/sci/ - Science & Math

first for Constructive Zermelo Freankel with just Exponentiation CZFE

second for Serre spectral sequence

>>11893127
>Elliptic curves

>S-cobordism theorem

Is basic mathematics by serge lang all the math I need before calculus?

Bros...

>>11893127
Bro, you...
You forgot to link the old thread.

>>11893197
Unironically, yes.
Ironically, use KhanAcademy.

I thought you couldn't add points man this is confusing

>>11893224
Why couldn't you? You can add whatever you want, I might just start adding manifolds if I feel like it.

>>11893227
>manifold with non-constant dimension
Please don't do it.

Kinda weird how lots of mathematicians never do anything related to geometry.

>>11889375 Old thread
Also is there a difference between ]0,1[ and )0,1( in the reals? I'm referring to the reals minus the unit interval.

>>11893227
I dunno, I just got told you couldn't. Like it would get messy or not make sense. That you really are dealing with vectors, and that really you only substract points (in the same way you would vectors too, which is even more confusing).

>>11893251
>)0,1(

>>11893251
>)0, 1(
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
BOURBAKIIIIIIIIIIIIIII WHAT THE FUCK IS THIS

>>11893251
some people use ]0, 1[ as notation for the unit interval. Never have seen someone use )0, 1( in any context

What are some good books on combinatorics and combinatorial an analysis?

>>11893254
There is a one to one correspondance between vectors at the origin and points. Adding one or the other is the same thing.

>>11893275

>>11893257
>>11893258
>>11893266
That is actually a pretty clever notation for [math](- \infty, 0] \cup [1, \infty)[/math], if the anon meant that. Otherwise we should just accept him and his... special way of thinking. I want to believe.

What makes Ramsey theory so incalculable so fast?

>>11893321
Combinatorial explosion

>Erdos needed drugs to think about proofs
bros...

>>11893277
Okay
But why is a vector a line segment then if it's a single point

>>11893321
the question "is it true that R(a, b)<n" is a question about edge-colorings of full graphs with n vertices
there are [math]2^{n(n-1)/2}[/math] such colorings which is obviously a lot
so it's not very surprising that they're hard to compute

at this point, most improvements require both mathematical insight and computer assistance, usually the papers are something along the lines of "we came up with this specific family of graphs via some heurestic reasoning, and then ran a computer search"

i wonder if anyone tried running some kind of neural nets to get counterexamples

Was it based?

>>11893412
How much graph theory is required to understand Ramsey theory?

>>11893413
I think so. My HS maths/physics teacher wrote an article on something he came up by himself on Wikipedia just to check if they let anything pass. After a month of not being deleted, he contacted them somehow and confessed it was all a fabrication.

>>11893418
do you know what is a vertex, an edge, and a graph? then you're good to go

Since I became an assistant prof (finally), I started to regain my interest in women somehow.
That's probably how badly Phd and Postdoc affected me in the past

>>11893425
>do you know what is a vertex, an edge, and a graph?
I do.

>>11893434
What classes are you teaching, doctor?

I really want to fuck the anime trannies.

>>11893437
ML

What are the most interesting Euclidean domains?

>>11893413
Yes. Cheaters can suck my dick.
>>11893453
Q[x]

>>11893439
I would like to fuck the anime trannies too. Any woman or a non-anime tranny would make me happy too.
>>11893453
[math]Z[i][/math]

>>11893439
>you will never gangbang yukarifag with all your /mg/ bros
I wonder if he's finally gotten married.

>>11893460
>implying he didn't kill himself

>>11893459
[math]f(\alpha+\beta i)=\alpha^2 + \beta^2[/math]

>>11893413
>caring so much whether the little shithead calc students you're paid to grade cheat or not that you're willing to engineer a big scandal just to prove a point that won't change anything
TA'ing is shit you have to get out of the way so you can get paid. Who the fuck takes it this seriously?

>>11893467
Pure schadenfreude.

>>11893467
Seethe harder. He did the right thing.

Topologically speaking, how many holes does a t shirt have? And why?

>>11893477
Three, because its 2-homology group has rank 3.

>>11893462
He probably didn't.
>>11893477
Three. If you flatten it, a t-shirt is a disk with three holes.

>>11893127
I'm halfway through my PhD and just now realizing I don't want to stay in academia. What should I try to learn in the next 3 years if I want to work as a quant or data analyst? I already have some experience coding in Python, Haskell, and some C++. Nothing much beyond Project Euler tier shit though. I have a working knowledge of stats and I'm currently reading about machine learning as well. Anything else?

>>11893477

>>11893485
>I don't want to stay in academia.
Is academia really that bad, bros?

