/sci/ - Science & Math

Previous thread: >>9967629

Is there any intuition for what the Jacobi identity means or is it just some weird state in between associativity and nonassociativity?

I hate this general.

>>9982736
the original motivation is probably the fact that the commutator satisfies it

>>9982724
my calc 2 and, later, analysis 1&2 prof was the personiication of that proof

even in calc 2 (which is 80%integration methods), he refused to do examples in class and didn't answer any specific or technical questions. it had to be a theoretical question to get an answer. otherwise he would say there was no time in class to explain/do examples and refer to the textbook. also he spent the first 2 class meetings going over corrections he had made to our textbook (rogawksi). and he took off points if you weren't forming "complete mathematical sentences"...to fucking calc2 students who had never had a theory class before.if you had a single arrow instead of a double arrow you got 3 points off. if you misused an equal sign (again, this was calc2 not a theory course), 5 points off.

How many of you use TeX for assignments? My handwriting is shit and have been looking into using texmaker for future work

>>9982783
>How many of you use TeX for assignments?
Takes too much time.

>>9982783
Just work on your handwriting, you're going to be writing anyway while you work on problems because no one is autistic enough to work in tex.

>>9982783
Did it once for the lols in vector calculus but haven't since. Only worth doing if you can TeX as fast as you can type.

>>9982783
it's not for assignments, it's for reports, websites, publications and anything else that needs to be easily formatted and/or updated later.

you can live without it but it is fun to know

What is your favorite math research area and what area you guys currently studying?
Has anyone here come up with a complex proof on their own?

Does this topic deserve its own thread?

redpill me on a theoretical differential equations book. we dont have one for my class and instructor spends half the class on example s, half the class on hmwk

>>9982842
>redpill me on
kys

Which one of you did this?

>>9982842
Learn functional analysis first

I really like this book on inequalities called "The Cauchy-Schwarz Master Class" and recommend it very much:
http://www.ma.huji.ac.il/~ehudf/courses/Ineq09/The%20Cauchy-Schwarz%20Master%20Class%20.pdf

Also gimme more of these weird cyclic inequalities or whatever they're called, like in pic related.

Give me a hard induction problem.

>>9982902
Let [math]f:\mathbb N\to\mathbb N[/math] be a function defined as [math]f(n)=n/2[/math] if [math]n[/math] is even, and [math]f(n)=3n+1[/math] if it is odd. Let [math]f^k(n)=f\circ f\circ ...\circ f(n)[/math] [math]k[/math] times, that is, composition by itself [math]k[/math] times.

Show that for all [math]n\geq 1[/math], there exists some [math]k\in\mathbb N[/math] such that [math]f^k(n)=1[/math].

>>9982912
Can we instate a rule that bans texlets?

>>9982912
If you can't solve this please leave this general and consider studying something more suitable for your level of intelligence

Which is the biggest known prime such that we also know which is the previous one?

>>9982928
I imagine it's the largest known twin prime, the really fucking big ones are all sporadic.

>>9982912
I said hard faggot

>>9982917
if you think im gonna spend over 5 minutes looking up the tex codes just to meme a guy then you're uncool

>>9982964
>pretending to be other people on the Internet
Yikes!

How much of /mg/ is also /trek/

>For now, we ask the reader to take the functions existence on faith

The other day someone at my uni told me that you could define a vector space over the set of n-gons and even define an inner product, do any of you know where I could read more about this?

could it be possible to find a pathology that undermines current established theories in say analysis that would trigger a complete re-haul of particular ideas?

>>9982736
It is an identity that is satisfied by the commutator [a, •] in an associative algebra, as >>9982766 said, as well as the Lie algebra of a Lie group.
It has then been generalized to the current notion of Lie algebra we know today.
Note that it is a fairly reasonable definition, as it precisely states that the brackets [a, •] are derivations with respect to [•, •] (not that it is a particularly natural property to consider, but it is a natural interpretation)

>>9983095
You would have to redefine logic

Good luck

I was having some trouble with this problem, I don't really know how to start. How are you supposed to figure out what values of a make the series converge? I know how to prove that the sum of two vectors in the first set is finite but not how to show it's in the set

I know one of you fuckers goes to ANU and has had Jim as a lecturer. Reveal yourself.

>>9982783
I use TeX for everything. My handwriting is awful though.
I figure everything out on paper then type it up. It's a pain though, definitely adds on a lot of time to my work.

>>9983211
>I figure everything out on paper then type it up.
get some fuckin alphabet tracers and practice. it'll save you hours...

>>9983244
Basically everyone here uses LaTeX. That's actually a really good idea though.

>>9983255
i use notepad++ only faggots use latex

>>9982902
Let n be a positive integer. Consider
S={(x,y,z):x,y,z in {0,1,...,n}, x+y+z>0}
Determine the smallest possible number of planes, the union contains S but does not include (0,0,0)

>>9982902
Let n>=2 is an integer. Denote P_n as the product of all distinct primes less than n

prove P_n<=4^n

>>9983265
>faggots
Why the homophobia?

>>9983211
>>9983244
>adds time
how does it add any time to your work at all unless you're technologically inept
just fucking practice it and after a few problem sets it's faster than writing because you can copy/paste and add stuff wherever you want without erasing a bunch of shit to fit in that clarification you want to make
problem sets become much higher quality too, it's basically an essential if they're graded for correctness

>>9983278
This is a joke, right? It's a tautology.

check em

>>9983255
i use latex too. but not for assignments. i use it for projects, reports and publications. not brief homework that will get quickly glanced over then trashed.

>>9983335
>>9983333

2 off

>>9983332
>It's a tautology.
How so?

>>9983190
The sum of two elements in H2 looks like (a1+b1,a2+b2...). Look at what happens when you expand the square of one of these.

>>9983317
are you offended?

can someone explain the intuition behind a guage integral?

