/mg/ maths general

[Anonymous No.12516628]

If your numbers aren't real just call them ideal edition
Talk maths
Formerly >>12505939

[Anonymous No.12516632]

>>12516628
>actually completed the K3 trio
Blessed OP.

[Anonymous No.12516648]

why do we let the paydoes, i mean anime posters, flood mathematics threads?

[Anonymous No.12516743]

>>12516648
ummm because anime posters are the best mathematicians? and also because anime trannies are really really hot?

[Anonymous No.12516869]

>>12516648
It can't be helped

[Anonymous No.12516880]

[Anonymous No.12517081]

>>12515018
I at least try read most posts or skim a bit
>>12515761
Fun

[Anonymous No.12517111]

I'm not expert, but lately I see lots of people talking about topoi and condensed sets. I don't understand a thing, what is it about? Some new unifying theory? Is something in algebraic geometry? And what's all the fuss about?

[Anonymous No.12517130]

>tfw poor working memory
how do I improve this? this has always been the source of my struggles with math

[Anonymous No.12517135]

>>12517130
I have that as well. Sucks really. I don't know if it's fixable.
Hearing about people like Ramanujan working on a slate they had to keep scrubbing over and over workling on a single problem until they got a result blows my mind.

[Anonymous No.12517147]

>>12517135
Indians have really good memory. It's why they become doctors and memorize shit like how to spell words and the digits of pi.

[Anonymous No.12517202]

What is the logical flaw in this statement 1.1.1 and its proof?

[Anonymous No.12517220]

https://i.4cdn.org/wsg/1609180928115.webm

[Anonymous No.12517252]

>>12517220
topkek

[Anonymous No.12517279]

>>12517220
Hah. I still love the Hitler Learns Topology video

[Anonymous No.12517280]

>>12517220
unironically true

[Anonymous No.12517285]

>>12517081
Very much fun.

>>12517111
So, toposes are basically generalisations of category of sets (and functions) if you want to approach it from the direction of logic, and then you get to do all sorts of logic stuff in them with their internal logics (which are usually non-classical). This allows one to formulate pre-existing theories in a new way using the language of topos theory, and then generalise those by replacing the underlying sets with objects in other toposes (like one can replace the underlying set of a group with a underlying space, smooth manifold etc.) and so on. According to my intuition, this would allow logicians to break weaker theories with a suitable choice of topos, but I don't know if they actually do that.

Alternatively, one can approach from the direction of sheaves. Given any topological space X, we can take its topology, order it by inclusion and consider that poset as a category. This turns any presheaf over X into a contravariant functor from the (category of the) topology poset. This, as well as the sheafification of a presheaf, generalises from topologies to small categories when one equips the category with a Grothendieck topology (thus making it into a site), and then takes the contravariant functors from that category to the category of sets. The Grothendieck topology will now allow us to sheafify those functors and we will obtain the category of sheaves over the given site, called a Grothendieck topos. These are a special case of toposes and generalise sheaves over a space.

If those approaches seem even remotely interesting, then please:
(1) google the jargon to find out if you are actually interested;
(2) ask additional questions from someone who actually knows the stuff instead of me;
(3) start studying that stuff.

>>12517202
To me, it looks like the reader does not know that any real numbers exist, so the problem is the existence of any such x. Assume there are no real numbers, then a real number x will exist by 1.1.1.

[Anonymous No.12517322]

>>12517111
>additional
Condensed sets are also a categorical abstraction from certain nicely behaved topological spaces and Schulzboi thanks that this is indeed more relevant than the standard pointset topology approach to them.
He uses this to cast measure theory is a very function/arrow heavy language and aims at making a lot of functional analysis algebraic.

[Anonymous No.12517343]

very ableist to use schizo as an insult

[Anonymous No.12517353]

>>12517285
>>12517322
my greatest fear is that the optimal way of doing mathematics is writing 5000 pages of completely abstract stuff and then you write 10 pages of inserting specific objects into the abstract theorems and you obtain all the powerful theorems
essentially this will lead to less and less people understanding everything, and more and more people getting filtered from math

[Anonymous No.12517378]

>>12517285
thanks anon, I'll google a lot of jargon then

[Anonymous No.12517410]

>>12517353
That is quite worrisome, yes. I'm usually triggered already by close nlab is to that every now and then.
>google a concept
>click on the nlab link
>find out that your thing is actually a cogroup object in some localisation of an [math](\infty, 1) [/math]-topos
No intuition provided.

