/sci/ - Science & Math

>>11800085
41 yourself tranny

>>11800085
rope yourself kike

>>11800143
>>11800199
NORMALFAG SEETHE

ignore this >>1800000

>>11800356
whoops meant this >>1800000

We already had this edition.

>>11800427
they never learn

>>11800085
I had a three day streak of non stop math. I could not do it today. I left some theorem half proved. Wish me luck tomorow.

>>11800793
glhf

>>11800085
Holy shit, this bullshit agIn, grow up kid

>>11800085
Made love to my gf nice and slow yesterday and she told me she loved me as she came

just did some maths

>>11800975
Thanks for sharing. We will all send evil thoughts and curses to you and your gf so you both die of cancer.

>>11801106
I redirect your bewitchment towards kurisu. Burn in hell weeb nigger.

>It was founded by a disgruntled physicist named Philip Gibbs. It is unclear who funds the website today, and it is not even clear who actually runs it today if you just read the website. Compared to arXiv, there is little moderation (according to the FAQ, a paper can e.g. be removed if it's not science/math, if it's plagiarism, if it's obscene and so on, but not if it is obvious nonsense unless it is computer-generated nonsense) and people can submit papers anonymously or under fake names.
>At the moment, there are 22,596 papers on viXra.
>viXra has no sustainability plans. If whoever runs it decides to take it down tomorrow or their servers catch fire, I'm not sure what would happen.
>Warranted or not, it has a reputation of being an alternative to arXiv for cranks and to host a lot of junk science, fake proofs or even outright nonsense. It is not recommended to submit something to viXra if you wish to be taken seriously, because the reputation of viXra would probably taint your paper's reputation by association. (Which is not the same thing as saying that arXiv "bestows credibility" on papers found there. It's rather viXra which bestows a bad reputation.)
Is that all true?

>>11801359
Thank you anon

Damn, that 'fuck normies' retard really singlehandedly killed all interest people had in posting in /mg/, congrats kid, you destroyed a great general.

>>11801508
I don't even know what got him so butt-flustered

I'm super fucking grumpy today.

What are some cool things in combinatorics that I could do a senior thesis on? There are a few people in my math dept who do random matrix theory which I think has some connections to combinatorics but idk. Any suggestions?

>>11801589
What's a senior thesis?

>>11801591
It's basically a senior project that you are expected to start your senior year and present at the end of the year.

>>11801591
It's when you write about something actual seniors and experts have written about and then poof, suddenly you're a "master".

>>11801375
https://vixra.org/abs/1612.0123

>>11801598
What level are we talking about?

>>11801589
Do you need to do any original work or is it just expository

>>11801591
It's a thesis you write in your senior year

>>11801589
necklace problem/4 color theorem are cool, if they don't care if it's original. You could always make up your own problem, since that is the joy of combinatorics. You could do something connected to dynamical systems (ergodic) if that interests you. idk there are tons of options

>>11801616
fucking kek

Since the thread is so slow I am gonna ask my terrible question.
What is the use of abstract algebra and why is it so important for other fields of math?
I don't mean the applications of it, although I would like to learn about that too but more importantly why does it seem so foundational that some other math fields also require it or make use of it.

Btw I don't know much about math so what I am asking might be nonsense so apologies if it is.

>>11801616
Man, wtf, why do they allow that? I mean, the whole idea of the site is awesome, you can post papers anonymously and without burocracy, but then again they should at the very least not accept utter garbage like that

>>11801620
it's just expository. The expectation is that you just look go into depth on some problem you are interested in. There are really no restrictions except that it should be in-depth.

They also offer the option of doing something novel but they don't really expect it.

Also this is at the level of a senior undergrad. I've taken all the normal courses (lin alg, abstract alg, analysis 1,2, complex analysis) as well as some grad level courses (abstract 1,2, real analysis 1,2, point-set topology). I also know combinatorics at the level of 'A walk through combinatorics' by Miklos Bona.

Again, any suggestions are greatly appreciated.

>>11801639
>grad level courses
>point-set topology
oh, anon...

>>11801645
Idk man the class was a grad-level class using munkres.

>>11801639
https://en.wikipedia.org/wiki/Permutohedron
Also why did you chose combinatorics, out of curiosity?
>>11801645
It's a grad level course at some institutions don't be mean

>>11801657
Thank you! I'll look into this.
Tbh I can't really tell you why I like combinatorics. I just think the problems are really fun to solve and the solutions are more often than not really satisfying. Also, reading/learning about combinatorics doesn't take long at all so I can get to solving problems quicker than most other subjects. Interestingly enough tho this is not a common opinion. A lot of my friends that I've talked abhor combinatorics and most of the professors I've talked to don't really like it either.

>>11801657
at shitty schools like mine yes, its by no means a grad level topic

Can any anons take pictures of the notes they take or point me to some guide on math note-taking? For some reason, I feel like I've lost the ability to take effective notes and I need some inspiration . . .

>>11801703
>taking notes
haha gay!

>>11801703
I tend to use throwaway pieces of paper or otherwise actually already write in TeX on the laptop

>>11801614
>master
Maybe they will change the name?
Github just announced it will rename git master branches to something else to not trigger people with thoughts of slavery.
https://www.bbc.com/news/technology-53050955

In other news, here's some free Springer books:

https://hnarayanan.github.io/springer-books/#Mathematics%20and%20Statistics

>>11801627
Whenever you have a mathematical object, you can usually associate half a dozen different algebraic objects to it.
To give the single most ancient example possible, just about everything has some sort of group of symmetries.

Would you rather be deaf or blind?

>>11801718
I assume the scrap paper is for working out details as you go? Would you mind uploading one of your TeXed-up note sheets, I'm just curious to see what the layout is.

>>11801858
I'd prefer dead over either of those.

>>11801858
deaf
>>11801869
I don't believe you.

>>11801871
>I don't believe you.
Why not?

Given a consistent formal system x, from which a proposition φ has been derived(x ⊢ φ, and hence x ⊨ φ), could it be that ~φ could exist due to the fact that there exists a proposition ψ not derivable from x?

>>11801703
http://cognitivemedium.com/srs-mathematics
http://augmentingcognition.com/ltm.html

bros... how do i get motivated to return to the university?

>>11801589
>There are a few people in my math dept who do random matrix theory which I think has some connections to combinatorics but idk
Some connections is a little modest, but as far as I'm aware the applications of random matrix theory to combinatorics are 90% big dick probability theory and hard analysis and 10% actual combinatorial stuff. Most of the applications I've heard of seem to be to getting asymptotics of large permutations and Young tableaux (which can naturally be translated into interesting stuff about representations of large symmetric groups).

If you are very comfortable with (and like) heavy analysis it would probably make an impressive thesis but it's also probably on the upper bound of what they expect the difficulty of anybody's thesis to be.

>>11801858
That's like asking, would you rather be an algebraist or a geometer?

>>11801978
I'd rather be an algebraist, obviously. Fuck geometards.

>>11801627
Studying the solution set of a polynomial over a field is very closely related to studying the finitely generated reduced algebras over said field. Properties of the algebra translate readily to properties of the solution set. For example, things like smoothness or singularities can be intrinsically defined on the algebra. The algebraic dimension of the algebra translates to the geometrical dimension of the solution set. Maps between algebras translate to maps between these solution sets.

In algebraic number theory, one of the goals is to understand the failures of rings having unique factorisation. For example, the integers, as you should know from school, can be uniquely factorised into primes. There are some ring extensions of the integers that also have this property, and it helps to solve diophantine equations by factoring - eg: solving [math]x^2+2y^2=n[/math] over the integers can be whittled down to finding the properties of the ring [math]\mathbb Z[\sqrt{-2}][/math] by means of factoring the equation into [math](x-\sqrt{-2}y)(x+\sqrt{-2}y)=n[/math] and finding the prime factors on either side. It turns out that these types of integer extensions don't always have unique factorisation, but have a weaker property of unique factorisation in ideals, but can still be used to find all the solutions.

In algebraic topology, you can give a sequence of rings to a topological space that can help differentiate it from other spaces. It turns out these rings have functorial properties, where continuous maps between spaces lead to homomorphisms between the rings, which is a powerful technique in general.

>>11802023
>In algebraic topology, you can give a sequence of rings to a topological space
Good thing you don't actually need any of those tools in topology because all it takes is an intuitive argument. Topologists don't deal with rigor.