>>11893477
I posted
>>11893483
but now that I think of it you could also say it has 4 holes because it's homeomorphic to a sphere with four holes.
It all depends on what you mean by a hole. Different definitions can lead to a different answer.

>>11893495
>a disk is a sphere minus one point, so a disk has one hole

>>11893495
>depends what you mean
what is the first Betty number of a t shirt

[math]S^1 \times S^1 = T^2[/math]

>>11893496
A t-shirt is arguably more similar to a sphere than a flat disk. Just because two shapes are homeomorphic doesn't mean they must have the same number of holes outside of a strict mathematical context.
>>11893504
>what is the first Betty number
The first betty number is a topological invariant that in some ways captures the intuitive notion of how many holes a surface has. However, this is by no means the only possible definition of what a hole is and you must be an autistic undergrad to believe this.

>>11893518
>that's only a topological invariant. You could have other definitions of a hole
I literally started the question with "Topologically speaking"

>>11893504
3.
Disk with k punctures (on the interior) always has first betti number k.

>>11893382
A vector is an element of a vector space.

>>11893525
I just offered another topological definition of the number of holes in which a t-shirt would have 4 holes, idiot.

>>11893484
yea, I think that answers it. thx

>>11893224
Blame Decartes.

>>11893485
Sounds good.
You could write a bitcoin trading bot as a hobby project, it's fun, easy and highly related.

>>11893536
Kek

>>11893534
Well duh. A geometric vector, you know what I mean. One where a point in n-space is a beginning and another is an end.

>>11893544
"Number of disks I removed from a surface" is not a topological invariant

>>11893127
Does anime make you better at math?

>>11893561
Not "from a surface", it's the number of points removed from a sphere. Of course, this is not defined for most topological spaces but so is talking of holes in the colloquial sense.

>>11893570
Yes.

Do any of you pacific northwest anons want a high IQ white gf to make genius babies with? I am friends with a girl who is a physics student at UWS and has an IQ of 143. She is too shy to find a boyfriend so I've decided to look here to find one for her. Pic related is her.

>>11893570
No, but it makes you more likely to become an undergrad category theorist tranny.

>>11893493
>Is academia really that bad

>>11893576
Math logic is for lazy mentally ill midwits. Prove me wrong.

>>11893570
indirectly, by reducing your circle of friends to [math]\emptyset[/math] so your distractions are fewer
>>11893575
that's a man baby

>>11893582
How do you make these infographics?

>>11893584
She literally has a vagina I have seen it.

>>11893571
>R^2 has a hole
nice definition you got there, faggot

>>11893590
>>11893536

>>11893587
wow. SRS really got better the last years

>>11893591
>he doesn't even try to come up with some riemannian geometric definition for a hole

>>11893587
Why don't you date her then?

>>11893493
Depends on your goals. If you just want to teach, you can land a job at a community college easily. Otherwise, you're looking at 10 years of postdoc work before you have a chance at a tenure track position. And R1 postdocs are pretty much impossible to get without connections.

>>11893553
Sounds interesting. Any resources I could read on this?

>>11893612
I don't have a dick.

>>11893622
And?

>>11893243
or is it? maybe you are just low iq.

>>11893622
Anon, don't try to advertise your prostitutes on a Taiwanese waterboarding forum. The only people you're going to attract are pajeets and incels, that is, unless you're OK with that. Now you've ruined the first half of a perfectly good /mg/.

>>11893621
I think of it because I played with it two years ago.
Take any active trading platform, put 100$ there, figure out the Python API (currency of choice => price, buy-order, sell-order) and query the price once a minute. You get a stream of prices and the algorithm when to buy and when to sell is really up to you.
The most naive way to make money is to do this on two exchanges and buy+sell whenever the price difference on the two is larger than the trading fee the exchange charges. Obviously, you need two exchanges with lots of volume and obviously, don't expect to make any money because if you find this situation, clearly other people are already doing it.

I ended up never making a nice summary about my hacky bot, but you'll definitely find youtubers or tutorials about it.
For something arguably related, I'm currently thinking if I should try to write a chess-bot from scratch and use lichess.org as interface. I had some fun with the NIST dataset and a rough idea of how chess programming works and I'd like to combine the two. I do C++ and Python at work too, and I even know rudimentary Haskell - although I'd say Python is the obvious choice for quickly doing the website interaction. If you're interested in something like that (or somebody else, for that matter), you can find my contacts somewhere on the user page here
https://youtu.be/z2aq21lMw40
https://youtu.be/YdKVqfKbXxg

>>11893660
>The only people you're going to attract are pajeets and incels
problem?

>>11893687
>I had some fun with the NIST dataset
by which I mean I can write a shitty ML routine without tensorflow, which is what I'd like to do (to get a grip on how everything works, not making use of external routines)

>>11893611
You can be extremely cheap and just say that it's a manifold which can be isometrically embeded in space and can hold still water (something something Navier Stokes).
>>11893622
Is there some sort of two cunnies for one deal going on here?