>>9983358
based and redpilled.

>>9983376
>if you only knew how bad things really are

>>9983317
In the USA, men who have sex with men account for 67% of all new HIV infections in the country despite representing just 4% of the population.

>>9982724
>Edition: think.
(Why?)

>>9982835
Currently working in Transcendental Number Theory, have two preprints but nothing published yet.

https://arxiv.org/pdf/1809.01841.pdf
A vanishing criterion for Dirichlet series with periodic coefficients
>Tapas Chatterjee, M. Ram Murty, Siddhi Pathak
>(Submitted on 6 Sep 2018)
>We address the question of non-vanishing of L(1,f) where f is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions f for which L(1,f)=0. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.

>>9982724
This isn't real, right?

>>9983190
You gotta use the Schwarz Lone Starr. Specifically use Cauchy-Schwarz for the first problem. Then the triangle inequality for the next one. To solve that last part let a_n=1/n. I'm sure you can take it from here.

>>9983490
It's real, I think it's from one of Shelah's books.

>>9982783
I tried to TeX everything as an undergrad, but I was too slow back then. Ever since I started grad school I just TeX everything. I make enough mistakes that it's easier to just type thing so that I can edit later. I handed in a few handwritten assignments because my algebraic topology profs wanted me to draw pictures.

>>9982835
Categorification of metric spaces.

>>9982996
Only you, it seems.

>>9983376
WHERE IS THE PROOFS

>>9983376
Man I feel bad for him now. I mean he's a fucking hack but still.

I was eating cereal this morning and I noticed that the balls that float to the surface of the milk clump up following a hexagonal pattern. I even stirred the bowl a bit to break them up but after a while they clumped up hexagonally again. They have to reach a critical mass/area on the surface of the milk to clump though.

Could it be that milk and cereal are the key to space packing problems? How cool is that?

I'm trying to acquaint myself with discrete mathematics after about a decade of no higher mathematics whatsoever. I feel like my brain is melting.

Is |p(X)| always |X| to the power of 2?

>>9983941
...Or 2 to the power of the cardinality of X, I should say.

>>9983941
Assuming you meant [math]2^{|X|}[/math] then yes, but it gets somewhat subtle once you start dealing with infinite cardinals. Eg, it's independent of ZFC what [math]2^{\aleph_0}[/math] (cardinality of the power set of the rationals) actually is.

>>9983941
In the finite case yes. This is a simple counting exercise. In the transfinite case, it holds only formally (that is, we denote it as such as an analogue to the finite case).

>>9983066
Huh? What field could this possibly be over?

>>9983959
>>9983960
Okay. I'm obviously going to have to retread this ground, but at least I can move on now with an incredibly tenuous sense of knowing something, maybe.

This is how I reasoned out a set theory exercise. Do I understand it?

Question: Is A a subset of p(A) for any A?

"No. A=(a,b), but the powerset contains {a,b}, which is a unique element. A contains a and b, and is therefore an element of p(A), but it does not contain {a,b} and is therefore not a subset."

>>9984051
I'm hazy on the difference between elements and subsets. In particular, I don't know how a and b can still be identified as elements if {a,b} is supposed to be non-equivalent to (a,b).

>>9984051
Your counter-example is ok, but you could have simply used a singleton. If A={x}, then P(A)={{x}, o}, and neither element is x.

>>9984062
That o was supposed to be the (/) letter. Anyway, o denotes the empty set there.

>>9984051
>>9984054
What do you mean by A=(a,b)?

>>9984062
Alright then. I'm saying the right things.

>>9984066
It's just the example from the tutorial I'm working through.

>>9984066
Wait, shit. I probably shouldn't have been using parentheses.

http://iopscience.iop.org/article/10.1088/1742-5468/aad6be/meta
>Uncovering multiscale order in the prime numbers via scattering
>S Torquato1,4, G Zhang2 and M de Courcy-Ireland3

Abstract
The prime numbers have been a source of fascination for millennia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor , proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor of the primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. However, the primes show an erratic pattern of occupied and unoccupied sites, very different from the predictable patterns of standard limit-periodic systems. We also identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.

>>9983190
Notice that triangle inequality and Cauchy Schwartz only works for finite quantities, and hence can only take a limit at the end

Tierney's book on categorical constructions in stable homotopy theory is quite pleasant to read.

>>9983190

So can we move /sq/ back to /sqt/?

>>9983211
>anu
what a terrible university

Is the correspondence theorem for ideals (that is, there is a one-to-one correspondence between [ideals of a ring R containing the ideal J] and [ideals of R/J] ) also true for prime and maximal ideals in a (commutative ring), with no other requirements?

As in: for a c.ring R, there is a 1:1 correspondence between prime ideals containing J and prime ideals of R/J, and ditto for maximals?

If it is true, I find it very weird that my textbook doesn't mention it even as an exercise when it has an entire chapter devoted to algebraic geometry.

>>9984319
Post yours. Put up or shut up.

>>9984328
Try a proof. It's not too hard.

>>9984494
yeah i know, i thought perhaps i was missing something

>>9984512
Post your proof and someone will giev feedback.
I'm too drunk to write something coherent

>>9984515
What I'm more interested in is:

Let [math]k[/math] an algebraically closed field, [math]V[/math] be an algebraic variety and [math]W\subset V[/math] a subvariety. The correspondence theorem says that every subvariety [math]W[/math] corresponds bijectively with a prime ideal of the coordinate ring [math]\Gamma(V):=k[X_1,...,X_n]/\mathcal I(V)[/math], where [math]\mathcal I(V)=\{f\in k[X_1,...,X_n]\mid \forall P\in V: f(P)=0\}[/math]. We also have that [math]W\subset V\implies \mathcal I(V)\subset \mathcal I(W)[/math], and as a consequence, there is an ideal [math]\mathcal I_V(W)\subset\Gamma(V)[/math] that corresponds to [math]W[/math] in the quotient.