>>11802043
A topologist will readily prove every handwavy technique under gunpoint

>>11802046
That's actually a common misconception. Topologists don't actually think in terms of proofs. Handwaviy techniques are all that it it takes, according to anons in the previous thread.

>>11802009
I would give half of my algebraic skills to have even a hint of geometric/topological intuition.
t. different anon

>>11801718
>thoughts of slavery.
Virtue signaling is probably the dumbest thing ever invented

>>11802046
>>11802055
For example, in topology to prove S^n and S^m are not homeomorphic it's enough to say that an embedding of it into R^k separates it into two components only for the appropriate k. You don't actually need to prove it.

>>11802055
Take the space. Thicken the edges if necessary so we can apply Mayer Vietoris. It is trivial to see that collapsing under the subspace in question is just [math]S^4[/math] with two handles. The result follows

>>11802094
That's a bad example because what you said probably could actually be justified with rigorous mathematics.
In topology, they don't deal with such nonsense as "rigor". Nothing needs to be justified. All you need is to name some lemmas which sound plausible (they don't even have to be true), and just explain how you think of it visually.

>>11801860
https://axiomsofchoice.org/kfv_._note

E.g. this was for a gig for a research project for the Austrian Road Safety board
(also made a Jupiter textbook to draw stuff from a database, but not shown here)
At this point I also have a coarse style guide in my head on how I approach books or papers.
I previously used dokuwiki, but now I just have a textfile parsed into html and render it in the browser (via a custom Flask server). The html part is explained in

https://youtu.be/bk0J7WGwk90

>>11801889
>Given a consistent formal system x, from which a proposition φ has been derived(x ⊢ φ, and hence x ⊨ φ)
That's a soundness assumption there, yes?

>could it be that ~φ could exist
What do you mean by "~φ exist"?
I suppose you mean a model of x which validates ~φ.
Looks like you shove down the "inconsistency" down to the model, which although, I'd say, also is in practice a synthetic construct. I think I only believe in syntax, so classically I'd say no that's not possible.
Then again, this might go in your direction, although it's fringe as far as math departments are concerned I suppose:
https://en.wikipedia.org/wiki/Paraconsistent_logic#Example

>>11800085
You're not a social outcast, you're not even a social misfit. You're just a social failure.

>>11801858
Deaf for sure.

Did somebody watch the OATS today?

>>11802301
I'd like to know if my conjecture was correct: >>11777312

>>11802132
>Jupiter
It's Jupyter

>>11802329
u rite
https://youtu.be/7Xf-Lesrkuc

cba

>>11802358
quasi trivial

>>11802381
>song about your my mom
Siblings posting about their mother?

>the group-theory

why do topologists call them 2d shapes when they're clearly 3d?

>>11802444
2d means that, locally around each point, it can be described using two parameters. It does not necessarily mean that it lies flat on a plane

>>11802444
if you put yourself on a sphere and forget how to jump, you can only move in two dimensions (forward/backward & left/right), there is no local third dimension.

>>11802444
So according to you a sphere is lower dimensional to a klein bottle?

I see so topology is more like glassblowing instead of sculpting clay

>>11802504
If there's no rigorous argument, there's no rigorous argument. We just roll with it.

Does anyone know if there's a way to tell the OEIS to search specifically for the _start_ of a sequence? So I'd punch in [x,y,z] and it would only return sequences with a_0 = x, a_1 = y, a_2 = z?

>>11802571
Email the creator and ask.

If someone proved the Riemann Hypothesis and posted it on viXra, do you think academia would accept it? Took them years to accept Perelman's proof of the Poincaré Conjecture, and that was only because he was already famous previously.

>>11802601
The attitude toward viXra is simply that stuff from viXra is probably schizobabble and not worth working through, not that viXra is a 100% stamp of death.
The only hump to getting it accepted would be finding one solitary mathematician willing to skim it and notice it's not obvious nonsense, which wouldn't be that hard if you were an otherwise sane person just using viXra for the meme, and then it would spread quickly.

It took years for Perelman's proof to be _formally_ accepted because there were details to fill in (and he didn't really care to do so himself) but I'm not aware of anyone who ever held the opinion that it was false or substantially incomplete.
Experts can usually very quickly appraise whether a paper is plausible or not and people were pretty confident his proof worked, or would work with some extra details, very quickly.

>>11802601
You remind me of someone with this question of yours. Would you say a proof is a social construct?

>>11801953
hmmm I do like analysis and am fairly comfortable with it. I think I'll ask one of the profs and see if it would be worthwhile. A quick google search brought up a book called 'combinatorics and random matrix theory' by baik, deift, and suidan. I am taking a measure-theory based probability course next semester so it could be interesting.

Thank you!

>>11802647
>Would you say a proof is a social construct?
Yes!

>>11802601
The should post it on 4chan.

>>11802717
Most intriguing. Would you rather have tea or coffee? Why do you dislike bananas? Which species of birds is knocking on your windows? Answer these correctly and I will bring you a bottle of beer this weekend.

>>11802717
who are you, based non non biyori poster?

>>11802601
no, you'd also need to realize some positive example with your proof (e.g. find some natural number counterexample so something that was previously unfeasible to find)

>>11802754
just take any number in the neighbourhood of [math]\infty[/math]

>>11802731
I should be the one asking question here, luckily for you I have only one: Why did you play with my feelings? You knew I loved you.
>>11802737
A shadow of my former self

If I could prove anything I would just create a /pol/ thread with Kurisu as the OP and dump it in there.

>>11802776
The positive version of me you were so fond of was merely a fantasy of yours based on a facade, and had nothing to do with my hideous inner self.

>>11802816
Anything in the sense of "give me any claim you wish and I will prove it" or in the sense of "I wish I could prove even one thing"?

>>11802836
I'll save you from yourself then, I know you have a lot of good deep inside of you.

>>11802836

meaning if I could discover any non trivial result

>>11802900
Stop wasting time on ancient and bitter drunks and make yourself the strongest poster of these threads.

>>11802907
Does it suffice if it something well known, but you derive it yourself? I remember when I found out that the classifying space of the circle is actually the complex projective space by accident when I was just playing around with the bundle G -> EG -> BG, and that felt somewhat pleasant.

>>11800085
i am reading Data Science and Predictive Analytics book https://link.springer.com/book/10.1007/978-3-319-72347-1 and there is example about predicting RealEstate prices from other features. Dataset
https://umich.instructure.com/files/416274/download?download_frd=1

I am surprised by their choice of feautures for predictors - weakly corelated to column we want to predict, some of those features even have negative correlation. So i made experiment and changed features to ones with highest correlatio to column i want to predict, but i got worse fit. Worse correlation between predicted and true values (by 0.03 worse) and worse rmse (0.03) also. Why is it happening? And how to choose predictors, if not by largest correlations?
Here notebook https://colab.research.google.com/drive/1MVoy40YJFxXgegvPZNLNk2VeUW_EjEQh?usp=sharing

https://twitter.com/byrunt/status/1271478802305212416?s=20

>>11802955
i am using Neural Networks btw

Good night, /mg/

>>11802952
How can I become the strongest when I can't even save the one I love?

What is being an actuary like in Canada?

If I want to prove that the n-sphere can't be written as the product of two non-zero-dimensional smooth manifolds, I can just use the Kunneth formula.
Are there any other tricks for showing that a manifold can't be written as a product?

>>11803270
>actuary
this is a mathematics thread anon

>>11803093
Good night lad.

If I have a succesion, [math]s_{n}[/math], where every term of the succession is determined purely by the natural number n, is it true that one can find a sucession such that [math]s_{n} = S_{n} - S_{n-1}[\math] for any s_{n}?, I'm not exactly sure if this is true but if it were It would be very interesting to see why, another interesting question is if one could create a "technique" to calculate [math]S_{n}[\math] given [math]s[\math]. Here's a cute dog too

>>11803461
Your terminology is weird, are you translating from a different language? What you are calling successions are called sequences.

Anyway what you want is the sequence of partial sums [math]S_n = s_1+s_2+...+s_n[/math] . Then [math]S_n - S_{n-1} = (s_1+s_2+...+s_n)-(s_1+s_2+...s_{n-1}) = s_n[/math] .

And a morning /mg/

>>11803129
You must embrace the pain of loss and use it as your fuel.

>>11803398
Cheers mate.