>>12517378
Also, I recommend you check signora Caramello's videos out. I like her accent a lot. https://www.youtube.com/watch?v=MNuaHchtrUw

[Anonymous No.12517490]

>>12517322
different anon here
I don't know shit aobut cat theory, but I know something about functional analysis and for my a claim that somehow you could algebralize it seems highly improbable. Some regular parts when you always do the same tricks? Sore. The whole thing? I don't see a reason why it would be possible.

[Anonymous No.12517621]

>>12517490
In my experience, every time these categorists claim to have algebraicized some analytic subject it's nothing that analysts would actually find helpful for the kinds of questions they're interested in.

[Anonymous No.12517651]

>>12517621
That's my impression too, for example :
>we have rephrased interpolation in the language of category theory
> we actually just rewritten Maricinkiewicz interpolation theorem in a fancy way
Thaths nice, but can your theory provide the weak estimates for the endpoint espaces?
>haha, no you have to know them apriori
Ok, does your theory say anything about interpolation in non-symetric function spaces?
>haha, w-what?

[Anonymous No.12517696]

Occasional reminder that there are large numbers of boomers out there who are completely mentally incapable of thinking in foundational terms other than set theory, and that they will propagate this disease onto the fresh minds that they have to teach, and all of this protects us from categoryshitters, typeshitters and various other foundation proposals I don't care about.
>>12517220
Nice.

[Anonymous No.12517722]

>>12517147
I'm 1/4 Indian, where's my good memory?

[Anonymous No.12517728]

Wolfram tells me that [math]e^{ e^{|x|} - e^{|y|} }[/math] is nowhere differentiable, but it seems like it's doable. What am I in for?

[Anonymous No.12517744]

>>12517728
Just [math]e^{|x|}-e^{|y|}[/math] is nowhere differentiable, so good luck.

[Anonymous No.12517754]

>>12517728
Bullshit, its differentiable for any point (x,y) such that neither x or y is 0.

[Anonymous No.12517772]

>>12516628
Sqrt(cos(i)/(pi mod(i))*arctan(4))

[not in Berlin, sadly No.12517830]

>>12517353
Well what's optimal, that's a normative phrase.
It might be effective for a time of discovering certain new things, but I guess it's a phase. Possibly you also mean that the affinity of popular algebraic geometers leads to a stream of texts that present math just like so, and that new students waste their twenties in learning.

In physics at least, where I dare to pass judgement, I have the feeling that a new speedup of progress can be achieved by a cleanup of terminology. I'd like for future book to stop with the habit and desire to carry along old conceptions, mixing them with new results and thus create those convoluted messes.

>>12517490
>>12517621
My vague understanding is that he looks at
[math]\int f\mu[/math]
with
[math]f\colon S\to R[/math]
as
[math] \mu\mapsto \big(f\mapsto \int f\mu\big) [/math]
with and his better topological spaces have some cartesian closed properties so that
[math]{\mathrm {Hom}}({\mathrm {Hom}}(S, R), R)[/math]
is a much better objects, some topological abelian group in itself. Something something profinite group, FinSet, something.

>>12517220
good effort

>>12517410
https://ncatlab.org/nlab/show/carrying

[Anonymous No.12517845]

>>12517830
I want to believe that is some nlab autoironomorphism. It just... It has to be, right?

[Anonymous No.12517918]

>>12517830

>μ(f∫fμ)
Nigga, isn't it just considering integration of f against an object m as an operation m on f's, i.e a linear functional, i.e the basic idea behind functional analysis?

[Anonymous No.12517953]

A teacher was telling us how a chinese group solved a problem before her. So, I was thinking, what if the chinese stole it from them and just filled the gaps?
Do you think it would be possible that this data stealing meme could actually affect that kind of stuff?

[Anonymous No.12517963]

>>12517845
I'm afraid someone just liked the 2002 paper linked there at the end.