My question is: is this ideal [math]\mathcal I_V(W)[/math] the one corresponding to [math]W [/math] in the correspondence theorem?

>>9983726
Kek what a dickhead.

anyone have any experience with this book? http://www.math.nagoya-u.ac.jp/~richard/teaching/s2017/Nelson_2015.pdf
is it any good?

Answer is... Jgdtfdghjkoncsffyjjhvxazvftyuuihdxcgh6852842684556

>>9982835
I'd like to go into Dynamical Systems and Manifold + their relations to physics but I lack some basics

Currently doing some undergrad general algebra (Polynomial theory mostly) and Complex Analyis, I'll move on to baby real Analysis + Rudin after.

>>9982842
Vladimir Arnold is pretty good, if a bit too "physics"-based and light on the proofs (like most of Arnold's books.)

>>9984705
From the looks of it, it's a good introduction before moving something like Papa Rudin (Real and Complex Analysis).
Rudin is by far the best book on Lesbegue integration, but it's best read with context and other books to help the study. The fact that he always skips a few steps in his proofs makes him a harsh read.

Is the prove of the well ordering principle just strong induction on the negation of the statement?

What is the status on the Scholze - Mochizuki fight?

>>9984927
Mochizuki tried to an hero over it so Scholze felt bad and backed down. Stix is trying to bully Scholze into release it anyway though.

>>9984958
Source on those claims?

>>9984927
Stix thinks there's an unfixable problem

>>9985046
I made them up anon jesus learn to recognize shitposting

>>9985051
Source?

>>9985105
Nothing official

>>9985055
Mochizuki is true samurai...
Wouldn't surprise me.

>>9984958

>>9984248
I don't get how this shows that addition is closed

>>9982783

any good university doesn't allow handwritten shit anymore

>>9985333
>his uni tells professors how to accept assignments from their classes
let me guess, you're british?

>>9985303
The sums must be finite, and that shows they are at most something finite (and at least 0).

>>9985303
The condition for any element to be in the set H_2 is that it is a sequence and that the sum of its elements squared converges.

So I took two sequences, each whose sum of its elements squared converges, added them term by term, then considered if the sum of its elements squared converges.

How does this proof make any sense? How does he get that since [math]z[/math] is not a unit, then [math]z=z_1t[/math]?

I would have used the fact that R being Noetherian is equivalent to "any family of ideals has a maximal element", and used the "family of proper ideals" as the family in question and they hence are all contained in the unique maximal ideal since R is local. Since it is principal, then it must divide any non-unit element..

>>9985442
forgot image

>>9985442
Hell, I don't even need the Noetherian property for that, I can just use (assuming C) that every ideal is contained in a maximal one and use local condition, but I guess I need it to show that it divides it only a finite amount of times.

>>9983920
That sounds intriguing

>>9985442
>>9985447
The elements not in the maximal ideals are exactly the units (since any non-unit generates an ideal which is not R and thus is contained in the maximal ideal, and the maximal ideal obviously doesn't contain any units).
So if z is not a unit then it is in the maximal ideal and thus can be written in that form.

>>9985646
>maximal ideals
Should be "maximal ideal" since obviously there is only one

>>9985646
hmm true, i saw the proof but the book only applied it to a specific local ring, thought it was just a special property of that example.

>>9985651
That annoying when they do that

>>9982724
t

-4/3 = Minimum Overlap

負四三分之二=最小的重疊

Negatív négy-harmada = minimális átfedéssel

Are f(x) = x and f(x) = 0 the only linear transformations from R to R? If so, how can I prove it?

>>9986051
>Are f(x) = x and f(x) = 0 the only linear transformations from R to R?
No, for any linear transformation T and scalar c, cT is a linear transformation (the set of linear transformations is a vector space).

>>9986056
Yeah I forgot those. My question still holds. How can I prove that those are the only family of linear transformations R->R?

>>9986072
>How can I prove that those are the only family of linear transformations R->R?
What have you tried?

>>9986090
It's easy to check on the integers, but I don't know what to do with the rational or irrational case.

>>9986114
>It's easy to check on the integers
what did he/she mean by this?

>>9986072
what is a basis for the vector space R?
what does an arbitrary linear transformation do to that basis?
what does that transformation do to the whole space?

>>9983376
I can't find any explanation for why he needs this on his webpage. Are people trying to murder him for his paper being crank or because he's Satoshi Nakamoto or what?

>>9984922
Yes, if you assume induction you can prove WOP and vice versa

>>9986164
Is it true that he is Satoshi? I've heard something similar but I can't recall where it was

http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/IUfaq_en2.pdf

>>9986464
I shitpost a lot about IUT but I still don't understand what it even is desu

>>9986368
A Jewtube video called "I know who Satoshi Nakamoto is" or something similar, by an old guy.

>>9986484
I have only recently started to read through it. I am moving at a snail's pace (1-2 pages a day).

>>9986558
>I am moving at a snail's pace
Mostly because Mochizuki references stuff I am not familiar with, so I have to backtrack and read other papers before I can continue.

Can someone here give me a good book on Synthetic Geometry ?
Another on projective geometry would be neat.

>>9984113
wew

I don't know why I keep reading stuff related to IUT. I'm not a mathematician, I'm a mere undergrad in economics. I don't even know what or who Teichmuller is. But there's something about this general's excitement that keeps me going...

>>9987295
You're allowed to be passionate about it, anon
I'd say that you want to build yourself a little roadmap before you can try to "attack" the IUT papers themselves, although they are not known for their clarity.

