world experience.

>>10255711
Is there a problem with it?
Sometimes I really just don't notice this stuff.

>>10255783
Didn't you guys just elect a crazed fascist who was somehow better than all the other candidates? No thanks.

>tf unironically thinking about moving to brazil

>>10255793
>why are you posting this here
>why do you even wanna come here
Is the job market actually that good and I just don't know it?

>>10255798
because of this post: >>10255723

>>10255783
I saw on academia stack exchange that a dictator had been elected and all funds were cut

>be undergrad
>work on research project with a couple of profs and another undergrad
>can't understand most of the math
>just compute stuff in sage and MAGMA
>even then the other student did most of the stuff
>tfw both our names are on a paper now

>>10255800
You do know you'll need to either take a test(more specifically, you participate in a competition to see who gets the best grade and gets the job) to work for a public university or get fucked over in a private one, right?
But the country's nice, feel free.

>>10255805
Apparently this is just what undergrad research is like. I'm doing a research project right now that basically consists of me fumbling out some incoherent thoughts that my advisor then turns into actual results.

>>10255805
>>10255813
after taking which classes can an undergrad ask to do a research project without being laughed out of the office?

>>10231040
terrance tao laughs at your brainlet curriculum.

>>10255815
Just basic familiarly with the broad area you'll be working in, you'll have to learn a ton of additional stuff to work on the project anyway. You might be turned down anyway though, professors are busy. If that happens hopefully they can recommend someone else for you to ask though.

>>10255792
>>10255803
Haven't heard about this, but I'm pretty sure that big boy institutes/unis are safe.

>>10255815
Idk, when I started working on this project, I had the basic lower division classes done and my first semesters of complex variables and real analysis. The project had a lot to do with algebra, and I ended up learning basic abstract algebra for the project. I learned a lot about a lot of different branches of mathematics, but the research goes so deep there's only so much you can absorb as an undergrad.

>>10255819
Dude, Tao is the outlier of outliers. I wonder how was his education, though.

>>10255902
I honestly wonder how much people like him are truly these superhuman outliers versus how much their early education (read: their parents) shaped them. Like apparently Tao was taking advanced math classes as a ten year old, I find it really hard to believe he was the one taking initiative for that. I was a """gifted""" child, maybe if my parents had forced me into college classes instead of left me in public school I would be a fields medalist too.

>>10255902
>hey kid, ya wanna buy some uppers?

>>10231506
It's because chemistry uses more than just the mathematical part of your brain. there's a bit of human type knowledge going on, so most people are able to use more of their overall brain power to understand it.
At least, that's my guess. Anybody who's done math and art before can tell you how different they are, so if you could blend them together somehow, it'd be easier for a layman than simply just math or just art.

ive always wondered if the field axiom 0 =/= 1 is redundant.

if you have the definitions for additive inverse, 0 and 1, distributivity and commutativity, then you get

[math]
1a=a,\space a+0=a\\
1a=a \implies (1+0)a=a\\
\implies 1a + 0a = a\\
\implies 1a+0a-1a=0a=a-1a=0\\
\implies 0a = 0
[/math]

this isnt the law for 1a=a unless a must be 0, but a is an arbitrary element of the field
so either a field has one element (which is 0 and 1), or 0=/=1 in the field.

the reason people seem to say that a 1 element field doesnt exist is because 0 and 1 need to be distinct, but in this context thats just circular

so it seems redundant to add this axiom since its inherent in any field with more than 1 element, but ive seen it in every single field definition. Do you guys know of any reason for that? or did i just fuck up in the steps somewhere

>>10255917
Well, don't make the same mistake your parents did if you find out your kids turn out gifted.