>>11803296
Just a random idea that will most likely be completely useless, but what if you assume your manifold can be written as a product of some spaces X and Y, then take the smash product of X and Y and get something completely non-sensical? Just a random idea coming from someone who just woke up.

>>11803485
Nah, forget the smash product. Instead of obtaining something non-sensical out of that, the idea itself is pure nonsense.

>>11803485
>And a morning /mg/
You sleep for 2h a day wtf?

>>11803536
Not much more without sleepy pills and/or booze. I think I got 6 hours of sleep between saturday and sunday. That was luxurious.

>>11803545
That's bad for your brain man, Einstein slept for 10h a day

>>11803548
Yeah, not the best way to do things. I even stopped using caffeine in order to sleep better, but nay. Though, having an already empty skull nullifies the brain damage.

>>11803569
Need to take care of that brain tumor

>>11803548
>tfw sleep apnea for years
I feel like I definitely got dumber

>>11803577
I just need 2 more years, then I can retire. I need to finish what I started, but it will take time. Have you made any progress with whatever you have been doing lately?

>>11803595
No, but I will tomorrow, I swear, good night fren

>>11803627
Sleep well and wake up happy.

De Stijl founder Theo van Doesburg's "Arithmetic composition" is a simple black-and-white geometric painting which invites a (very) elementary recreation, and reflection on the notion of tangrams. First, assuming the picture to be a unit square, its main diagonal, [math]\sqrt{2}[/math] is trisected to give the side of the large black square, [math]\frac{\sqrt{2}}{3}[/math], producing an area of [math]\frac{2}{9}[/math] - the nice "base" value, were it not to be iterated. More interestingly, the larger hypotenuse and smaller edge along the unit edge are in the simple [math]\frac{2}{3} : \frac{1}{3}[/math] relationship.

The thing is scaled, iterated, and so on-in the picture, only to its fourth iteration, but with the suggestion to infinity of course. This conceit leads to the amusing series

[eqn]\frac{2}{9}\sum_{k=0}^{\infty} \frac{1}{4^k}= \frac{8}{27}[/eqn]

Where, intriguingly, the numerator and denominator are again in a simple [math]2^3 : 3^3[/math] relationship. The notion of scaling length involves halving (or doubling), and the related notion of scaling area involved quartering (or quadrupling). Ad infinitum, a term involving simple cube numbers is found. Several simple, nice (yet distinct) relationships obtain.

Suggested exercises: generalize van Doesburg's construction to the cube, and higher dimensions. Provide insight on resulting, simple numerial relationships.

>>11801375
>arXiv "bestows credibility" on papers found there

Of course it does (in the eyes of the scientific community). You need authorization by a current member to get in which makes it an exclusive club to join.

>>11801627
what cartoon is that image from?

>>11802094
based

>>11802444
it's only the surface of a sphere or whatever, it's not "filled in"

>>11803785
You're misreading what he's saying. "it's not on arXiv so it's probably wrong" is not the same thing as "it's on arXiv so it's probably right."
False shit gets posted to the arXiv literally every day, nobody would ever claim that something must be credible solely because it was uploaded to arXiv.

>>11803858
>"it's not on arXiv so it's probably wrong" is not the same thing as "it's on arXiv so it's probably right."
yeah it is, by Bayesian reasoning.

>by Bayesian reasoning.

>>11803485
>completely useless
No, not really. I had already thought of just looking the classifications of 1-manifolds/2-manifolds to prove that it can't be any of the possible products, but it's kind of a pain in the ass.
By the way, another idea I had, and this one might inspire you, is looking at the subbundles of the tangent bundle. For example, an oriented manifold with Euler caharacteristic nonzero doesn't split as the product of a one-dimensional manifold with an n-1 dimensional manifold, since the kernel of the projection onto the n-1 dimensional component is a one-dimensional subbundle of the tangent bundle, and we can use a Riemannian metric to construct a section.

Unless orientability actually isn't enough for the section to exist, in which case by diff top is substantially worse than it should be.

>>11803840
A pretty shitty one. Vinland Saga.

>>11804294
Realized five minutes later that you can just use the fact that the product of the Euler characteristics is the Euler characteristic of the product.

>>11802055
You () are very wrong

>>11802046
is correct

It is helpful to talk about things in terms of intuition when doing high level topology and geometry. But this intuition is *not* the intuition you think you have. This intuition is an intuition that was built from extracting ideas from rigorous proofs. They are apple to easily translate between this new learned intuition and rigour. Read tao's blog on post-rigour. When talking intuitively a topology will say stuff like a "nice" space, this is not being un-rigourous, the context tells you how "nice" this space has to be.

Brainlets do one course in abstract algebra and think topology is unrigours because of stupid memes about donuts and coffee cups.

>>11804294
>>11804321
Luckily/sadly it splits, yes. What if you tried using Poincaré duality (https://en.wikipedia.org/wiki/Poincar%C3%A9_duality)) after assuming you have a product? This would probably just reduce to your original Künneth argument. Alternatively, you could write a point p as (x, y) and then the tangent vectors at p should split as the direct sum of those at x and those at y. Fixing either and letting the other vary could lead to a contradiction, but I don't know. Maybe Künneth is simply the best way to go. Sorry, I only took the veeeeeery basic diff geo course years ago, so my rambling is built on foundations extremely shaky.

Let a circle roll without slipping on a line. It's obvious that when the circle returns to its original position the point touching the line will be the same as before.
The same probably isn't true of the sphere rolling on the plane, is it? Can any point of the sphere be made to touch the plane after some motion?

>>11804354
>You () are very wrong
Well then your opinion is radically different from the vast consensus among the topologists on /mg/.

>>11804413
Mark out a great circle on the sphere that goes through the point touching the ground. Do a fourth of a rotation along the great circle. Turn the sphere 180 degrees on the horizontal axis. Roll the sphere back to original position.
The antipode of the original point is now touching the ground.

>>11802088
why do you act like you know anything about topology when clearly you don't ?

>>11804415
can you please say what is the vast consensus among the topologists on /mg/ ?

>>11804443
Are you saying that's not a proof? You must be autistic retard who can't into visualisation.

>>11804413
Yeah just map a continuous path on the sphere from starting point A to end point B, find the path length, and draw on the plane a loop of same length as the path, roll the sphere so that the path always touches the loop. Might need some differentiability conditions. Wa-lah

>>11804413
Fun question.
For one, it's clear that you can make the ball (of unit radius, says) touch different points at the origin:
Consider the ball on the sharp end of a Pytagorean triangle with side length (2n+1)*pi for any n. If you roll it to along that edge, you end up on the antipode. Then you go the 90° route and turn it back. Now if you close the the triangle you end up back on the original point on the plane, but with the ball at a point with an arch length of sqrt(2)(2n+1)pi mod pi.

>>11804432
I think we should disallow rotating the ball on a point, for a non-slipping situation.

This kind of situation already pops up in configuration spaces e.g. of winter sleds, you get funky Kleinian'ish tangent bundles

>>11804457
Do you think there is some other surface instead of a plane where the configuration of points on the sphere to which you can return to is discrete?

>>11804443
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

>>11804450
it's not a proof if you regard sparating R^k by S^n as something that can be taken for granted or proved by drawing a picture. this is not obvious at all.

>>11804415
it seems that by
>vast consensus among the topologists on /mg/
you really mean your own
>interpretation of opinions of people who have little formal training in mathematics

find me any professional topologist who uses "handwavey techniques" that cannot be backed up by rigour

you are trying to win a stupid conversation, if you have any trainning in mathematics you will know that topology is rigorous

>>11804478
>>11804484
These really read like posts written by people who don't really enjoy topology much.
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

>>11804457
You can just induce the rotation with a smooth path, tho. Choose a tangent vector, choose another tangent vector that you want to map it to. There's a path on the sphere that has the first vector as its initial speed vector, and the second one as its final speed vector. Choose a path on R^2 of the same length and rotate the curves alongside each other, as >>11804453 pointed out.
>>11804468
Nah. The above argument lets you rotate it about a point however you want, and then a modification of >>11804432 tops you off.

>>11804491
Wait, no, you have to also consider the initial and final speed vectors of the curve in the plane.
Still works, I think.

>>11804491
I don't really see why the path of choice on the sphere and the path of choice on the ground would always match up

>>11804468
I don't know why you ask that as if we have already established that? At least I don't see it.