I'm a memegineer in Materials Science and I want to study Dynamical Systems & differential Geometry, this is what I did, I built myself a roadmap which I try to follow

For IUT, assuming you're already familiar with Calc & you know how to read a proof, I'd say you need a very solid background in abstract algebra & linear algebra first, then basic real/complex analysis and topology, as they are more or less required to understand any modern field of math.
After that, you'll want a solid dose of Number Theory and related topics.
You can take a little break and read about Synthetic Geometry, then move on to Analytical Geometry.

Serious things start when you'll read about Algebraic Geometry which requires probably a lot more than what I listed here

Frankly, what you need is to learn about the roadmap. Then you need to get yourself motivated.
I can study up to 2am reading books to achieve this goal because I just love the feeling of progressing.

>>9987372
You don't know what you're talking about my man.
Please stop.

>>9987387
Fuck you I'll do what I want

>>9986118
It's easy to check that f(x) = cx is the only linear transformation from Z to Z.

>>9987474
What's different about the non-integer case?

>>9986866
Berger Geometry 1 and 2 for projective geometry.

>>9982794
i do :)

>>9987372
Thanks fren. Your assumption is right, now I need to work on all of that you just mentioned.

Boys, I'm getting into ML, but i'd like to get into the Maths behind it. I know it's mostly linear algebra and multi variable calculus. What books do you guys recommends ?

>>9986866
Coxeter: Geometry revisited

>>9987295
it's not worth it

>>9986051
do you mean R-linear (ie. f(x+ty) = f(x) + tf(y) for all real t,x,y) or additive (f(x+y) = f(x)+f(y) for all real x,y) ?

>>9988249
https://mml-book.com

>>9982835
>What is your favorite math research area
Classical algebraic geometry (invariant theory, intersection theory, complex geometry)

>what area you guys currently studying?
Algebraic groups and rigid analytic geometry

>Has anyone here come up with a complex proof on their own?
Nope, just starting out

>>9987295
economics is rough. id switch to math honestly

>>9984492
unsw

>>9983211
If you have some disposable income you could try mathpix, basically converts handwriting to latex. It's not bad, I've tried it myself, unfortunately you gotta pay for it which I'm not really willing to do. I'm not that lazy. But if takes you a lot of time then maybe it would be worth it. I just checked the app and the first 50 scans are free, maybe try it out. Or not.
>>9982783
I don't use TeX for assignments but I do use it for notes. I use overleaf/sharelatex mostly.
>>9982794
>>9983211
Working on your handwriting is important, but something I tend to do to make my life easier is use a whiteboard first. I bought myself one and it's been a big help since I can first complete the problem on it fairly easily and then take my time writing it down neatly instead of having to constantly erase on paper whenever I make a mistake or need to rewrite something.

>decided to take Analysis 1 part time while working
>think because I was hot shit at business school math I should breeze through this

unironically one of the hardest things I've done. Being able to coast through business school is absolutely true. I couldnt imagine doing 4 years of math straight. still an incredibly enjoyable experience

>>9988329
What do you mean? The coursework or the competition?

>>9988550
Undergrad analysis turns boys into men

>>9988836
The coursework. I switched to my major to math because economics math was too hard for me

>>9988280
Based Cheng
>someone in a blue shirt?
>just stare at it

>>9988495
I don't use non-free software :^)
Whiteboards are pretty based. I do a lot of work in uni classrooms with other people on whiteboards.

>>9988550
>>9988843
My school's analysis course is a piss-take. The real analysis course before it is way harder and the lecturer spends all his time pointing out that nothing you do is specific to real numbers anyway.

>>9988884
>>9988888
>Whiteboards are pretty based. I do a lot of work in uni classrooms with other people on whiteboards.
Same, my ability and speed for doing work has increased a lot thanks to them, they really do make life way easier.

>>9988863
What exactly was hard in your case? Econometrics? Quantitative Methods? I'm legit curious.

>>9988998
he's trying to bait people you fucking retard, look at the name of the thread

>>9986051
Let T:R->R be a linear transformation. Then T(x)=T(1x)=xT(1) for all x in R since it is linear. Denote T(1)=c. Then T(x)=cx.

>>9988843
That’s a funny way to spell Number Theory

>>9984113
no one's going to comment on this?

So.. is this paper still censored?

https://quillette.com/2018/09/07/academic-activists-send-a-published-paper-down-the-memory-hole/

https://arxiv.org/pdf/1703.04184.pdf

>>9989649

Its a very interesting result.. Here is the Priceton source with link to the paper.

https://www.princeton.edu/news/2018/09/05/surprising-hidden-order-unites-prime-numbers-and-crystal-materials

>>9989649
Plato was right about everything AGAIN. Randomness is an illusion.

>>9989649
You drive people away with the anime pic.

>>9989728
There is literally nothing wrong with anime.

>>9989493
Yeah, I kind of knew that. That's why pushed him more, just to be certain.

>>9989728
>You drive people away with the anime pic.
hello r*ddit!

>>9984113
>Reddited by 5
>no 4chan option
FUCKING WHY

>>9989748
I didn't say there was. I said you drove people away with anime.

>>9989863
How am I a redditor for pointing that out? Have you not noticed how these threads and the posters are essentially Reddit tier, and most of them are even afraid of anime. I don't care personally.

why do algebraic geometers get all the girls

>>9982794
I had to TeX a lot for grad school and it eventually became easier to TeX than write or use word. I have a writing disorder and my handwriting is atrocious. Most profs email in pseudotex anyway so if youve been chatting its easy to copy paste and fix it up quickly.

>>9988283
i've been indulging in a hobby of arithmetic geometry for a bit, but desu its most beautiful when the results have analogies to classical results. I would have to agree :)

>>9990422
>writing disorder
nice meme

>>9990380
Cohomology.

>>9990047
>anime
LOL

>>9982766
That is not a meaning

>>9982783
You should definitely try it, it's easier to change mistakes than writing in pen and not as annoying as writing in pencil.