>>11893707
Maybe...

>>11893712
Unfortunately, I'm not american.

>>11893712
Move to the East Coast and be barefoot and pregnant in my kitchen while I start grad school in Fall 2021. Deal?

>>11893687
based botezposter
i want to have sex with her

>>11893741
Based basedbotezposterposter.

Threadly reminder to make passionate love with physicists.

>>11893786
>>11893741
Her sister is better, sorry

>>11893790
You mean philosophers?

>>11893687
This is photoshopped, look at the king.

>>11893786
>playing chess with alex on rainy afternoon
>you get legit pissed because you realize she's letting you win
>she starts hugging u and shit and trying to make u feel better, but also giggling because ur being ridiculous
>you make
>then you facetime eric hansen while having sex so he can see how hard he's getting cucked

>>11893792
andrea is a basic zoomer sloot you ephebe

>>11893687
man, having her lick my dirty anus and suffocating her with my dick while holding her nose closed would be lovely

What's your favourite axiom?

>>11893127
pic is the telegrapher's equations
can someone break it down for me in general terms and understanding?

>>11893812
choice

>>11893812
infinity

>>11893812
Axiom schema of specification

>>11893812
V=Ultimate L

>>11893812
Pair set

Is the fact that ab=ba for natural numbers really obvious to you? I mean I can prove it but I'm kind of surprised that there aren't more people questioning this. How do you think of it?

>>11893850
https://en.wikipedia.org/wiki/Peano_axioms

>>11893850
>>11893850
prove it from the peano axioms

>>11893575
Fuck it, I wanted to meet new people this summer anyway.

herpderp097@yahoo.com

>>11893857
Why did you link this? Do you think I don't know what they are? I already said I am able to prove it.
I was talking about whether or not it was intuitive.

>>11893858
Read the post before responding to it, moron.

>>11893850
Most people, mathematicians included, couldn't define + and * without using their fingers.

>>11893862
What about it wouldn't be intuitive?

>>11893812
Union

>>11893467
Some people like teaching. Other people like fucking over cheaters. When you post things like this, it makes it sound like you have some embarrassing story you are hiding.

>>11893850
If I have a rectangle with vertices [math](0, 0)[/math], [math](a, 0)[/math], [math](0,b)[/math] and [math](a, b)[/math], it has area [math]ab[/math].
The isometry [math](x, y) \rightarrow (y, x)[/math] maps this rectangle isometrically onto the rectangle [math](0, 0)[/math], [math](0, a)[/math], [math](b, 0)[/math], [math](b,a)[/math], with area [math]ba[/math].
However, since isometries preserve area, we have that [math]ab=ba[/math] Q.E.D.

>>11893910
Based.

>>11893908
>random projection out of nowhere

>>11893850
Yes. Imagine you arrange marbles in a grid with a rows and b columns. Now, if you turn the grid by 90 degrees, it has b rows and a columns but the total number of marbles obviously doesn't change

>>11893850
If you have a rectangle, you can turn it 90 degrees and it covers the same amount of area of course.
Or for naturals, if you have a square lattice of points in a rectangular shape you can turn that lattice of points 90 degrees.

>>11893913
Oh honey, if you want a random projection, you'll have to give me a vector space with a finite measure and a desired dimension of the image.

>>11893910
Based and Cartesianpilled.

>>11893812
extensionality

What fields of math does this represent?

>>11894040
Combinatorics.

>>11894040
This >>11894046. Specifically graph theory.

>>11894046
Never met an idiot who liked combinatorics (or math at all, for that matter)

>>11894051
>wow! It's the based Haruhierino problem solved by 4chan! It's easy to state, but it's so hard!

>>11893251
>is there a difference between ]0,1[ and )0,1( in the reals?
obviously. The first includes the real numbers 0 and 1, while the second excludes them

>>11893493
>piled with paperwork
>publish or perish
>rush to get tenure
>no time to think about maffs
>only way you can ever do research is through incompetent graduate students
>required to teach freshmen
I legitimately feel bad for them

It's been almost 10 years since the meme haruhi superpermutation stuff. I say it's time we do some progress on another obscure combinatorics problem.

>>11894040
number theory

>>11894118
I honestly don't think anyone else here is simultaneously skilled at combinatorics and willing to post relevant results as a joke.

>>11893243
do you mean olympiad geometry?
because yeah, its literally used nowhere and competitions should stop making shit geo questions that teach nothing useful
if you mean topology, then youre wrong
>>11893812
Don't have one, but i do hate Foundation with a passion.