Maybe there's some nice trigonometric tricks to figure out what points can be found.
These situation have, in any case, been extensively been studied.

(E.g. the the mentioned slew which on an infinite snow field has the configurations [math] {\mathbb R}^2\times {\mathbb S} [/math] but a confined tangent bundle, e.g. for each [math] ((x,y), \phi) [/math] , the motion in [math] {\mathbb R} [/math] along the ray with angle [math] \phi [/math] is unrestricted (linear motion) but the other two degrees of freedom are couples and somehow bound. That's not your example since you can roll in any direction, but it's the same math.) Maybe we find the motion on the plane as projected onto the ball.

>>11804490
This reads like someone who doesn't know much math.

"Intuition" is good in topology. This is a very different kind of intuition than natural intuition. Topologist may sometimes talk in a way that seems handwavey, but this seemingly handwavey talk is rooted in an ability to translate between rigor and learned "intuition".

Read tao's post on post-rigor.

>>11804513
>I don't really see why the path of choice on the sphere and the path of choice on the ground would always match up
Because the sphere is rolling without slipping, the distance traversed on the plane must equal the distance traversed on the sphere. If their tangent vectors match up at every point, the paths must match up.

>>11804491
it depends what does "roll without slipping" means to you
the way I understand it, there is no "freedom of choice" - once a path on the sphere is selected, it produces a unique path on the plane.

>>11804516
We're talking about topological manifolds, which are extremely intuitive objects with almost no pathology. If you want something to be true about manifolds, it's true. Yes, proper rigor is important when your spaces don't behave locally like Euclidean space. When they do, rigor is a crutch.

>>11804518
Sure, but the situation gets complicated by the fact that a path-intersection on the sphere doesn't necessarily correspond to a path intersection on the plane.

>>11804526
Just take a geodesic then?

Draw a 90-90-90 degree triangle on a sphere.
Roll the sphere around this triangle. This corresponds to rolling along a square on the plane. This proves that you can change the tangent point from x to y, as long as the arc length between x and y is pi/2 (we assume the ball is a unit ball)
Now any two points can be joined by two arcs of length pi/2. So the answer is a yes.

>>11804528
yes, I think I can see the spin on a point construction now

>>11804525
I am not talking about exercise that is being discussed. The intuition that is being used there can be translated into something rigour.

I am talking about the anon that says topologists do not care about rigour, and they are not able to produce the rigour.

>>11804490
>an embedding of S^n into R^k separates it into two components if and only if k = n+1
you honestly think "topologists" prove this by drawing a picture and appealing to intuition ?

>>11804525
>which are extremely intuitive objects with almost no pathology
LOL this is so not true. topological manifolds (without smooth structure) are weird as fuck. or do you think the horned sphere or space-filling curves are intuitive ?

>>11804513
FUCK, that's true.
To be honest, my intuition is screaming at me that I can always rotate it because, if I'm not literally rolling the sphere on another sphere, I can exploit the difference between their holonomies to produce rotations.

I think you can actually straightforwardly argue that, if the set of points is discrete, then that induces a local isometry between the sphere and the surface, which implies rolling the sphere along the sphere.

>>11804542
>>11804545
>>11804550
I've localized the topological nastiness from the problem into an extremely particular and uninteresting situation. Such is the goal of topology. Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

>>11804556
Can anyone second the isometry argument for me?
Assume that the set of points is discrete. Then, there's a smallest path that moves you from one point to another. Take a small enough ball, connect every point to every other point in the ball by geodesics. The corresponding curves on the surface are all isometric, and the map is injective by assumption, so it's locally isometric to the sphere, and thus also a sphere.

>>11804562
You are a fool.

This argument is using rigour, you reduce the situation to a familiar one that you have already worked through the details of.

>>11804580
And this really reads like a post written by someone who doesn't really enjoy topology much.

>>11804562
I accidentally read this as a response to the Sphere problem and lol'd in this establishment.

>>11804556
Yeah, I accepted you can effectively spin the ball on a point.
And certainly, for each path you draw on the sphere, by the act of rolling along the drawn path, there's a corresponding (isometrically mapped) path on the plane. What's also clear is that if you roll and wiggle in one general direction with the ball for 3 days, you'll have a ton of intersection of paths on the ball but none on the plane and the question as I understand it is if we can map each point [math] p\in {\mathbb S}^2 [/math] to the origin [math] (0,0)\in {\mathbb R}^2 [/math] by a rolling motion.

I'd like to know now: If we just consider general triangles in a small disc (say of ratius [math] \pi [/math]) on the plane [math] {\mathbb R}^2 [/math] and roll along those, which points on the sphere can be bring to touch [math] (0,0)\in {\mathbb R}^2 [/math]

>>11804572
It's a bit too much work to follow where you points and paths are if you're no explicit

>>11804539
I think this works

>>11804589
obviously you get some symmetric neighbourhood of the original point of plane
the actual question which requires some computation is how big is it

>>11804586
I love topology. I dislike when normies debase my favourite subject with nonsense claims.

All field in pure mathematics care about rigour. A good analogy is between mathematicians and athletes. Practicing your sport is building intuition. Eating healthy and strength training is rigour.

Athletes enjoy actually playing and practicing their sport. Eating a healthy diet is required to do this and so elite athletes care about it. Nutrition is not the interesting part but its a needed foundation.

Saying that topologist don't care about rigour shows that you are not that far along your mathematical career.

Read. Tao's. Blogpost. On. Post-Rigor.

>>11804541
You don't need to spin on a point though? Just stop the ball and change direction.

>>11804413
Cute question
Nothing else to add

I'm starting to hate topology.

>>11804562
answer the question >>11804545

How does /mg/ feel about stochastic processes?

>>11804545
>>11804612
>you honestly think "topologists" prove this by drawing a picture and appealing to intuition ?
Yes. We've localized the topological nastiness from of the problem into an extremely particular and uninteresting situation. Such is the goal of topology.
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

>>11804606
Well spinning brings you more points on the sphere you be force to touch certain points on the plane.
Alsp
>spining is a good trick

Googled and found this, should have some relevant machinery
https://www.researchgate.net/publication/224626993_Control_of_a_Sphere_Rolling_on_a_Plane_with_Constrained_Rolling_Motion

>>11804609
I wonder how many arguments in topology are secretly lattice theoretical results.

>>11804614
based.
The fact that one at the very least mostly just ever uses Wiener/Brownian processes is sad. That's like doing arithmetic and restricting oneself to the multiples of 17 because the other number are intanglble.

>>11804609
Dumb animeposter.

/sci/ - Balls

>>11804618
>I wonder how many arguments in topology are secretly lattice theoretical results.
Most of the pointless ones at least. Locales and frames are nothing but lattices.

>>11804619
No need to point out the obvious.

>>11804589
>It's a bit too much work to follow where you points and paths are if you're no explicit
Essentially, we consider two discrete equivalence class of points on the sphere which can be rolled into each other, one associated to the point p on the surface, and the other to q.
Now, rolling the sphere from p to q induces an isometry of the sphere that maps one equivalence class to the other.
This means that there's globally a smallest path on the surface which swaps some two points in an equivalence class, and then by taking small enough geodesic paths, we see that we have a covering of the sphere.
There aren't any coverings of the sphere, tho, since it's simply connected.

>>11804629
>something something covering space
I have no idea what possesses me to do this stuff sometimes.

>>11804617
>Yes
why do you act like you know anything about topology (or topologists) when clearly you don't ?

>>11804649
For (You)s.

>>11804629
>>11804640
FUCK, I'm not in the state of mind to be thinking about this stuff, this is way too stupid.

My bad for wasting you lads' time.

>>11804626
My thinking is motivated by the fact that
>Topologies on a finite set X are in one-to-one correspondence with preorders on X.
https://en.wikipedia.org/wiki/Finite_topological_space

There's a fair amount of constructive Hilbert space theory and by extenion on metric induced topology, but the constructive point set topology is exhausting. The question arises if there's actually good math beyond those. Algebraic Geometry is a bit of a mystery case for me, since all the good results and nogo results about numbers should be more or less explicit. I'd like to know if the really funky constructions are necessary.

(
There's also an icky secret that free constructions, such as those related to limits and colimits - that are represented as infinite sequences of sets of higher and higher rank - need the Axiom of Replacement and doesn't even take place in Zermelos first theory or elementary topos theories.
TL;DR this math has quite some demanding logical requirements in the language we currently use.
https://mathoverflow.net/questions/41118/axiom-of-replacement-in-category-theory
)

I am an CS undergrad, will I need both volumes of Stewart's Calculus or just the first one will do?