>>9983015
>for proof, see Appendix A

>>9990380
you need to worship the matarajin

>>9982783
Never, it's a pain in the ass. Only use it when it's absolutely necessary (ie. writing a thesis or something substantial).
I used to have shitty handwriting in HS but after a few years, it has turned very pretty imo

>>9982724
Help.

>>9990595
Help what? This is so incoherent I don't even know what you're trying to do.

>>9983358
Nice quote

>>9984006
Think about it this way: you get two options for each element of the set: either put in the subset or don't. So you get 2^(number of elements).

>>9983095
Yes. The field with one element is such a thing. (It's not a pathology but it does require a re-haul.)

Another is the sporadic simple groups.

>>9986866
Hartshorne's "Geometry: Euclid and Beyond"

https://arxiv.org/pdf/1809.02518.pdf
>The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures
>Terence Tao, Joni Teräväinen
>(Submitted on 7 Sep 2018)
>We study the asymptotic behaviour of higher order correlations
>E_{n≤X/d} g_1(n+ah_1)⋯g_k(n+ah_k)
>as a function of the parameters a and d, where g_1,…,g_k are bounded multiplicative functions, h_1,…,h_k are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g_1⋯g_k does not (weakly) pretend to be a twisted Dirichlet character n↦χ(n)n^{it}, and behave asymptotically like a multiple of d^{−it}χ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t=0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture E_{n≤X}λ(n+h_1)⋯λ(n+h_k)=o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.

https://arxiv.org/pdf/1809.02278.pdf
>An E-sequence approach to the 3x+1 problem
>San Min Wang
>(Submitted on 7 Sep 2018)
>For any odd positive integer x, define (x_n)_{n⩾0} and (a_n)_{n⩾1} by setting x_0=x, x_n=(3x_{n−1}+1)/(2^{a_n}) such that all x_n are odd. The 3x+1 problem asserts that there is x_n=1 for all x. In this paper, we focus on (a_n)_{n⩾1} and call it the E-sequence of x. We prove a remarkable fact that the divergence of all non-periodic generalized E-sequences implies the periodicity of (x_n)_{n⩾1} for all x. Moreover, we prove that several classes of non-periodic E-sequences are divergent. These results present a possible way to prove the 3x+1 problem which we call the E-sequence approach.

>>9990595
You can't cancel the factor of (x+h) from the numerator and denominator like that

>tfw too dumb to understand advanced mathematics

Atiyah's book on K-theory is quite pleasant to read.

>>9992048
Atiyah's books are pleasant to read in general.

>>9990977
>field with one element
isn't this contradictory by definition?

>>9991890
of course you are with that attitude anon.

>>9982934
Can someone explain why this is. I thought you had to count from the last one to get to the next.

>>9990595
[math]\frac{6(x+h)}{(x+h)^{2} + 7} \neq \frac{6}{x+h+7}[/math]
try expanding the denominator instead. Also, this is not the place for such questions, try >>9992198 instead. Come back when you finish high school!

>>9991890
Too dumb to understand you don't need to be smart for advanced shit.

>>9992209
https://ncatlab.org/nlab/show/field+with+one+element

>>9992209
It's the trivial field with the additive and multiplicative identity being the same. Let's call it 0.
If 0*0 = 0, then this identity is its own symmetric element according to both laws.

>>9982724
How do I calculate yellow are with definite integral? It's in polar coordinates. Will half of the area be integral of upper function minus integral of bottom function?

Hi, Anon! I've just entered the university for a master's degree in CS, and planning to devote my master's thesis to robotics. Swarm of robots in particular. The problem is, in the robotics lab here no one has worked narrowly with swarms of robots before.

Do you know any math/CS problems within the field that remain unsolved and can be possibly done by undergraduate? Maybe you know and can suggest some good readings, recent papers and lectures on swarm robotics worth to read. Every advice will be appreciated.

>>9992405
First, you only need to calculate a quarter of it due to elementary symmetry considerations.
And then, you can try to get an explicit form for the curves in regions 1 and 2.

That's how I would approach it anyway.

>>9992430
http://4chan-science.wikia.com/wiki//sci/_Wiki
You sound like a huge cunt. Make your own damn general.

how do you understand "implications" in logic intuitively? I just can't wrap my head around the fact that "false implies anything" result.
I would have thought false implies nothing because I could say, if pigs fly, then this world is of magic. which by this framework is actually a true statement i.e. this world is of magic.

>>9992505
That's why regular languange isn't considered a formal language.

>>9992505
google EFQ
Prototypical example:

Assume a false statement: All horses are brown.

Then:
>All horses are brown
is true, hence
>all horses are brown or unicorns exist
is true, but all horses are brown is false since there are black horses so
>unicorns exist
is true.

>>9992539
That's the principle of explosion that comes from contradictions in an axiomatic system. What the material implication refers to is a vacuous truth.

>>9992505
Part of it comes from the fact that we've simply defined it that way. It makes our lives easier, even though it seems a bit odd when translated to everyday language. When proving stuff we normally only concern ourselves with cases where the precedent is actually true, so having false imply truth means that we can simply show that P being true implies Q being true, and then say that the statement P -> Q to be true in general.

Although, if pigs did fly, then the world would surely be magic. So that example is fine.

>>9992309
>Borger’s absolute geometry
Hey I have this guy as a lecturer.
Based Jim

>>9991890
you don't actually need to be smart to understand math, that's a meme. most of us here are sub 100 iq

>>9992657
Fuck don't tell them

>>9992505
>I could say, if pigs fly, then this world is of magic
If this statement were false, then there would be some counterexample. But (hopefully) this is pretty clearly an absurd idea; finding a counterexample to this statement would involve finding a flying pig. Since there are no flying pigs, a counterexample cannot exist.

>>9992505
Venn diagram-wise, a statement that is true in less places will imply one that is true in more places. So the "less true it gets" the more it will imply, until you get the empty set which is contained in any set.