>>11894040

>>11893816
Assuming:
V = electric potential; I = current; L = inductance; R = resistance; C = capacity and G = 1/R.
Then it states clearly that the potential change over the body about the same as the negative of the change of the current multiplied by the inductance (obv since a change in current implies a magnetic field and the inductance is related to magnetic field...). i.e. consider a static current then no magnetic field then the change of potential is only related to the current passing in the body.
The second one has similar explanation, left as an exercise to the reader

>>11894149
I'm pretty sure we have at least 1-2 people good at putnam or imo here. Any such person is capable of solving a low-tier combinatorics problem, i believe.

[math]\textrm{Let} \ p \ \textrm{be a prime number, then} \ (p-1)! \equiv -1 \ \textrm{mod} \ p [/math].

>>11893449
you’re not a mathematician

I'll leave here my result, maybe someone is interested in this obscure theorem:

There are infinitely prime whose distance is [math]2[\math].

>>11894268
Didn't someone post that problem a month ago? I still recall the solution.

Anybody here from Prague?
I'm in Prague next week, wanna show me around?

>>11894040
geometry and number theory

>>11894149
any anal autists here want to channel their inner chloe? might be an alternative to combinatorics questions https://mathoverflow.net/a/44933

>>11894300

>>11894268
Wilson's Theorem?

>>11894309
Correct.

I'm pretty new to math, and I am trying to get a better grasp on what it really means for things to be isomorphic.
Does the bijection between naturals and integers imply that any arithmetic theorem on integers holds for naturals and vice versa?

If not, why? Counterexamples would be greatly appreciated.

>>11893277
On affine spaces. Multivariate calc is a good place to indtroduce the distinction. Afer all you just need to be mindfull how y ou add your tuples and what the tuple actually represents. You could just add points to points in many cases and notation and computation wise it would work out as simple tuple addition. However without the distinction you get a mess if the space is not affine.

>>11894268
yes, did you want a proof?

>>11894336
An isomorphism of what structures?

>>11894305
Disgusting problem.
Even if I somehow had a solution, I wouldn't publish it, since I wouldn't want people to name it after me or to associate it with me.

>>11894354
When I said arithmetic, I meant operations that can be derived from the successor function and addition.

>>11893184
I can't believe it. Elliptic curves anon.... you lost!

>>11894361
So we are talking about models? Notice that the naturals satisfy the claim "there is a least element" but the integers do not. If I am not completely mistaken, this should suffice in proving they are not isomorphic. If we approach it completely algebraically, there is also a problem as 2 does not have an additive natural inverse, but it does have an integer inverse, so that breaks it also as an isomorphism of groups.

>>11894374
I never lose.

>>11894379
Gotcha, I still need to develop my model theoretic and group theory knowledge, but what you said makes sense.

>>11894404
Take the Galois pill, lad.

>>11894410
It's been on my itinerary. Any suggestions on a good introduction at an undergrad level?

>>11894336
isomorphism is just renaming stuff and keeping the structure as it is. for example logarithm and exponential map is an isomorphism between R with the structure of addition and multiplication, and (0,inf) with the structure of multiplication and exponentation. it means that if someone throws at you these two sets equipped with the mentioned operations but doesn't tell you which is which, you wouldn't find any difference.

What's the most entry level source to read about elliptic curves?

>>11894416
https://www.springer.com/gp/book/9781852339869

>>11894404
To put it more shortly: A mere bijection is not enough, but it also has to respect whatever structure there is. Since the model stuff is a bit rusty, I'll explain my algebraic counter-example a bit better. If we let f: N -> Z be an isomorphism, then there has to be a natural number n such that f(n) = -2, but this forces 2+n=0, and this is not the case for any natural number. Have fun learning!

>>11894426
Cubics and Conics can literally be read by freshmen who have only done calc 1.

>>11894305
nah that problem must have been attempted by multiple people by now
the point is you need to find barely relevant, obscure problem

>>11894426
https://www.amazon.com/Elliptic-Modular-L-functions-Student-Mathematical/dp/0821852426

It only assumes knowledge of calculus, linear algebra, basic number theory, and basic algebra.

>>11893812
i dont know i just can't choose

>>11894437
heh

>>11894437
Unironically kek'd at this one.

Are elliptic curves the future of cryptography?

>>11894447
Aren't most modern standards already using elliptic curves? I know AES does

>>11894447
elliptic curves are the present of cryptography
quantum memery is the future

>>11894447
No, the future of cryptography are Calabi-Yau manifolds.

The schizo in the start of the thread has got to me. Is the complement of any finite interval in the non-extended reals undefined?

>>11894458
>Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used.
Can I get an explain like I'm a first year grad student explanation?

>>11894419
I see, so it's all about the operations you are given less so the sets. I assume that cardinality is a necessary, but not sufficient condition for isomorphism then?