>>12517918
>ωa∫fμ
The point being that the function spaces are better behaved when not using the standard topology concept. As in pic related

I like this view by an outsider, it hits the right notes immediately
https://youtu.be/UWZH-CHzFKQ

[Anonymous No.12517970]

>>12517830
> [math]\int f \mu[/math]
Wild animal lost in the concrete jungle tier notation, just write [math]\int f \ d \mu[/math]

[Anonymous No.12517982]

>>12517970
I took it directly from the Scholze monkey bear

[Anonymous No.12517989]

What's the group for translations of a d-rectangular grid of cubic lattice points in an infinite d-dimensional lattice?

[Anonymous No.12517996]

>>12517989
To clarify: I just need the fancy name and notation

[Anonymous No.12518005]

>>12517996
[math]\{T_k\}[/math] or something? Smells isomorphic to [math]{\mathbb Z}^d[/math]

[Anonymous No.12518009]

>>12517982
Scholze proved wrong in the head.

[Anonymous No.12518053]

>>12516628
So you have a function of two variables X and Z and Z is the time derivative of X, is dX/dZ=0?

https://www.youtube.com/watch?v=M05ixbSOY80&ab_channel=FacultyofKhan

[Anonymous No.12518074]

>>12517989
Everything is determined by the image of the unit cube, so [math]\mathbb{Z}^d \times H_d[/math].

[Anonymous No.12518086]

What can't Coq do?

[Anonymous No.12518097]

>>12518053
When this gut took the derivative of the Legrangian with respect to X, he said that the derivative of the kinetic energy is 0 because Kinetic Energy only depends only depends on X and not Vx , but here is how I think about it, since they are both a function of t, Imagine if X= t^2 , then Vx = 2t
dX/dVx = d(t^2)/d(2t) and t^2 = 1/4(2t)^2 hence if we set 2t = u, we have dX/dVx = d(t^2)/d(2t)= d(1/4u^2)/du = 1/2u= t.
Sorry for lack of latex, someone please clarify this to me it's driving me insaane!

[Anonymous No.12518109]

What are all values of [math]i^i[/math]?

[Anonymous No.12518223]

>>12518086
be Turing complete?

>>12518097
The Lagrangian here is a map
[math]L\colon {\mathbb R}^d\times {\mathbb R}^d\times {\mathbb R} \to {\mathbb R} [/math],
i.e. "a function [math]L(x,v,t)[/math]"
The derivative "[math]\frac{\partial L}{\partial q'(t)}[/math]" is [math]\left(\frac{\partial L}{\partial v}\right)_{v=q'(t)}[/math], so you don't need to worry about correlations between [math]q'(t)[/math] and [math]q(t)[/math] here.

There's the ubiquitous habit of also denoting then t-parametrized value "[math]L(q(t), q'(t), t)[/math]", a function
[math]{\mathbb R} \to {\mathbb R} [/math]
by [math]L[/math], so don't get confused.

Also don't mix up partial derivatives, as in [math]\left(\frac{\partial L}{\partial v}\right)_{v=q'(t)}[/math], with total ones, as in [math]\frac{d}{dt}\left(\frac{\partial L}{\partial x}\right)_{x=q(t)}[/math]

Lastly, there's variations of functionals [math]\frac{\delta S}{\delta q}[/math], where e.g. [math]S\colon ({\mathbb R}\to {\mathbb R}^d)\to {\mathbb R}[/math], so don't confuse those either.
If you read a good of physics textbooks, you'll get very confused with d'Alembert's principle (and if you watch random undergrads on youtube you'll get away with the idea that it's just a trivial rewrite of Newton, so I'm already sceptical). If you read a math textbook you'll have some work bringing those together with the physics books but if you look into the post Arnold slash control theory slash theoretical engineering, say, diff geo formulations that make it rigorous, you get

https://en.wikipedia.org/wiki/Virtual_displacement#Definition

so just learn the rules and don't mix up [math]d, \partial, \delta[/math].

And with that said, watch out for L's that are neither maps on [math](x,v,t)[/math] nor [math]t[/math] but actually functionals eating [math]t\mapsto x[/math].