Or, looking at it intuitionistically, there is a function from the empty set to any set.

>>9992505
Keep in mind that implication =/= causality. Conditional statements that have false antecedents aren't true because the false statement would (hypothetically) cause the consequence, they are (vacuously) true because there is simply no way to proof otherwise. Even if the statement doesn't really give any information or make sense, it is considered true by virtue of not being able to be proven false.

https://online.math.uh.edu/HoustonACT/videocalculus/index.html

>>9983317
>phobia
lol

>>9983317
>Why the homophobia?
Why the phobiaphobia?

>And although we cannot explicitly construct one, we can use Zorn’s Lemma to prove their existence

does there exist a "fundamental" proof of the fundamental theorem of algebra or is it just one big meme?

>>9984576
>tfw Shelah has like 10,000 articles and no one will remember any of his results after he's dead because of his terrible exposition and also because he only proved theorems about set theory shit that like 3 other people care about

>>9993142
Fah
G

>>9993251
nonconstructivism was a mistake

should i study spivak or apostle for the calculus? im looking for a book that explains motivations and provides analysis of the calculus.

or are both memes and i should just use stewart?

>>9993399
>>9993251
Everyone says this until they prove a theorem using the axiom of choice and suddenly they are AC's #1 fan.

>>9993543
https://en.wikipedia.org/wiki/L._E._J._Brouwer

>>9993526
I have only read Stewart and Apostol so I'll give you my perspectives on just those two.

Apostol: This book should be regarded as [math] \textbb{the} [/math] calculus textbook. A pure classic. However, I'd say that it is starting to age. Many sections could be removed and it would still cover the topics in the modern calculus curriculum. This is why, if you are a university student, I'd recommend using a more modern textbook.

Stewart: This book is also great but for other reasons. I would call this book a complete turn your brain off read. You see, I read this book when I needed to review the main theorems of multivariable calculus for a higher course. For that purpose, Apostol and textbooks in analysis were simply too wordy for that purpose, and because of their rigorous nature I had to really turn my brain on to read through tedious proofs that I had already read years ago. For this purpose, I turned to Stewart to just quickly go through all of multivariable calculus without needing to really think. Every section can be understood in the first reading, not all of the exercise are bullshit, and if you do want proofs he has those at the end.

TL;DR:
Apostol: The most rigorous calculus textbook that exists
Stewart: The least challenging calculus textbook that exists

>>9993526
Apostol, fuck spivak.

>>9982724

undergrads (read: everyone here), what are you minoring in? Not sure whether to do CS or Philosophy. Assuming I already have a active github, etc, will the CS minor help any?

Mostly, I want to learn to think well. I’m open to anything.

>>9993526
Ive read half of both Apostol and Slovak and all of Stewart and would definitely go with Spivak. It’s simply enthralling

Is there such a thing as fractional arithmetic operators? Like an operator that's half way between an addition and a multiplication?

>>9993750
I'm not minoring in anything because it's a administrative ball-ache at my uni. My personal opinion would be to do philosophy if that's something you're interested in, I have no formal CS education and haven't had an issue getting software dev jobs.

>>9993974
My intuition is to say no fucking way but fractional derivatives exist, so maybe? I don't know what property it would capture though, generally when abstract arithmetic ops are considered it's in terms of repeating the successor function over and over and over again. I don't know what a fractional version of that would be.

>>9982724
how to pick up calc quickly? willing to work but not very hard

>>9993750
A minor is basically irrelevant for practical purposes. The knowledge provided by it is extremely shallow, and everybody interested in hiring you is aware of this. It's not worth giving yourself any extra workload for the purpose of acquiring one.
That said, if you have any interest in studying CS you will probably end up with a minor in it anyway. They're really easy to get.
I graduated with two minors in CS and economics without any planning to get either of them.

>>9993974
Gotcha back boyo: https://www.hindawi.com/journals/mpe/2016/4356371/

What does /mg/ think of Bourbaki?

>>9993974
Lookup "fractional hyperoperation"

Also, apparently there are complex (?) hyperoperations

Mathematicians are monsters

>>9994032
extend to finite fields, maybe even arbitrary rings?
>my boner can't handle this

>>9994026
>https://www.hindawi.com/journals/mpe/2016/4356371/
WTF?

>>9992293
>>9992221
>>9992657
I have enough self-awareness to realize that the height of my potential is to prove a couple middling theorems by recombinatorially mashing together known methods from the literature. Truly novel insights, like whatever Peter Scholze is up to, are beyond me. Complex pattern-matching will get you far, but not that far.

>>9994231
Also I have a job now so chances are I'll never get past the second-year graduate level even given an entire decade.

>>9994235
Also I want the kind of panoramic understanding where I can look at qualifying-exam-level problems and instantly intuit what the right answer should be, but even that is probably beyond me.

My friend is stupid and posted some naughty stuff on a blueboard so he asked me to post his question.

"how do I intergrate"

Given a pair of points X and Xp, I need to find the set of all points P so that the pair appears under a constant angle when viewed from P. It can't be a circle, ellipse, parabola or hyperbole. Can someone give me pointers as to what's going on here?

>>9992505
>by this framework is actually a true statement i.e. this world is of magic.
That's not what we can deduce from the proposition "if pigs fly, then this world is of magic". You are correct that, since pigs do not fly, the proposition is vacuously true. But all the proposition says is that "*if* pigs fly, *then* this world is of magic". The key meaning (and the source of the truth value of the proposition) here is that pigs' ability to fly implies the world being of magic. This is very different to the proposition "this world is of magic" which is a basic proposition, whose key meaning and source of truth value is found by relating the proposition to our actual world. It is true if it is in accordance with the actual world, false otherwise. Therefore it is false that this world is of magic.

>>9994245
Replace all the dumber numbers with arbitrary constants and after that, integrate by parts

x^0.21 = exp(x*ln(0.21))

>>9994262
My naughty friend sends his highest of regards, as do I.