>>11894427
Thank you.

>>11894429
Thanks for the counterexample spelled out. It gave me an inspiration for a few ideas that I want to try and prove now. Thank you!

>>11894460
>Is the complement of any finite interval in the non-extended reals undefined?
why would it be ?

Unpopular opinion: elliptic curves are boring.
>wow group law so cool
>wow lattices so cool
>wow L functions so cool
Complex multiplication is based though.

>>11894473
>I assume that cardinality is a necessary, but not sufficient condition for isomorphism then?
true

>>11894463
When a manifold has a Riemannian metric, it has a Levi-Civita connection. This connection lets you do parallel transport, that is, you can drag tangent vectors along curves. All the contractible closed curves on a base point generate a group of parallel transport automorphisms of the tangent space at the point (which is usually non-trivial because most manifolds aren't parallelizable) (the identity component of the holonomy group). When this group is contained in SU(n), and the manifold is compact, you have a Calabi-Yau manifold.
There are also half a dozen other non-equivalent definitions, but I don't really care about those.

>>11894473
>cardinality is a necessary, but not sufficient condition for isomorphism then?
As long as you are dealing with functions of any sort, yes. An isomorphism always has an inverse, and so it is a bijection, and hence the cardinalities coincide.
>It gave me an inspiration for a few ideas that I want to try and prove now
Nice!

>>11893127
what does this thread think of NJ Wildberger

>>11894475
go back to twitter please

>>11894491
>When this group is contained in SU(n),...,you have a Calabi-Yau manifold.
any explanation why would anyone care about this (doesn't matter whether mathematicians or physicists) ?

>>11894495
Never went there, tranny.

>>11894495
This.

>>11894506
Do you want some attention maybe a head pat for speaking your mind, sharing your learned opinion with the thread? Fucking faggot.

>>11894505
>any explanation why would anyone care about this (doesn't matter whether mathematicians or physicists) ?
IIRC string theorists have models where the universe is the product of R^4 and a six dimensional Calabi-Yau manifold and also models where the universe is a product of R^4 and a 7-dimensional G2 manifold.
Mathematicians care because we care.

>>11894268
p=2 or p odd
p=2 trivial
p odd, then Z_p has p elements, 0_p, 1_p,...,p-1_p(=-1_p)
as p prime, (p-1)! =/= kp some k in Z
invertible elements Z*_p are Z_p \{0_p}
p-1_p is self is its own(unique) inverse
1 < k < p-1 must have 1< k^-1 < p-1
x^2=1_p only has 1_p and p-1_p, so k =/= k^-1
so product 1_p*[2_p*...*p-2_p]*p-1_p = [k_1*k^-1_1*...k_(p-3/2)^k_(p-3/2)]*p-1_p = [1_p]p-1_p

>>11894525
Yikes. Who hurt you?

>>11894528
thanks
>the universe is the product of R^4 and a six dimensional Calabi-Yau manifold
is it possible to explain how does this work ? I know diff geo, you can talk about holonomy and shit

>>11894531
Jesus-Christ, learn how to use LaTeX.

>>11894531
Based and Wilsonpilled.

>can you explain string theory to me desu :3
don’t do this!

>>11894536
>is it possible to explain how does this work ?
I couldn't, because I don't know. All I know about string theory comes from skimming Joyce's chapter on String theory, and Joyce openly claims in the chapter to not understand string theory either.

>>11894538
think big J has better things to do with his time 2bh

>>11894460
>schizo
?

>undefined
sounds like you just defined it

>>11894552
okay, thanks anyway

>>11894531
The way you write make me mad
Take the field Z/pZ, then product of every nonzero element is (p-1)!, The elements that are their own inverse satisfy x^2 = 1, so theyre +-1 since field
So every other element in the product besides +1 and -1 cancel out with their inverse.

>>11894586
your brevity obscures illumination

>>11894596
Your illumination obscures the proof.

>>11894536
We want supersymmetry from our compactification, which means we need the manifold we compactify on to have a covariantly constant spinor. Since we want N=1 SUSY in 4D, because this is closest to known particle physics, we need to break 3/4 of the SUSY. Compactifying on a manifold with SU(3) holonomy will work; this is because the SO(6) holonomy of a generic 6-manifold is basically the same as SU(4) (locally, at least), and holonomy rotates the tangent spinors into each other, so if we have SU(3) instead of SU(4) we will leave 1/4 of the possible spinors invariant under holonomy.

>>11893382
"from the origin" is the important part. The first point of the vector, if it's from the origin, is fixed(At the origin, 0,0), so it carries as much information as a point would in that case.

>>11894596
No, introducing a bunch of unnecessary symbols and notation like an undergrad learning to write proofs for the first time obscures illumination. Anon stated the main idea of the proof (the inverses thing), all the other details can be filled in rotely.