(So [math]\frac{\delta q(t)}{\delta q'(t)}[/math] is not that you want to compute, but you could probably give meanint to it via some diract delta functionals)

[Anonymous No.12518228]

good old

[Anonymous No.12518356]

>>12518223
So in short you don't consider q(t) and q'(t) as explicit functions of t but as any arbitrary function of t?

[Anonymous No.12518368]

Why do infinitist schizos and boomer set theorist think it's fine that their shit can't be run on a computer? Do they propose we have trained slaves completing and verifying all the papers submitted every day, or that we keep the current system, where the validity of complicated proofs are based on renown?

[Anonymous No.12518377]

what is the best way to into math? I'm a bit of a brainlet when it comes math and stuff, but I think I could improve a lot if I tried. how much of math is actual computation of numbers compared to understanding how they work and the theory and concepts behind it?

[Anonymous No.12518408]

>>12518356
I'm not sure what the difference between explicit and arbitrary would be here, but no that's not the short version of it.
I try to be shorter the second time around: You take the derivative of "L(x,v,t)" with respect to v at the value q(t). Whether v=q'(t) is determined by x=q(t) does not play a role in the derivative here, since you take the derivative of L in which x and v are independent parameters (independent dimensions). So the derivative ∂L(x,v,t)/∂v, is blind to whatever is going on in x (even if for whatever reason you'd eventually evaluate x at v). The fact that this expression is finally evaluated x=q and v=q', which are evaluated at some values t is not relevant, and thus the functional form of q doesn't matter.

To understand which derivatives in the Euler-Lagrange equations are taken, you might want to follow d'Alamberts principle (at the risk of getting confused by yet another notion of derivative or variation, [math]\delta [/math]), or you look at spatial calculus of variation examples and have faith that something that makes sense for lengths also makes sense for a kinetic-minus-potential energy quantity.

[Anonymous No.12518412]

>>12518377
Math is like japanese except that there's no media to coomsom.

[Anonymous No.12518415]

>>12518377
Read all of Lang then read Rudin.

[Anonymous No.12518431]

>>12518412
>what are manga guides
He can even read the manga guides in moon and kill two birds with one stone.

[Anonymous No.12518433]

>>12518408
I think I can kind of understand, it's like you're constructing random functions by giving a point, the value at the point and the derivative at that point. You could fix the Point and value for example and change the derivative however you like. They are all independent in the langrangian.

[Anonymous No.12518458]

sup bros i wanna teach myself first year undergrad maths which textbooks should i use?

[Anonymous No.12518472]

>>12518458
refer to
>>12518412

[evil No.12518617]

>>12518433
>it's like you're constructing random functions by giving a point, the value at the point and the derivative at that point. You could fix the Point and value for example and change the derivative however you like. They are all independent in the langrangian.
Yeah that sounds about fine.

More accurately speaking, if you write down the Lagrangian equations like you do, the evil thing is that your x is not the position and v is not its velocity - although that's a special case. It's a ruse!
E.g. if you got an analytical mechanics problem where the (actually, world space) position is [math]r[/math] and you can parametrize it as [math]r(\theta):=r_0\sin(\theta)[/math] for some fixed [math]r_0[/math]. Your x can then be taken to be the angle, and it can also be weirder. The kinetic energy at a point in time can be take as [math]T=\tfrac{1}{2}\dot{r}^2[/math] but that's in real space. The analytical mechanics principle is
[math] \sum\ (-\frac{\partial \Phi }{\partial r} - m\, \ddot{r} )\cdot \delta r = 0 [/math]
with some potential [math]\Phi[/math].
with the possible [math] \delta r [/math] fixed by the geometric model of your situation for each time and the only reason why things even get as complicated - why something like [math]\frac{\partial T}{\partial x}[/math] is even a thing - is because [math] \delta r [/math] involves a Jacobian [math]\frac{\partial r}{\partial x}[/math]. You're doing geometry here after all.
Not sure if this confuses more than it helps but that's why the dependencies are as they are in the Lagrangian differential equation in the first place.
Speaking about position x and velocity v might be harmful.