>>11804659
your lecture notes should be fine

>>11804657
Please don't post my wife here.

>>11804659
What does your syllabus/professor say?

>>11804671
>my wife
I will kill you.

>>11804671
how many fucking wives do you have

>>11804676
>not rejecting your professor entirely on philosophical grounds

>>11804676
I asked the professor (calculus 1) what book should we use and he said this one, I just want to know if I'll need vol. 2 for the other calculus classes

How does /mg/ keep track of their notes? I like loose leaf paper but find it hard to keep track of.

>>11804657
Yes and those finite spaces are Kolmogorov iff the preorder is partial. You get all sorts of strange paths between x and y, like constantly x for t < 1 and y at t=1. It can be applied to the subgroup lattices quite nicely. Or, so I thought. For some reason all this topology posting is making it seem unpleasant... Any way, I suggest you take a look at for example Barmak's writings on the matter, assuming you are interested in it & haven't already done so
https://www.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf
Oh and I remember someone mentioning that you don't need the axiom of choice for Tikhonov's theorem if you move from spaces to locales years ago in one of these threads. Basically, you could have a constructive version of that theorem mayhaps. Sounds like something you'd be interested in looking at?

>>11804698
>>11801703

>>11804698
You simply can't beat pencil and notebook.
I'll usually rewrite my notes more formally in a 'master book' where I have my notes from all subjects.

>>11804657
Finite topological spaces are pretty much equivalent to doing stuff on graphs and posets. So it's essentially combinatorics - and as everyone knows, combinatorics is the most valuable field of maths. So maybe topology isn't as bad.

>>11804698
I write on loose paper, then use a clip.

>>11804704
I'm mostly interested in Lie group theory and hope the algebraic structure pins all the good theorems down.
Thanks very much for the link, looks based, pointed and compact.

>>11804707
>master book
Highly problematic, see >>11801718

>>11804707
For undegrad this worked for me, but in grad school and getting into research the volume of pages i am going through makes these seem unfeasible. I don't want to have to rewrite notes.

>>11804697
Vol 2 is multivariable? If so yeah, probably. I think there's also a huge book by him that has everything in one tome, but that may be cumbersome.
In any case, ask a Calc II professor or read the syllabus for that class.

>>11804707
Notebook as in the spiraly ones or like composition

>>11804724
Spiral, obviously.

>>11804717
Enjoy!

>>11804717
Any good introductory books on Lie theory that you can recommend, anon?

>>11803840
Vinland Saga. It's about Vikings. It is somewhat historically accurate when it comes to the bigger picture but the story itself is fiction of course. I really liked it, would recommend if you have free time during the quarantine.

I sharpen both ends of the pencil

>>11804744

>>11804728
That's what I thought yeah. Do you not use pens? Or graphed notebooks?

On a related note, I never understood why americans use composition books. Is it just that the desks are tiny?

>>11804748
I don't use pens or graphed notebooks, but the latter is something I've often thought about using. It's just that I'm so used to using lined.

As for your second point, I've never understood the appeal of comp books either and I'm American. The fact that you can't actually fold the paper over and have it stay out of your way is the biggest pain.

>>11804733
I think "Lie Groups, Lie Algebras, and - Hall, Brian" does a fair job. Depends on what you want to get at. I see it pops up as pdf on page 1 if you google it. There's also an Errata online.

>>11804763
Thanks, I'll check it out. I just want to get a sensible overview of the topic, nothing too in depth.

>>11804716
>I write on loose paper, then use a clip.
This is what I do. I can't stand writing math on lined paper since math writing doesn't really fit into lines well and any diagram looks like shit, but at least where I live it's completely fucking impossible to find bound notebooks of blank paper that aren't $5 a piece. Muji sells nice ones but in order to get them cheap you have to be physically at a Muji, which I'm not often.

>>11804763
Hall's book is (IMO) the best _introductory_ text that's been written for the subject by quite a bit. Lie theory (especially of algebras) unfortunately suffers from not really having many actually good textbooks. The situation for Lie algebras is so sad people actually still _read_ (not refer to, but actually read) Bourbaki for some things.

What's it called if you integrate a function but every time you integrate it more integrals show up and it keeps getting worse and worse forever and ever?

>>11804763
>>11804776
Thoughts on Knapp's book?

>>11804779
per partes but differentiating what you were supposed to integrate and vice versa

>>11804744
I do this too, but I always end up writing with only one end and chewing the other end. The purpose of sharpening both ends was to discourage me from eating my pencils, but fuckit.

Is Lie theory the most aesthetically pleasing branch of math?

>>11804413
Smells like Cartan geometry. Check out Appendix B of Sharpe's "Differential Geometry".

>>11804780
Knapp's writing is very easy to follow, and the second half is stuff that's not in many other textbooks (which is why it's called "Beyond an Introduction" even though it contains an entire intro course in the first half) but there seems to be a trend with all of Knapp's books where he rambles a lot and rather than deciding what to include he simply tosses in everything he can think of.
His Lie theory book isn't 800 pages because it needed to be, it's 800 pages because every book he's ever written is 800 fucking pages long. He can't do anything shorter.

>>11804779
I would call that calculus. Integrating even once adds the integral sign and pain.

>>11804779
Stupidity.

>>11804791
Visually? Sure. Galois theory otherwise.

>>11804821
imagine if evariste didnt gunduelicide himself to death

>>11804827
Imagine if Abel didn't die of tuberculosis.

>>11804843
Imagine if Ramanujan didn't shit himself to death

how is the sum of all natural numbers -1/12?
if you consider that all natty numbers to be sum of ones. 1+2+3+4... = 1+(1+1)+(1+1+1)+(1+1+1+1).... then you can say that it's just 1*x where x is infinity (because you are just adding ones) which means the answer is infinity. i know how they came to the answer -1/12 but it doesn't make sense

>>11804851
Imagine if Teichmüller didn't go get himself killed in the war.

>>11804851
Imagine if anon didn't become 24.

Are there many applications of homological algebra outside of algebraic geometry/topology?
Most modern algebra textbooks make you do all the boring-ass diagram lemmas and exact sequences and all that, but I've never seen them actually used to _do_ anything and whenever somebody asks it's always fluffed off with "oh well you'll need it in topology".

>>11804865
what if galois offed himself because he knew he had barely 3 years left?

>>11804876
Doesn't hold water, if that was the reason he would have waited until his 24th birthday

>>11804863
>how is the sum of all natural numbers -1/12?
It's not.
>which means the answer is infinity
Yes. The sum is divergent to infinity.
>i know how they came to the answer -1/12 but it doesn't make sense
They prove that if the sum converges to some finite number, then that finite number is -1/12. But it doesn't (under usual assumptions).

>>11804876
kek

>>11804863
My general pitch of sanity is that

[math] \dfrac {1} { \log(z)} = - \dfrac {1} {1-z} + \dfrac{1}{2} - \dfrac{1}{12} (z-1) + {\mathcal O}((z-1)^2) [/math]

[math] \implies [/math]

[math] \sum_{n=0}^\infty n \, z^n - \dfrac {1} { \log(z)^2} = - \dfrac{1}{12} + {\mathcal O}((z-1)) [/math]

and a hint for why a relation between [math] \log [/math] and [math] x_n := n [/math] exist is the periodicity of the or related functions w.r.t. that sequence in the imaginary axis direction.

Here's some properties of summation methods that you might want to check your compare your blockwise summation against:
https://en.wikipedia.org/wiki/Divergent_series#Properties_of_summation_methods

any good websites to refresh memory on high school-college math? i can't write notes, i am too autistic to ask someone for their notes and i don't want to read 200 pages of explanation that i already know when all i want is a brief summary to refresh memory

>>11804698
loose paper; it's not half bad t.b.h, just keep folders

>>11804925
Khan academy?

>>11804698
for research: loose paper + numbering the pages + "themed" symbols around the numbers, like a circle, triangle, box
for learning: cheapest paper I can find, will be thrown away anyway afterwards

>>11804918
Anon, for the love of god, stop posting pictures of my wife!

>>11805135
Whacha gonna do white boy

>>11804925
patrickJMT

>Statistics isn't mat-

>>11804873
Not sure what kind of answer you're looking for. HA is like point-set topology. It's pretty dry and not really studied for its own sake, but any working mathematician has to be familiar with its language and basic results.