>>11894658
not a physicist here, juat have a random question
how shaky is susy at this point?

>>11893812
1 =/= 0

>>11894638
IM PROOOOOOOVVVING AHHHHHHHHH

>>11893812
riemann hypothesis

>>11893536
Topologists BTFO

>>11894794
Why is it out of order?

>>11894819
The order of elements in a set doesn't matter.

>>11894832
alphabet is ordered

>>11894839
[math]A=\{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\} \\ B=\{q,w,e,r,t,y,u,i,o,p,a,s,d,f,g,h,j,k,l,z,x,c,v,b,n,m\} \\ \pi:A \rightarrow B, \pi(a_i)=(b_i) \\ \textrm{Since} \ \pi \textrm{is a bijection, it follows that} \ A=B.[/math]

>>11894882
what are you smoking man, alphabet is linearly ordered
also
>>11894882
>it follows thatA=B
no.

>>11894888
>linearly ordered
Actually, it's alphabetically ordered.

>>11894895
Based.

>>11894888
The order of the alphabet is completely arbitrary. If we were to recite our alphabet in a different order, would it have any meaningful difference on the nature of the letters that compose it and how we use them?

>>11894910
>The order of the alphabet is completely arbitrary.
Yes.
That's also the case for every single ordered set.

>>11894910
>The order of the alphabet is completely arbitrary
yes. that doesn't make it unordered.
>>11894910
>If we were to recite our alphabet in a different order, would it have any meaningful difference on the nature of the letters that compose it and how we use them?
no. that also doesn't make it unordered.

>>11894910
It would affect it. Suppose we were to list things using the first letters. It would look like (q), (w), (e) etc.

>>11894339
This is sorta what I had on my notebook. Multi calc you use vectors located at a certain (x,y) (Or whatever) point, right?

I have no idea what an affine space is btw. I didn't quite get it from wiki

>>11894493
old /sci/ meme

was feeling suicidal, opened up my textbook and did some problems, now am not suicidal
thank you math

>>11894946
Read the informal description.
https://en.wikipedia.org/wiki/Affine_space#Informal_description

>>11894970
this, but the opposite

>>11894970
Such is the power of elliptic curves.

So i understand what's going on but how the hell do you come up with the formula and choice for q ?

Is there hope for you in mathematics if you can't solve and can never hope to solve an IMO problem ?

>>11895169
Just staring at it, I think he used Newton's method.

>>11895192
Yes.

>>11895197
Nope, never mind.

>>11895202
Even the A1s that asian 13 year olds do for fun ?

>>11895208
Yes. Your mathematical future is not contingent on the IMO.

>>11895192
literally no IMO geometry problem is ever useful
competition math is just autofellatio

>>11895169
For example if [math]p^2<2[/math], then you want to add a rational that depends on the length from [math]p^2[/math] to [math]2[/math]. So for example, [math]p-\frac{p^2-2}{f(p)}[/math], where [math]f[/math] is an integer valued function of [math]p[/math]. You can make this function as large as you want, the only thing that matters is that when you square the fractional term, it is small enough. The author here decided to pick [math]f(p)[/math] so as to cancel out terms in the expansion, specifically the [math]p^2[/math] term, and it turns out nicely

>>11895192
Depends what you mean. Not every mathematician is/was capable of developing the skill to score gold on an IMO. Probably most aren't. But anyone good enough at math to be a mathematician can do IMO problems.
Olympiad problems are by design never impossibly hard. If you fiddle with them for an afternoon, or maybe a couple days if it's really sticky, you should be able to get most of them eventually. The part of the olympiad that takes real talent is being able to consistently get the insights you need within 20 minutes on most problems so you can actually finish all the problems on the competition in time.
>>11895222
Who the fuck does math because it's "useful"?

>>11895416
Actuaries, physicists, engineers, computer scientists.

>>11895420
not math

>>11895416
not useful in the sense of >>11895212
It might help due to exposure to problem solving, yes, but it's ultimately inconsequential in learning actual rigorous mathematics.

>>11895416
why the fuck would you call competition geometry math?

>>11895426
>all these sour grapes because he got filtered by triangles

>>11894536
Any recommendations for String Theory aimed towards mathematicians? My HEP background isn't enough to keep up.

>>11895169
Don't worry about it, literally the most unintuitive page in all of Rudin. It's there to scare arrogant people away.

Is Zorn's lemma too powerful for its own good?

>>11895475
I got filtered here

>>11894170
>i do hate Foundation with a passion.
Why?

>>11895497
Well then you'd better stop and draw it out. That's an important idea to understand. It can be drawn in a picture.
Basically, the idea is, scale x and y so that the difference between them is bigger than 1, then pick an integer in between the scaled versions.