It's like calling both
[math]T+\Phi[/math]
and the Hamiltonian
[math]H:=\frac{\partial L}{\partial \dot{q}}\cdot \dot{q}-L[/math]
by the name "energy" - you might get cut.
And I don't wanna see you hurtin, babby.

[evil No.12518626]

>>12518433
>it's like you're constructing random functions by giving a point, the value at the point and the derivative at that point. You could fix the Point and value for example and change the derivative however you like. They are all independent in the langrangian.
Yeah that sounds about fine.

More accurately speaking, if you write down the Lagrangian equations like you do, the evil thing is that your x is not the position and v is not its velocity - although that's a special case. It's a ruse!
E.g. if you got an analytical mechanics problem where the (actually, world space) position is [math] r [/math] and you can parametrize it as
[math] r(\theta) := r_0 \sin(\theta) [/math]
for some fixed [math]r_0[/math]. Your x can then be taken to be the angle, and it can also be weirder. The kinetic energy at a point in time can be take as [math] T = \tfrac{1}{2} \dot{r}^2 [/math] but that's in real space. The analytical mechanics principle is
[math] \sum\, (- \frac{\partial \Phi }{\partial r} - m\, \ddot{r} ) \cdot \delta r = 0 [/math]
with some potential [math]\Phi[/math].
with the possible [math] \delta r [/math] fixed by the geometric model of your situation for each time and the only reason why things even get as complicated - why something like [math] \frac {\partial T} {\partial x} [/math] is even a thing - is because [math] \delta r [/math] involves a Jacobian [math] \frac{\partial r} {\partial x} [/math]. You're doing geometry here after all.
Not sure if this confuses more than it helps but that's why the dependencies are as they are in the Lagrangian differential equation in the first place.
Speaking about position x and velocity v might be harmful.

It's like calling both
[math]T+\Phi[/math]
and the Hamiltonian
[math]H:=\frac{\partial L}{\partial \dot{q}}\cdot \dot{q}-L[/math]
by the name "energy" - you might get cut.
And I don't wanna see you hurtin, babby.

[Anonymous No.12518650]

>>12517220
Worth the watch.

[evil No.12518678]

and [math]r(x)[/math] or [math]r(t)[/math] and all derivatives as parameter-dependencies of coordinate maps, and those geometric Jacobians have nothing to do with trajectories apriori.
The idea of even introducing what's usually written "[math] \frac{\partial }{\partial \dot{q}} [/math]" is used because [math] \frac{\partial }{\partial \dot{q}} r(q, t) = 0 [/math] and it makes some rewrites possible.

Yes I think half of your code should be type signatures, then all confusions will be solved

[evil No.12518690]

and [math]r(x)[/math] or [math]r(t)[/math] and all derivatives are parameter-dependent coordinate maps, and those geometric Jacobians above have nothing to do with trajectories apriori.
The idea of even introducing what's usually written "[math] \frac{\partial }{\partial \dot{q}} [/math]" is used because [math] \frac{\partial }{\partial \dot{q}} r(q, t) = 0 [/math] and it makes some rewrites possible.

Why yes I think half of your code should be type signatures and full of explicit substitutions, then all confusions will be solved.
Also relates to the "cleanup terminology" note in >>12517830

[Anonymous No.12518696]

Why do we stop at triple integrals?
Couldnt we just keep going on forever?

[Anonymous No.12518708]

>>12518626
You got any brainlet freindly book recommendations?

[Anonymous No.12518732]

>>12517651
Isn't that kind of thing what Arnold was bitching about

[Anonymous No.12518759]

>>12518377
Fellow brainlet here. Linear Algebra was fun. Get something like Friedberg and work through it. Calc is cool too but the proofs are harder.

[Anonymous No.12518777]

>>12517989
The group algebra [math]\mathbb{Z}[\Lambda][/math].

[evil No.12518812]

>>12518708
No, I think those still have to be written. It also depends on what you study. Ironically >>12518732 is mentioning Arnold and I know his Mathematical methods of classical mechanics is online. I don't like it too much I think, I only like the German books, but it will work.