>>11804873
key word: cohomology. If you're deep into it, you're gonna stumble upon it

>>11805192
Wow, I've never seen a large flowchart before. I'm convinced now!

>>11805192
>posts a bunch of meme distributions which statisticians just memorize and don't understand
-h

Retarded question: Assume 10 large numbers are randomly generated. I want them to be in the range of 1-50, as such I mod the number by 50. Is this considered mutation? Is the randomness of these numbers preserved?

>>11805219
No, because 50 isn't prime

>>11805227
So if I mod by 53 it is preserved? And just disregard the 3 irrelevant numbers should they arise?

>>11805219
You literally can't randomly generate unbounded integers in a way that every integer has equal chance to be picked to begin with.
But if you randomly generate an integer between 1 and 50n you can mod out 50.

>>11805230
14 is prime. Mod by 14, then multiply by 6, then disregard the rest of the numbers

>>11805238
Do not post Megumin in this general.

>>11804876
>galois offed himself
He got himself killed over a thot.
At that point his math ability was zero anyways.

>>11804744
I sharpen both ends of my mechanical pencil.

>>11805294
oi m8 u got a loicense for that assault weapon

>>11805294
>mechanical pencil

>>11805298
Huh?
I only started to use mechanical pencils in Uni.

Also, why are you posting the rules of that weird, transphobic 4chan site? This is 4channel...

>>11804873
Take two topological spaces. We want to find out if they are homotopy equivalent or not. Take one of the spaces. We can define a sequence of abelian groups for every [math]n[/math] natural consisting of formal sums of [math]n[/math]-dimensional "patches" on the space. There are two types of patches - the ones that consist of genuine, [math]n[/math]-dimensional structures on the space - ie, the real deal, and the ones that actually are just the boundary of a larger [math]n+1[/math] dimensional patch. The latter isn't useful to studying the [math]n[/math]-dimensional structure, so we quotient out of the formal sums those that consist of these [math]n+1[/math] dimensional boundaries. For example, on the circle, the points consist of 0-dimensional patches, but any two of these patches (read: points) are the boundary of an arc. Therefore, they are quotiented out of the group, because essentially, collapsing this arc still leaves us with a circle - the information of more than one point on the circle is "useless" up to homotopy.

So now that you have a formal group of sums for each set of [math]n[/math]-dimensional patches, quotiented out by the "useless" information up to homotopy, you get what's called the degree [math]n[/math] homology of the space. Spaces that are homotopy equivalent will have the same homology, so we can use it to differentiate spaces. There are special types of sequences that arise from homological algebra that help to actually compute the homology of a space - for example, the Snake lemma gives us the long exact sequence of homology, which can be used to reduce the problem to computing the homology of simpler subspaces. Further techniques can be used to give us certain exact sequences that reduce the problem to a smaller dimensional case - for example, through the use of the [math]\text{Tor}[/math] derived functors, we achieve a so called "universal coefficient theorem" that is very practical.

>>11805315
Mechanical pencils are peak zoomer.

>>11805238
>>11805234
Thanks, quick follow up, a 10000bit hexadecimal string is randomly generated, are any substrings of this string also random?

>>11805318
>"oh well you'll need it in topology"

>>11805376
Well what exactly does he expect, that I'd write out an entire textbook on the applications of homological algebra?

>>11805384
something tells me he expects applications outside of topology which is quite literally the opposite of your post

>>11805384
>that I'd write out an entire textbook on the applications of homological algebra?
Yes.

>>11805405
oh i missed the "outside of topology" part, thought he just meant he was tired of seeing, "you'll see it later" without any actual later

>>11800085
To who ever is going through college or finished it, did you ever cheat in your math class? I'm taking an online course this summer and the temptation is just to much. I can just set up one tiny sticky note with formulas and nobody would know. Fuck guys. I'm so conflicted.

>>11805446
Just do a million examples so you can derive the formulas from scratch. If you cheat you will be exposed as a fraud sooner or later.

>>11805446
>did you ever cheat in your math class?
No.

>>11805446
>sticky note with formulas
Sounds like whatever class you're taking isn't worth not cheating in.

>>11805476
Was about to say the same thing.
What's the last class where you genuinely rely on rote memorization of formulas? ODEs?

>>11805201
I suppose there are two things that would answer what I asked.
What I intended was that for something supposedly "not studied for its own sake", I haven't yet ever seen it used to do anything other than prove more stuff in the homological algebra chapter, and the only applications I know about are "well you need homological algebra to do homology theory". So I guess I'm asking if it's JUST a topological tool or if it has some applications to algebra proper, some non-homological question best answered by homological techniques.

The other thing which isn't what I intended but would also be a nice answer is some cute application of it to something elementary. Your comparison to point-set topology reminds me of the topological proof of the infinitude of primes; not really the "purpose" of topology, but it's something fun you can do with it.

>>11805318
Other than responding with two paragraphs of topology to an "other than topology" question this is basically what I was hoping for, i.e. "here's somewhere we use diagram lemmas/derived functor bullshit to do something interesting".

>>11804873
You can study rings via the categories of modules using homological methods.

>>11805491
Oh, and groups of course. Group cohomology is the choice of champions.

>>11805493
What's the choice of losers?

>>11805503
Sorry I mistyped champignon.

>>11805503
Leftism.

>>11805503
Any applied math

>>11805503
Logic.

>>11805503
Linguistics, a.k.a. category theory.

>>11805516
Silence, lib.

>>11805503
Honestly, abstract non-sense for the sake of being so cool and abstract and smart. If you think abstraction is cool per se, I hope you grow past that stage soon.

>>11805524
Applied math is the thinking man's choice.

>>11805545
This.

>>11797108
I read through the Feferman paper today.

At one point he points to the work of the late
https://en.wikipedia.org/wiki/Edward_Nelson
who also published a book on Predicative Arithmetic
https://web.math.princeton.edu/~nelson/books/pa.pdf
which strikes me as quite Wildbergerian (except that guy would reject formalism).

Tell me what you think of the argument in the intro (pic related)

Here's some more or less informal notes, on more than just the paper (WIP)
https://gist.github.com/Nikolaj-K/aae1f4bd582e60e6b7e5b5431fee054c

>>11805573
tl;dr of it in my own words would be:

Predicative Arithmetic critique of Induction:
Numbers are defined by axioms involving the axiom scheme of induction:
"The natural numbers are those objects which fulfill all induction schemes."

In the scheme, a predicate instance may be one involving a universal quantifier
over numbers (which are only defined in terms of the predicates they fulfill), which is circular

See also
https://en.wikipedia.org/wiki/Vicious_circle_principle
In the notes above I give some more common such cases, although few go as far as to use the line of argument against Peano/Heytin arithmetic

The PA schema in pic related for reference.
Although for some reason none of the Wikipedia pages wants to actually post the Schema it seems, all pages give me the second order [math] \forall P [/math] version

Category theory noob here, I ran into this statement in Maclane's book and I don't quite understand what he means for a functor to be isomorphic to something. Can anyone help me understand this?

>>11805699
Read the definition of a natural transformation, which should be one of the first things in the book

>>11805699
topology was established to study continuous functions

category theory was established to study natural transformations

>>11805735
>to study natural transformations
But why?

>>11805727
do I therefore consider it in in the category with functors as objects and natural transformations as my arrows and get a natural transformation (isomorphism) from the category of D homs with a fixed object r in the domains into the above category?

if you feel integers and their relations are interesting, you can study number theory
studying even the simplest diophantine equations leads to complicated and fascinating things, such as galois theory, theory of elliptic curves, this in turn connects to algebraic geometry, literally most complicated math currently in existence

if you are interested in combinatorics, there is a lot of research going on, literally a layman can understand the problems there

if you have a knack for geometric things, you may choose differential geometry or algebraic topology

my point is, with all those appealing stuff in existence, i cannot understand what goes on in a mind of an undergrad who chooses to study category theory

>>11805748
A topological example: take any space X and apply the nth homotopy and nth (singular) homology functor (integer coefficients for both). Suppose you know something about one, how could you relate that somehow to the other? Well, it turns out that there is the Hurewicz homomorphism from the nth homotopy group to the nth homology group (for any n>1), and if your space happens to be (n-1)-connected, this Hurewicz morphism is an isomorphism for all indices up to n. It can be shown that this is actually a natural transformation between those functors, and provides an example of when they are used. There are, as you most likely have guessed, more than just this natural transformation out there, and so people realised that it is good to have a theory of those so that we can just refer to this or that property of nat transformations when encountering one.