>>11895572
and then you divide by the original scaling factor. it takes time to internalize this and really understand it.

>>11895517
dunno

>>11895590
Yes. The first time anyone sees such a scaling argument it will seem brilliant. The second time it will seem clever. The third time, pedestrian. All this stuff will come with practice and some experience.

>>11893807
>Then you facetime Eric Hansen
Died

>>11894176
>Lean
There was a professor at my uni who shilled this super hard. He was fired for mysterious reasons. Only the staff know and students received an email asking us to respect his privacy. Did the opponent's of formalism get to him bros???

>>11894176
formalized proofs are literally 100% the domain of pretentious philosotard undergrads and boomer profs who don't want to work anymore but don't want to look like they're not working

Dude just keeps finding the most complicated way to say the simplest things.

>>11894176
What do the diamonds mean /sci/?

>>11895592
Foundation is equivalent to every set having a rank in the cumulative hierarchy, over ZF(-foundation). It is super nice, I hope you reconsider.

>>11895773
Mathematics is inherently philosophical.

>>11895831
its possible if i ever get around to learning Set Theory more seriously
theres Jech and Enderton back on my campus, ive always wanted to get into them, also Bourbaki but we dont talk about that

>>11895840
https://www.amazon.com/Discovering-Modern-Set-Theory-Mathematics/dp/0821802666/ref=sr_1_4?crid=1SCQBYOEETRZO&dchild=1&keywords=axiomatic+set+theory&qid=1594617845&s=books&sprefix=axiomatic+se%2Cstripbooks%2C311&sr=1-4
Good intro book to axiomatic set theory.

>>11895773
Exactly, it’s end of their career researchers that spend the rest of their effort into sucking their own dick by making their proof better looking and not contribute anything better to their field.

>>11895796
imagine getting filtered by Rudin in the first chapter, jesus christ

Do we have any people here who genuinely, unironically think they're smart enough to make it (Fields medal-tier)? Don't answer if you're still an undergrad.

what is some comfy math I can learn without leaving my bed?
preferably something that doesn't make me want to try stimulating dopamine receptors with a gun

>>11896144
Interuniversal Teichmuller Theory

>>11896144
You should learn probability by playing russian roulette in bed, you can invite your friends to a discord call so they can join you in the fun :)

>>11895840
Jech is a book in the field of Set Theory. It's about transfinite induction, V=L and large cardinals and such. It states UFC axioms on page one and then makes 5 pages of comments on classes and well ordering. You don't learn what you mean by set theory in that - for that you want an intro to logic book.

How do I get more perspective in maths? I tend to get too caught up in the details while missing the bigger picture. Talking to other people, I noticed that a lot of them do this even more than me.

>>11896385
try not to get too caught up in the details and focus on the bigger picture

>numerical analysis

>>11896385
What helps me is keeping the bigger picture in mind

>>11896385
If you cannot into higher level abstraction, by this I mean have not been always capable of such a thing since early childhood without instruction or environmental pressure to do so, you’re absolutely under no circumstances going to learn to do so from others telling you how they think they do it. People are awful at explaining metacognitive processes, this includes extremely intelligent people like mathematicians.

I propose a new number, S.
Please treat S like a real number, even though it has the property that for any real number x, S > x.
We could also say 1/S = d, and that for all real x except 0, |x| > d.
Please be kind to S, I find it a really useful number.

>>11895796
It's formal thinking. You're not meant to be able to read it fluently at the beginning. As you get practice it will become natural to think that way about common slightly imprecise concepts you know.

>>11896385
Always, always, always, draw a picture or a diagram of your problem. Even if it's just a dumb scribble, or if it's arrows and letters, it helps you to piece together the bigger picture and to focus in on small parts. That way you don't need to keep it in your head, you can just look and there's the big picture developing right in front of you.

>>11893812
>In thebeginning was the Word, and theWordwas with God, and theWordwas God.

>>11896841
Based.

>>11896142
I'm smart enough to know that I'm not

>>11896886
That's not saying much. Anyone with an IQ over 80 knows whether or not they're smart.

>>11896142
Nope. "Smart" enough to get a PhD position but by no means skilled enough for any prizes. I still have a few years to prove myself wrong and get it by accident, though.

>>11896142
No way lmao. Not unless I randomly luck out and stumble across an important conjecture from some field which coincidentally can be solved by this one obscure technique that I happen to know about.
>>11896841
Baste.

Bros, how did people like Abel, Galois, and Gauss make such profound discoveries in their teens and early twenties?

>>11896912
Dedication and excellent work ethic.

>>11896912
You literally just had to know less to do maths at the time, and there were more low-hanging fruit.
Which isn't to say that you can't reach the forefront of some field and start doing research in your late teens by studying day and night, it was just easier back then.