>>12518696
>Why do we stop at triple integrals? Couldnt we just keep going on forever?
Ostensibly, but don't tell the finitists
https://en.wikipedia.org/wiki/Product_integral

[evil No.12518866]

come to think of it, is there a function
[math]f\colon {\mathbb R}\to {\mathbb R}[/math]
such that
[math]I_1[f]:=\int_0^x f(t) dt[/math]
and
[math]I_{n+1}:=I_1\circ I_n[/math]
for which
[math]\lim_{n\to\infty}I_n[/math]
is defined on f and give a finite nonzero value?
Seems like f must have f(0)=0 and polynomials get killed off factorially, but I might not think clearly

[Anonymous No.12518878]

come to think of it, is there a function
[math]f\colon {\mathbb R}\to {\mathbb R}[/math]
such that
[math]I_1[f]:=\int_0^x f(t) dt[/math]
and
[math]I_{n+1}:=I_1\circ I_n[/math]
for which
[math]\lim_{n\to\infty}I_n[/math]
with the final x at 1 or something
is defined on f and give a finite nonzero value? Seems like monomials get killed off, but I might not be thinking clearly

[Anonymous No.12518881]

>>12518878
Why would you ever need anything like this?

[Anonymous No.12518887]

>>12518878
Exponentials maybe? e^-x

[Anonymous No.12518889]

>>12518878
Assuming f is continuous, no, since you can bound it above by an affine function for any interval [0, a], and those get killed off on said interval.
Not assuming continuity? Probably still no but shittier to prove.

[Anonymous No.12518912]

>>12518889
Never mind, you can pass from f to |f| and then bound [math]\int 0 ^x |f(t)| \ dt[/math] by some constant function on any finite interval.

[Anonymous No.12519109]

bros i don't want to study for the math gre

[Anonymous No.12519136]

>>12519109
Are schools still requiring it? Will there even be math GREs offered next year? I didn’t have to take it because all of them were cancelled this year and I applied for fall 2021.

[Anonymous No.12519311]

>>12518881
Well that's an argument against a lot of math

[Anonymous No.12519337]

>>12519109
just say it was raycist in your application

[Anonymous No.12519341]

>>12517220
Did you make this anon? Found it quite funny even tho most of it went over my head

[Anonymous No.12519395]

>>12519136
im applying for grad school next year, and i'm assuming they'll be offered by then..
honestly you guys are lucky that you were exempted from this waste of time

[Anonymous No.12519399]

Guys please I NEED to know why if a polynomial's Galois group is not solvable then the polynomial's roots can't be expressed via radicals. What's the best way to develop an intuition for this? I've read through the theorems and proofs but it still hasn't "clicked"

[Anonymous No.12519443]

What is the oldest unresolved problem in mathematics? Has anyone found an odd perfect number?

[Anonymous No.12519476]

>>12519443
The subgroup-subset problem.

[Anonymous No.12519482]

Math is formal logic but more autistic.

[Anonymous No.12519499]

>>12519443
the definition of a set

[Anonymous No.12519609]

>>12519476
can you provide a google-able reference for this? all I get is definitions

[Anonymous No.12519656]

>>12517202
Looks fine to me

[Anonymous No.12519670]

>>12519609
I can state the problem. Ready? Here it comes:
If G is a group and H is a subset of G, is H a subgroup of G?

[Anonymous No.12519699]

>>12519670
I was wondering if it would fail for finite groups. Let's say we have a prime number p dividing the order of G, and let H be any subset of G. Let [math]O^p(G)[/math] be the normal subgroup of G generated by the elements whose order is not divisible by p, and let [math]K = O^p(G) \cap H[/math]. I've been trying to show that K need not be normal, but approaching it from the direction of G/K is way over my skills.

[Anonymous No.12519742]

>>12519699
dude...

[Anonymous No.12519781]

>>12519742
Yes, darling? Any ideas? Obviously, if H contains that normal subgroup, then G/K is a p-group, but but what if their intersection is not the whole [math]O^p(G)[/math]? Then G/K is not a p-group and I'm stuck.

[Anonymous No.12519875]

>>12519781
have you considered the possibility that H is empty?

[Anonymous No.12519892]

>>12519875
No. I can tell you why. For the empty set to be a subset of G, all its elements would have to be elements of G. Since there are no elements in the empty set, it cannot be a subset of G. Or can you give me an element [math]x \in \emptyset[/math] such that [math]x \in G[/math]? Undergrads nowadays...