>>11805791
It is in the category of functors from a category D to the category of sets, and their natural transformations. Essentially, you just want natural transformations going both ways and composing to identity, just like any isomorphism.

Remember to stay hydrated, /mg/.

>>11805814
Because its the synthesis of all mathematics.

>>11805319
>Mechanical pencils are peak zoomer.
Why?
Mine is from a traditional Manufacturer who produces them since 1928, the specific model has existed since 1989.

Zoomer use fucking ballpoint pens like the degenerate Freaks they are.

>>11805699
Have you tried googling, my friend?

https://en.wikipedia.org/wiki/Representable_functor

Functors can be isomorphic, e.g. in a Cartesian closed category (e.g. a category of sets) consider the adjunction between the function spaces
[math] X\times Y /to Z [/math]
and
[math] X \to Z^Y [/math]
realized by curring.
E.g. [math] f(x,y) := x\,\sin(3y) [/math] being mapped to [math] g(x) := y \mapsto x\,\sin(3y) [/math]

The same adjunction is given in logic by
[math] (P \land Q) \implies R [/math]
having the same prove value as
[math] P \implies (Q \implies R) [/math]
(if P and Q together imply R, then also: P implies that Q implies R)

Or on a completely decategorified level in arithmetic
[math] z^{x \cdot y} [/math]
is the same number as
[math] (z^y)^x [/math]

In category theory, you abstract away from all of those, and realize that in the right categories
[math] \text{Hom}(-\times Y, -) [/math]
is the same functor (by a natural iso) as
[math] \text{Hom}(-, -^Y) [/math]

which is admittedly a bit wild.
Think of the two as "one map" but then with an image gliding between different representations. (Although I guess some people would argue that in some categories you don't want to think on the object level)

The representable functor are those iso to something which maps from a given object into an internalized homset.

>>11805814
I don't disagree that all of this is nice, but I also don't see why that would one discourage from that latter?

>>11805821
Actually, I was thinking of starting a fast. Salt and water for a few days.

[math] X\times Y \to Z [/math]

>>11805446
>did you ever cheat in your math class?
No.

Not only was there no reasonable way to (most of my exams are oral anyway), I also would never do it, even if possible. Besides obvious moral considerations, after all *this was my choice*, the risk is far too great. Getting caught means getting expelled.

>I can just set up one tiny sticky note with formulas and nobody would know. Fuck guys. I'm so conflicted.
If you can not fit the contents of that stick note into your head, then you might as well just drop out.

Does the category of categories contain itself?

>>11805866
just follow the identity arrow

>>11805866
If Epstein didn't kill himself, then who did kill him?

>>11805857
>Actually, I was thinking of starting a fast. Salt and water for a few days.
Same. I've gained weight during this lock down. Let's do 72 hours or... 3 times... 3 times you know what...

>>11805903
can you stop fagging up the thread
thanks

>>11805920
No.

[math]f(\varphi)=cos^2(\varphi)+sin^2(\varphi)\\f'(\varphi)=-2cos(\varphi)sin(\varphi)+2sin(\varphi)cos(\varphi)=0\\f(\varphi)=f(0)=1^2+0^2=1[/math]

>>11805903
> Let's do 72 hours or... 3 times... 3 times you know what...
No I don't.
I only know the snake juice meme, but I like it. Haven't had a day without eating in 2 years.

>>11805699
Also, I suppose the adjunction actually had a representable functor baked into it and that might be the simples example:
For given objects [math] Y [/math] and [math] Z [/math], the Set valued functor "[math] \text{Hom}(-\times Y, Z) [/math]" mapping any object [math] X [/math] to the set [math] \text{Hom}(X\times Y, Z) [/math] (and arrow mapping you can find out yourself), is representable.
Namely, this functor is isomorphic to the functor "[math] \text{Hom}(-, Z^Y) [/math]" mapping "[math] X [/math]" to [math] \text{Hom}(X, Z^Y) [/math] (because that image is iso to the previous one).
Makes sense?

>>11805946
man you're good at maths
you have a nice macbook with an anime girl
you've made it respect

>>11805961
Anime is inherently mathematical.

>>11805961
>category autism
>"good at maths"

>>11805946
>Mac
>Kraut
>Looking at Anime
>Posting disgusting thots
Ach Anon ... ein wenig Selbstrespekt kann man ja wohl haben.

>>11805966
nah he posts all kinds of math
big respect

>>11805971
>he

>>11805971
>he

>>11805975
>>11805985
>she

>>11805970
>Kraut
no
>Posting disgusting thots
it's a German girl, actually
>Looking at Anime
not really, but I accept it as part of the Zeitgeist.
This is me posting cute faux anime:

https://youtu.be/EbgAu_X2mm4

>>11805990
I thought that was a picture of Harold, for a second.

>>11805990
>no
Jaja, bist aus den Bergen oder was?
>it's a German girl, actually
oof...
>not really
I have photographic evidence of you committing that sin!

>https://youtu.be/EbgAu_X2mm4
Uhm, based!

>>11805489
>if it has some applications to algebra proper, some non-homological question best answered by homological techniques.

You can discuss group cohomology purely in terms of the group operation if you'd like, without referring to classifying spaces. Also check out topological combinatorics where homological arguments are used to prove results about discrete objects like graphs.

>>11805489
I see, I think you might be confusing homological algebra and homology theories. My answer was mainly about the former.

Homological algebra is basically suped up linear algebra. It's concerned with computing information about abelian groups/modules and homomorphisms thereof in terms of kernels and cokernels. It's not at all apparent that this approach and things like the Snake Lemma, 5 Lemma, etc. are useful if you're studying HA in a vacuum. But it turns out that nowadays things like chain complexes and exact sequences show up everywhere in math, so it's necessary to have a theory on manipulating and computing with them. This is how you should think about HA in general, and really it's better to learn something like singular homology (see below) beforehand, or at least concurrently, so you can appreciate why these results were needed.

Homology theories are gadgets in topology and the main reason people first became interested in HA. The most basic example is simplicial/singular homology which >>11805318 gives a quick rundown on. To put it another way, singular homology associates a space [math]X[/math] to an abelian group [math]H_*(X)[/math], and the theory of algebraic topology is concerned with extracting information about the original space from the group [math]H_*(X)[/math]. It turns out that there are actually many different homology theories, but they all share the same idea of turning a topological problem into a HA problem.

As for a cute application, maybe this will make you happy: Once you know the homology groups of spheres, you can easily prove that [math]\mathbb{R}^m \not\cong \mathbb{R}^n[/math] when [math]m \neq n[/math]. And computing the homology of spheres can be done inductively using elementary HA and a long exact sequence.

>>11806038
>question asks for applications outside of topology
>write a big post about topology
>"whoops missed that part" when it's pointed out
>proceed to write ANOTHER big post about topology
what did she mean by this

>>11806078
>she

>>11806038
>proving that R^n isn't homeomorphic to R^m by comparing the homology of their one-point compactifications
I hate it.

>>11805920
Ok.

>>11805941
Most true.

>>11805946
Good luck with any number of days anyway.

>>11806084
Oh no no! Instead of doing that, try the sequence of inclusions [math](\mathbb{R}^n\setminus \{ 0\}, \emptyset) \to (\mathbb{R}^n, \emptyset) \to (\mathbb{R}^n, \mathbb{R}^n \setminus \{ 0\})[/math] and use reduced homology.

>>11800085
Anybody know how I can prove an exclusive disjunction? I’m in bridge to abstract mathematics course, we are constructing all of math from a few axioms, and doing homework exercises to practice proof writing in two column format. I have to show that for any reals, a,b,
a = b ⊕ a < b ⊕ b < a

I have gotten to here:

b - a = 0 ⊕ b - a ∈ R+ ⊕ b - a R-

I can show
b - a = 0 ⊕ b - a ∈ R+ ≡ a = b ⊕ a < b

So then I end up with

a = b ⊕ a < b ⊕ b - a ∈ R-

I can prove b < a from b - a ∈ R-,
but am I a little confused on what next. So far I have used substitution only after showing two expressions are equivalent. However, for b - a ∈ R-, I don’t have equivalence. I can prove b < a if b - a ∈ R-, but this wouldn’t be sufficient to state equivalence between the two, would it?