>>11896918
Pretty sure this is not actually a big factor. Sure, we know much more than we used to but also the knowledge we now are aware of is MUCH more organized and concise. There used to be whole books written about subjects that modern textbooks only take a chapter to cover. The irrelevant subjects (which were thought to be relevant in the past) have been discarded or adapted as a small part of another subject.
I think if you put Galois or Abel in today's world they would be able to reach the forefront of research as quickly as they could back then.
Look at Abel's wikipedia page
>When Abel entered the university in 1821, he was already the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library
>all the latest mathematical literature
How many books do you think that is? I'd wager it's certainly over 20.
On the other hand, let's look at how many books you need to read to reach the forefront of a subject today, e.g. Algebraic Topology. The knowledge is so well systemized that you could certainly do it in way less than 20 books. Ask pretty much any expert in any field today what books a beginner needs to read to get to their level and I'll wager it will be less than 20 books.
So, you have no excuse for being dumb. A lot of you could come close to Abel if you just put in the time. Read, read, read. Do the exercises.

>>11896929
>I think if you put Galois or Abel in today's world they would be able to reach the forefront of research as quickly as they could back then.
They also wouldn't have died so young.

>>11896940
Nah dude, even if dueling were illegal, Galois would still find another dumb way to die for sure.

>>11896912
It's very unhealthy if you let yourself think in that direction. Do anything for your entire lifetime, at any age, and you will have done extremely well.

>>11896942
i wonder what he could have done(or not done) if he didnt listen to his dick so much

>>11896942
KEK.
Reminds me of that dude who died in one punch during a protest, although Galois would probably die over a woman.

>>11896929
>Abel
Highly cringe idiot who got mad at based Cauchy.

>>11897070
Why?

>>11897113
While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). The paper was, however, misplaced by Cauchy.[12]

While abroad Abel had sent most of his work to Berlin to be published in the Crelle's Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. The theorem was put aside and forgotten until his death.
but also
His royalism and religious zeal also made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic"[23] and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician

>>11896641
what you've proposed are the hyperreal numbers

>>11893807
>andrea is a basic zoomer sloot you ephebe
yeah it's great

>>11894437
I wanted to make this joke fuck

>>11897224
What's this IR?

>>11897224
Me on the bottom.

r8 this paper
https://www.dropbox.com/s/rkhzouzivdx0d9h/How%20to%20find%20the%20angle%20of%20Isosceles%20Triangles%20-%20easier%20than%20right%20triangles.pdf

>>11897229
[math]/mathbb{R}[/math]

>>11897234
Thanks. Now I feel like an idiot.

>>11897229
An array of numbers.

>>11897248
Wait, it's all arrays of numbers?

>>11897231
Why would you dox yourself like that?

>>11897250

>>11897252
I don't know, I'm not the author

>>11897224
But the real numbers are a construct. Like most mathematical concepts, they don't exist in physical reality, but it doesn't matter since their useful.
>>11897248
Unironically this when thinking of reals as Cauchy seqeunces of rationals.

>>11897256
Basado.

>>11897258
They're what Leibniz called necessary existence.

Wait... what's an array?

>>11897289
That is a good question. I suggest we dedicate the rest of this summer to formulating axiomatic array theory to avoid paradoxical arrays of Russellian sort.

>>11897289
Everything is an array.

>>11897289
{1,2,3,4,5,6,7,8,9}

I don't know if this is too obvious but could someone explain why exactly this is equivalent?

>>11897289
ordered, finite list of numbers

>>11897531
I'll give you the general idea, you can check the details if you want to. So, the reflector is described by the text segment under the one you have highlighted, so we would like to have some sort of unit (or counit? I always doubt myself with these) [math]1 \to R\circ E[/math] as the [math]r[/math] is actually going to work as our counit (or unit). However, if we just look at what happens if we do it the other way around and start with an [math]\mathbf{A}[/math]-morphism [math]g\colon A\to B[/math], then this is actually just going to the same in [math]\mathbf{B}[/math], and so the natural transformation we are after is merely the identity. This is a sketch of the proof. Please do feel free to correct me if there is a mistake. I'm a bit tired, so there could very well be.

>>11897289
Finite list same type elements

>"What is the reciprocal of 3/4?"
>"Well the multiplication by x/y is an Abelian group automorphism of the additive group of a division ring, sending y to x, therefore its inverse must send x to y, and therefore it is y/x. So the answer is 4/3."

>>11898028
1/3/4 = 1/12

>>11898098
>Sorry, sir, but [math]x,y \in \mathbb{R}[/math]

>>11898101
[math]\mathbb{N}[/math]*

>>11898107
but 1/12 = -1-2-3-4...

>>11898148
Kek