>>9993750
what's your major?

>>9994017
>A minor is basically irrelevant for practical purposes.
no it's not you nutsack

i majored in economics and minored in history-public policy track. it taught me very important information on how to craft economic policy

>>9994334
Why don't you go to /r/neoliberal then you fucking nerd

>>9982172
>https://www.seas.upenn.edu/~cis515/linalg.pdf
>"Fundamentals of Linear Algebra and Optimization"
>Was expecting LA, KKT and some LP and NLP applications
>Got a fucking 1000 page tome on vector spaces, topology, functional analysis, spectral method, finite elements(???) and a SVM chapter thrown in for extra randomness.
>No preface or anything to justify it.

I mean, I understand the desire to have self-sufficient textbook all, but isn't this taking it way to far? It is far better to learn those topics in dedicated textbooks than to dump several chapters of basics in an applied textbook.

If I have learned one thing while doing math it's that we're all intgumented eidolons seeking the source of all things to which to return to. I [math] am [/math] the mathematics.

>>9994083
Mathematics was a mistake

>>9994384
There’s nothing with having extra stuff in your book, as long as it’s related and well written. Bitch, the author worked hard producing that monster of a book. Be grateful.

>>9993717
I'm thinking he's back

>>9982724
>studying applied math and physics
>Uni Class in physics starts
>Prof says math is more humanities than actual science

>>9994577
I fucking wish

>>9994231
A. Scholze would probably tell you that an essential part of his work was recombinatorially mashing together known methods from literature in a way that was intelligible to him.

B. Scholze is not your average mathematician. He's not even your average Fields medalist. You cannot give up just because there are people that are more gifted than you are. Just do you.

>>9993369
yes, with galois theory

>>9994030
the first book is terrible

>>9994250
i just started working it out but from classical results in geometry, the case where alpha is 90º is clearly a circle.

>>9994687
*mild correction* a circle with two antipodal points removed

>>9994461
No, look listen I respect the author tremendously. I know how difficult it is to write a book like that and in addition it appears to actually be pretty well written with a decent number of examples and figures.

However, this is not a precedent anyone should be setting. Of course it is related, but everything in math is related. Applied textbooks should not start including their foundational fields because then in a field like optimization you would try to include all of math in a single textbook (I mean that text is actually a large chunk of typical undergrad topics).

Besides the worst part about all this is that I will probably end up skipping those chapters entirely or even using another textbook since I don't have a lot of other textbooks I need to get to.

>>9992665
Kek

>>9994250
Consider [math]\alpha [/math] is between 90º and 180º. We can assume without loss of generality that the two fixed points are on the x-axis of the xy-plane, and we can fix them so that they sit on (f,0) and (-f,0) (f for "foci"). An arbitrary point in the curve is such that [math]\alpha=\arctan\left(\frac{|x+f|}{|y|}\right) +\arctan\left(\frac{|f-x|}{|y|}\right) [/math]. Using the identity [math]\arctan a+\arctan b=\arctan \frac{a+b}{1-ab}[/math], we get that [eqn]\tan\alpha=\frac{|y|\cdot(|x+f|+|x-f|)}{|y|^2-|f^2-x^2|}[/eqn]
The are constraints for the valid values of [math]x[/math], which are of course [math]-f<x<f[/math].
Notice that in the limit [math]\alpha\to\pi/2[/math], which is equivalent to the denominator tending to 0, we get a circle again.

>>9994815
forgot image.

The acute angle case is similar but there's an added complication

>>9993988
this is partly my reasoning, do philosophy to enjoy learning to think well, and just keep my github updated for employment prospects. but

>>9994017
I think I'll end up here, as I almost have the requirements fulfilled while having nothing in philosophy or anything else. What's your take on double majors? worthwhile, or just a scam?

>>9994329
math, duh

>>9994030
Good as a reference, ridiculously hard exercises, terrible textbook

What's a good book for someone 100% new to category theory?

>>9994815
>>9994250
and actually the complication is completely resolved by using the cosine rule instead. Let [math]\alpha[/math] be acute (perhaps works with non acute too but cba), and the set up as before. Then the cosine rule says:

[eqn]\cos\alpha=\frac{2y^2+|x-f|^2+|x+f|^2-4f^2}{2\sqrt{y^2+|x-f|^2}\sqrt{y^2+|x+f|^2}}[/eqn]

Pic related shows how the tan solution's complication at [math]|x|>f[/math] is resolved very neatly, I must say

>>9994983
fuck... and a simple check also shows that it also works for obtuse angles very neatly too... anyways, it was a good timewaste

>>9994976
If you have little to no prior experience with higher maths y recommend reading Conceptual Mathematics by Lawvere, otherwise you could use Categories for the Working Mathematician but it assumes you have a solid undergraduate knowledge in math

Friendly reminder that if you are not using at least 20 notebooks a year while studying math you are most likely just wasting your time

Is it just me or is the captcha getting worse?