>>10256024
>the reason people seem to say that a 1 element field doesnt exist
https://en.wikipedia.org/wiki/Field_with_one_element

>>10256271
>The name "field with one element" and the notationF1are only suggestive, as there is no field with one element in classicalabstract algebra

Hey /mg/ I am trying to enjoy using math since I hated it in school, I am retarded compared to you but have had awful teachers that did not nurture my mathematics and did not teach me, so I want to do this without school. What are some good books to start learning math? I am reading "how to not be wrong" and am halfway through, would like to know where to progress for a general introduction to practical useful math, I am interested in probability, and I have interest in economics (I have Sowell's basic economics). Thank you!

>>10256411
>What are some good books to start learning math?
Precalculus Mathematics in a Nutshell by George Simmons. Buy it and dont look back.

>>10256717
important thread
>>10256717

>>10256375
>no reading comprehension

Are there any operations that are closed under the quarternions but not under the complex numbers with only complex inputs? If not, is it possible to make one, or will that lead to a contradiction?

>>10256941
https://en.m.wikipedia.org/wiki/Algebraic_closure
Then again, I'm not sure exactly what you mean eith operation.

>>10256941
leads to contradiction, since the complex numbers can be seen as a subset of the quaternions, so anything that works for quaternions (algebraically) must also work for the complexes

>>10255792
>>10255803
Ok, so first of all
>fascist
>dictator
Sounds like fear mongering to me.
Second of all, can you link me the where it was said that ALL funds were cut? Because as far as I know, the dude didn't even went through the presidential inauguration yet. He has done precisely 0 work aside from gathering his ministers.

What I know: some "funding" will be cut everywhere (education, Healthcare, security, welfare, etc) for the next 20 years because if we don't do it the country's economy will finally collapse due to mismanagement of our social security. We need this reform, and this was decided earlier this year by our Lord Vampire President.

Now, the education funds: sadly, our base (elementary and middle school) will be the ones to tank that shit. We've always spent a lot more money on superior education than in our kids, when it should be the other way around. The unis will be mostly safe, but I'll tell you what kind of people in superior education that won't be safe: the ones that pump out useless "research" and pursue higher education for higher salaries alone, namely the people in humanities and music, the latter I'm more familiar with because my brother is a musician and PhD student (something to do with music and technology), and he said that at least 80% of the "research" done in music belong in the trash bin. He saw people getting alway with garbage thesis saying shit like "b-but that's what I FEEL!". This in music, where there's some level of objectivity, now imagine the social sciences with their "ass-sniffing as a method to approach new people" kind of thesis. A lot of money is going to the drain because of shit like this. The situation is so critical that there's a "pseudoscience hunting" taks force that cut funds for bullshit graduate students that fail to show the efficiency of their work (namely, the homeopathy faggots that waste our Healthcare money on their fairy-tale)
I could go on, but this is not the time, nor the place.

>>10255727
>if you're not a foreigner (and those foreigner are *much* less qualified than the french ones, but you know, affirmative action, thanks to the jews)
I'm French and I have literally never seen this happen

>>10257003
Where are you studying? Also, are you studying maths or physics? I don't know about other subjects. Also maybe you are still currently in class prépa. There you won't meet many incompetent immigrants because they purposely let the useless ones teach at universities because they don't care about university students. You'll see in third year when you leave (even in Grandes Ecoles) that the phenomenon is real

>>10256024
No it is not redundant (ie. it does not follow from the other axioms of a ring/field) but a field where 0 = 1 must be the zero ring, which has shit properties and is completely irrelevant (it has no modules, no arithmetic, nothing going on).
Hence, it is assumed in the definition of a ring that if there is a 1, then it must be different from 0, since the other case does not yield anything interesting and leads to all kinds of shitty case distinctions that only regard the zero ring.

>>10257016
I'm doing a PhD in math and I have studied both in a grande école and at university (for my M2). Okay now that I think about it there *was* that one russian professor that no one could get rid of. It is still a mystery how he got there. I suspect he might have been good and burnt out or realized that it was not his thing (happened to some French profs as well).
Other than that, all the profs I have had (french or foreign) were brilliant, and most were good teachers.