Basically, I am stuck at a = b ⊕ a < b ⊕ b - a ∈ R- , and don’t know how I will proceed.

>>11806078
Not the same girl

>>11806122
Why are there so many femanons in /mg/?

>>11805489
Some of the more advanced problems of group theory and representation theory are classification problems, where using homological algebra seems to be the only technique actually capable of giving some sort of halfway satisfactory result. As an example, take a look at the classification of blocks of finite groups. Classification up to isomorphism is pretty much out of the window from the start, so people have been introducing weaker and weaker forms of classification. First there was Morita equivalence, which was basically equivalence of categories, but that also seems to give too many distinct blocks. Most of the newer attempts use homological equivalences, like derived equivalence, which seems to have a better chance of yielding a meaningful classification, see for example Broue's abelian defect group conjecture.

>>11806110
Why would you use a sequence of reduced inclusions when [math]S^n[/math] is homotopy equivalent to [math]\mathbb{R^{n+1} - \{a \}[/math] for any point?

>>11806160
Because homological algebra and a categorically contaminated mindset makes you a worse mathematician, unironically. Not him btw.

>>11806160
Because I can. Literally no other reason. The way you would do it is unironically the best I can think of, then there is also the method of assuming they are homeomorphic and then removing a point from one and the corresponding point from the other, then you get supposedly homeomorphic spheres with non-isomorphic homologies, oopsie. The way I posted is probably the dumbest way to do it, and I like thinking about dumb ways to do stuff.

>>11806173
This could be true. It is, for sure, if one is pulled into it by some reddit tier "omg abstract = for smart ppl ^___^" mindset.

>>11806131
This is a nice example, pretty much exactly what I was trying to ask for.
The Wikipedia article rattles off representation theory as one of the areas homological methods are used but there are no examples there, I didn't know it was seriously needed in modular representation theory. Thanks.

>>11806184
Just to be clear anon, my response "cats=dumb" wasn't just meant as a mean insult. But I unironically hold the opionion that many of my collegues who slipped into higher structure stuff (because higher=better amirite lololololol) are wasting away their talent. Right now it's increadibly easy to get papers accepted in higher structure/cats in """""""""""physics""""""""", but mark my words, forming a self-proclaimed elite circlejerk that already at this point no other field in math can either understand nor use will lead to demise, fast.
Also, it's just not fun. Luckily you got some turboautists on the n-lab trying to actually unravel the wankery and interpreting higher structures in physics in a meaningful way but fuck me that's far too late and also the wrong direction.
This needed to be said, fuck the current school of abstraction=good starting point-point of view in math. And again, don't get me wrong, abstraction CAN be good, but when starting out with it will cripple you.

Reminder that in 100 years when the rest of mathematics is drowning in derived lax quasifunctors on locally categorical braided infinimonoidal bicategories, us combinatorichads will be the last bastion of mathematics that actually means anything.

>>11804741
>It is somewhat historically accurate
HAHAHAHAHAHAH TELL ME THAT'S A FUCKING JOKE PLEASE

>>11806265
>animefag can't tell the difference between wild fantasy and reality
do you really think there's a chance he's joking

>>11806234
I agree with the sentiment to an extent, although I wouldn't draw conclusions from the value judgement is a bit of. My argument being that all subjects are hard and seemingly slow progressing. The abstract one isn't singled out as inaccessible.
Have you tried getting into the structures in stochastic processes that fields medals are handed out for, or the seemingly anti-relevant results in partial differential equations theory?

At least category theory is a bit hip and the academics in the field stream their lectures and you can chat live with pic related or Schulman and all those guys, if you like.

>>11806255
based.

Though what about
https://en.wikipedia.org/wiki/Combinatorial_species
?

>>11806255
Should I take a combinatorics class?

>>11806278
The problem with abstract nonsense is not that it cannot be useful, but how it is tought and used in research. A good course in homological algebra must imo always present geometrical intuition and examples before introducing a very abstract concept. And I just do not see any application of higher structures, even though it's constantly claimed that it is "useful" in, say, stringtheory. A more honest reason would be it's useful to pull in fundings because everyone says it's great.
I didn't know about the streams in cats stuff, most people make their lectures not publicly available so it's definitely nice. Will tune it, why not. I mean, I could be completely wrong...

>the virgin [math]\tau [/math]
>the chad [math]\frac{h}{ \hbar}[/math]

>>11806278
Species theory is a fat meme. It's not actually good for anything, it's a prototypical example of category autists spending hundreds of pages establishing the Right Framework™ to think about problems people already know how to do but in the Right Way™.

>>11806234
>my response "cats=dumb" wasn't just meant as a mean insult
It doesn't actually matter whether you intended it as an insult or not, I do agree with your post to some extent. It may be this physics aspect that bothers you with (or in, whichever preposition) homological algebra in a way it doesn't bother a wnb homotopy theorist like me, but
>my collegues who slipped into higher structure stuff (because higher=better amirite lololololol) are wasting away their talent
this is easily relatable. I can't say for sure whether this is true or false, but I have not personally seen much use for any of the n-categories for n>2. Maybe it helps with some stable stuff, but I'm too unstable to know that.
>forming a self-proclaimed elite circlejerk that already at this point no other field in math can either understand nor use will lead to demise
I have attended several seminars/conferences and all but one had a nice atmosphere. If I say the genres were AT, RT, GT and one purely CT, I'll let you guess which one had the nasty vibes in the air.
>abstraction CAN be good, but when starting out with it will cripple you
A strong agree. If only there would be more time spent on having students develop some intuition about things instead of just rushing through a list of topics and make the students check some minor details like, say, the exactness of the Mayer-Vietoris sequence. Sure, it's good to know how to deal with the abstract stuff, but I would rather have known that [math]\Omega X*\Omega Y[/math] is a homotopy fibre of the inclusion [math]X\vee Y\to X\times Y[/math] years ago instead of learning that in late April or early May this year. No intuition what so ever. Only ability perform some dumb diagram chases, which is obviously also partly my fault for not learning it myself but partly the Zeitgeist's.

>>11806311
This.

[math]F(x)=\sum_{n\geq0}CardF[n]\frac{x^n}{n!}[/math]

>>11806333
begone frog

>>11806322
ic

>>11806270
Sorry bro. 4channel is an anime website.

>>11806404

>>11806324
Interesting response. In fact, I'm teaching a course this semester and have been trying to follow the rule "if you cannot sketch an example, change how to introduce it". I sometimes got feedback like "I understand the geometric intuition, but not the formulas." Could be worse I guess. But yeah, maybe some day I'll do a course in some sort of homological algebra focused stuff and watch the purists scream in agony as I'll force them to sketch EVERYTHING in the exercises or otherwise they won't get into the exam hah!

>>11806427
>if you cannot sketch an example, change how to introduce it
I think this is a good rule. Pictures are extremely important for developing the intuition and should be used way more often.
>some day I'll do a course in some sort of homological algebra focused stuff and watch the purists scream in agony as I'll force them to sketch EVERYTHING in the exercises or otherwise they won't get into the exam hah!
Can you actually do that somehow? Some minimum number of exercise points which you would only give to the artists? Anyhow, it would be good if you showed your students how the space itself determines what happens to the cup products and when a cohomology operation is forced to vanish et cetera. They would most likely appreciate it a lot. For some reason I want to call you Greg.

Anyway, nighty night /mg/

>>11806505
Night, lad.

>>11806505
>Greg
nice name, sounds like grog, but not mine :^) nighty-night anon

You do actively read and study philosophy, right, /mg/?

>>11806540
for me, its black science man

>>11805545
>I hope you grow past that stage soon
I know I did! There was a time where I wanted to be a category theorists just because it's 'le hardest math field' kek

>>11805946
You know some psycho can discover your location and identity just from that photo, right? Stop posting personal shit online, especially on 4chan

>>11805900
applied mathematicians

>>11806745
4chan scrubs exif and there's nothing really that sensitive there

>>11806760
>sensitive there
That's what you think, there was a hacker who posted a picture of his foot here once, FBI found him the next day.

>>11806757
This.

>>11806764
That's because 4chan gives info to the FBI at request

Hello agent Johnson
please can you help me solve my homework
n is a positive whole number
give all integer solutions for x^3+x+n=0