>>9995090
If you don't know at least a little algebra category theory isn't going to seem very motivated desu

>>9995093
>he doesn't use loose sheets which slowly build up into a gigantic pile on his desk corner
You're not gonna make it

>>9995093
>needing to write to learn math

send halp pls

>>9995261

>>9995277
In these kind of problems where the tightness of the bound is unimportant it will suffice just to find a rough estimate for epsilon. You can usually get some intuition for what it should be by looking at the expression, ie, if you have 1/x epsilon will probably resemble 1/delta

>>9995135
I've read about half of Lawvere book and it shows how category theory is interesting on its own

>>9995277
In general, you want to work backwards so as to select the correct delta. Since in the question they want you to consider [math]x[/math]'s that are "close" to [math]1[/math], in that you are bounding above the term [math]|x-1|[/math], then you want to somehow get only terms with [math]|x-1|[/math] in your expression. It is easier shown:

Idea: we want to find some [math]\delta[/math] such that when [math]0<|x-1|<\delta[/math], then [math]\left|\frac1x-1\right|<\epsilon[/math]. We want to use the latter to find the former. Note that:
[eqn]\left|\frac1x-1\right|=\left|\frac{1-x}x\right|=\frac{|x-1|}{|x|}[/eqn]
We want this latter expression to be smaller than [math]\epsilon[/math]. However, it turns out it is helpful to have more "space". To create this space, I use the reverse triangle inequality ([math]|a+b|\geq ||a|-|b||[/math]) by doing the trick [math]x=(x-1)+1[/math]:
[eqn]\frac{|x-1|}{|x|}\leq\frac{|x-1|}{||x-1|-1|}=^!\frac{|x-1|}{1-|x-1|}[/eqn]
The latter equality only being true for small enough [math]|x-1|[/math], which is what we're considering anyways. So in summary, I'm saying that I want the following to be true, given some [math]\epsilon[/math]:[eqn]\left|\frac1x-1\right|\leq\frac{|x-1|}{1-|x-1|}<\epsilon[/eqn]
Notice how this last expression only has [math]|x-1|[/math] terms. Now "solve" for [math]|x-1|[/math]:
[eqn]\frac{|x-1|}{1-|x-1|}<\epsilon\iff \frac1{|x-1|}-1>\frac1{\epsilon}\iff |x-1|<\frac{\epsilon}{1+\epsilon}[/eqn]
So now by construction, you get a good enough delta by defining [math]\delta:=\frac{\epsilon}{1+\epsilon}[/math]. Check:

[eqn]\left|\frac1x-1\right|\leq\frac{|x-1|}{1-|x-1|}<\frac{\frac{\epsilon}{1+\epsilon}}{1-\frac{\epsilon}{1+\epsilon}}=\epsilon[/eqn]

QED

>>9995456
>doing the trick

So, what's the latest /math/ meme guide?

>>9995462
cmon you know these undergrad shits get assblasted every time they see it done

>>9995464

During my undergrad I focused mainly on algebraic stuff and next term is going to be my last one, should I take a course in Riemann Surfaces even tough I don't have a course in differential geometry under my belt? Is it going to make me look better rounded in my applications to grad school?

>>9994620
Yes, fair enough, fine, good points.

>>9995504
More graduate courses on your transcript is always a good thing if you can do well in them.
That said, I don't think "well-rounded" is exactly what admissions committees are trying to find. Solid foundational courses are important (topology, measure theory, group theory, etc.), but when you get into more specific topics courses like you are describing it all just kind of blurs together into general experience with graduate coursework. There is no reason to think a guy who has taken 12 random niche courses all over mathematics is going to be any better of an algebraist than a guy who just took a bunch of algebra courses.

Considering you're missing some reasonably important background I personally wouldn't take it unless you're specifically interested in them for whatever reason.

>>9992048
It's not pleasant anymore. I hate this book.

>>9994976
>What's a good book for someone 100% new to category theory?

>>9995464
>So, what's the latest /math/ meme guide?
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.

>>9995610
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

>>9995625
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).

>>9995628
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.

>>9995632
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).

>>9995635
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).

>>9995637
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.

>>9995638
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)

Hey guys, this is my first post on this board.
I have a Calculus question

I'm being asked to find the slope of the curve y=4x -x^2 at the point P (1,3) and the equation of the tangent line at P.

I know how to do this when you find the derivative it becomes 4-2x and then you plug in 1 and get the slope of 2. However, I'm being explicitly asked to do this the P and Q way (x+h, (x+h)^2)

So here is my work
4 • (1+h) - (1+h)^2 - 4•1-(1)^2 / h
Becomes
4h - h^2+2h / h

I then factor out an h
h(4-h+2) / h

The h's cancel and I'm left with h-6
Then you plug in 3 for h and get 3 when the answer should be 2

I can find the line with point slope it's 2x+1, but this p and q has me messed up. Did I miss a one somewhere? Or another way I found you could get 2 but I don't think it's correct is

Plug in 3 in this step
4(3)-(3)^2 + 2(3) / 3
12-9+6 / 3 = 2

Anybody care to help?

>>9995691
>the P and Q way
what?

>>9995699
It's retarded, pic related

>>9995691
On that first line the last part should be +(1)^2. You didn't carry the negative sign all the way through.

You should end up with 2-h as your final result, becoming 2 as h goes to zero.

Now apply your y=mx+b skills to find a line that has that slope and crosses the point (1,3).

>>9995704
There's a reason they're asking yyou to do things this way

>>9995704
[math]\frac{4(1+h)-(1+h)^{2} - 4(1)+(1)^{2}}{h} = \frac{2h-h^{2}}{h} = 2-h[/math].

>>9995717
I figured it was something dumb, I'm probably going to try a couple more of these now that I know what's going on. Thanks!

>>9995093
Bitch I prove everything in my head like blind Euler.

what does /mg/ think of pic related?

>>9996682
Ugly

>>9995588
lol, did you get to the proof of Bott periodicity or what?

>>9996828
That (sub)chapter, yes. I can't follow it at all.

>>9996829
Hatcher's is good too believe it or not

>>9996838
I'll give it a try.

>>9996854
Bott periodicity proof is going to be ugly still though, maybe a bit more modern

>>9996857
As long as I don't have to guess what the author has written. My version of Atiyah's book is so shitty a scan I can't always be sure if I can guess what each symbol is. I hate reading low quality scans of books with the typewriter font.

>>9996875

I'm currently a calculus tutor but will start tutoring linear algebra in about a week. Last time I took it was 3 years ago.

What's the fastest way to brush up on it?

If I have the homogeneous system of with n equations and n variables and I get the identity matrix what does that imply from the original system solution( before I made it homogeneous)