>>10257025
Unital rings don't form an abelian category anyway, so there's no use for a zero object.

>>10256411
>What are some good books to start learning math?
see >>10231160

>>10257040
So you studied either at Polytechnique or at ENS Ulm. Obviously they won't put the shitty teachers there since you guys are the reason why we get cucked. You kike

>>10256974
I have no idea about internal Brazilian politics, let alone education, but based on what I have seen Bolsonaro say and advocate I am confident calling him a fascist. I hope he's not for the sake of all the brazilanons.

>>10256941
multiplication by j ?

>>10257134
You could have studied there too if you worked for it.

>>10257157
I wasn't made aware of them other than them being for engineering shit which didn't interest me in the slightest. Good larp though, nigger

>>10257162
>Good larp though, nigger
Nice cope

>>10257143
>fascist
>implying that's a bad thing
>onions-drinking intensifies

>>10257166
As much as I hate you I do thank you for being open about your ideology instead of hiding behind seven levels of irony and memewords.

>>10257172
>As much as I hate you I do thank you for being open about your ideology instead of hiding behind seven levels of irony and memewords.
fascism is the only logical way forward out of the regressive leftist ideology that has swamped most western nations

>>10257172
So are you a communist or a neoliberal pawn? We may have more in common than you may think.

Also, tell me what your mathematical specialization is because in curious now.

>>10257143
Whatever dude, this wasn't even my main point.

>>10257143
>Bolsonaro
>a fascist
I fucking wish.
>be Bolsonaro
>have two divorces and three marriages while claiming to be catholic
>reach the absolutely embarassing position of captain in the army before dropping out for politics
>initially pretend to be an extremist for popular support
>recently pretend to be a moderate to reach presidency
>encourage family members to also get into politics
>say some disgustingly stupid shit

I'm out of paper

GOD FUCKING DAMN IT

Fuck everything.
>alienated by my entire department
>had a huge fight with my colleagues
>had a fight with my family
inb4 shit9k
I don't give a shit. I'm fucking drunk call it a hissyfit or whatever.
fuck humanity
fuck ideals
fuck /sci/
fuck /mg/
and fuck (YOU)

>>10257318
>>>/r9k/

>>10257282
Carve the proof of your theorem in your skin and write your suicide note in blood over your other papers before you bleed out. You'll instantly become more famous than 99% of mathematicians for being the absolute madman who killed himself in the name of mathematical progress.

>>10257282
type ?

>>10257318
>>alienated by my entire department
>>had a huge fight with my colleagues
How are you so autistic as to feud with your /math/comrades

>>10257219
Fascists are hypocrites pretend-traditionalists You basically verified that he is in fact a fascist. Hitler and Mussolini where exact like that.

>>10257583
>Hitler and Mussolini
[math] \mathcal{C~O~R~N~E ~L~I~U~~C~O~D~R~E~A~N~U} [/math]

>>10256534
Thank you, do you think the paper version is better than the digital? I found the digital already, but the paper version is only $4. The online version sucks because its a scanned pdf.
>>10257124
I asked my math friend about this and he told me its too much pure math, which is less of a practical approach. Can you recommend me more for probability or applied mathematics?

>>10257636
I like the paperback because it just feels nice to hold and read out of. I used that book as a refresher after high school to prepare for college, and I wrote all over it. Placed into calculus. Currently finishing my BS in Applied Math. Im somewhat old school. Unless it's one of those giant fucking tomes that are terrible to carry around.

>>10257636
But if you mean "better" in terms of the print quality, I wouldnt say so. The printing was kinda blotchy, which is probably why you can find these books so cheap. But the content is great, especially for noobies trying to get up to speed fast.

Is there a mathematical way to be happy? Assume graduate knowledge in algebraic topology and geometry

>>10257843
>algebraic topology
You're already fucked mate

>>10254738
a vector space is a duo of a field and a set of elements (vectors)

happy new year mg

>>10258156
Happy New year, buddy.
New thread when?