[Anonymous No.12520032]

I'm sorry, but you guys will never reach the levels of the impossible 100 heads in a roll. It's all pointless now.

[Anonymous No.12520053]

>>12517410
I wish there was a 1lab that explained stuff using regular old categories.

[Anonymous No.12520074]

>>12520032
I just realised the head side of a coin is where you have the head of the monarch and that is why it is the head side. I am a genius.

>>12520053
That would be nice, but nice things don't happen. They probably have a list of nice things and then the self-deified higher category theorists branded those sinful. The Hyperdoctrine dictates that one must not do nice things, only higher categories and try to convert as many people into the dark cult practices as possible.

[Anonymous No.12520094]

>>12519670
kek

[Anonymous No.12520265]

[math]\text{The Weierstrass }\wp\text{enile-function}[/math]

[Anonymous No.12520436]

>>12517220
>i forgot the definition of a set again

topkek

[Anonymous No.12520582]

>>12519699
ì was wondering if you can stop being a peedo

[Anonymous No.12520714]

>>12520436
That ruined the last thread

[Anonymous No.12520750]

Do you anons have a powerful Mind's eye? Mine is so weak I can bearly rotate a shape in it.

[Anonymous No.12520964]

>>12520750
Nope. I have no spatial reasoning capabilities whatsoever. I really envy those who can visualise shapes and stuff properly.

>>12520582
I don't know. I have to start it first, and only then can I tell you if I can stop being one.

[Anonymous No.12520995]

>>12519670
No. Consider the group of integers under addition and the subset {1,2}. The subset is not a group, therefore any H is not necessarily a subgroup.

[Anonymous No.12520998]

>>12520750
It's pretty alright I think. But then again I remember things like visualizing (a+b)^3 kind of annoying, I kept missing pieces

[Anonymous No.12520999]

I fucking hate mathematical induction

[Anonymous No.12521000]

>>12520995
undergrads

[Anonymous No.12521048]

>>12520750
>Mind's Eye
Do you lads have to use the faggiest terms possible?

[Anonymous No.12521055]

>>12520995
Thanks. I think that looks correct, which is quite sad. I hope you don't mind if I borrow your solution in a paper of mine. An immediate corollary is that H is subgroup iff it satisfies the conditions of being a subgroup.

[Anonymous No.12521074]

>>12520995
i can't believe we tried to solve this problem for like 5 /mg/ threads until this guy posted the solution
god bless you anon

[Anonymous No.12521079]

>>12520999
why?

[Anonymous No.12521095]

>>12520995
{1, 2} isn't a group though, moron

[Anonymous No.12521134]

>>12521048

>Do you lads have to use the faggiest terms possible?

It comes from the Cicero's mentis oculi. Turns out you're the heterosexual schizo here.

[Anonymous No.12521150]

>>12520750
Are there any fleshed out thoughts on how the Minds Eye relates to mathematical practice?

[Anonymous No.12521171]

>>12521055
Ironic shitposting is still shitposting -_-

[Anonymous No.12521174]

>>12520995
Good job anon

[Anonymous No.12521178]

>>12519670
You said it kind of wrong but
>>12521055
kek

[Anonymous No.12521187]

>>12521171
I wonder if the original question was an honest attempt to grasp things and we bullied someone to kyssing them shelf...

>>12521178
Did I misremember the subset-subgroup conjecture? It must be the threshold of 24 years and all the daily brain damage after that age.

[Anonymous No.12521197]

>>12521187
I think it was something like if H is a subgroup of a group G and A is a subset of H, is A a subgroup of G
Also yeah I think it was posted in earnest

[Anonymous No.12521198]

>>12521187
This is the original.
I borrowed the cap from the previous thread.

[Anonymous No.12521250]

>>12521197
>>12521198
My bad. Apologies sent to your direction(s).

[Anonymous No.12521277]

>>12521250
Apology rejected, post a properly apologetic anime girl instead of a smug one.

[Anonymous No.12521282]

Are there any blogs or stuff of that sort that you regularly come back to to see if there's something interesting?

[Anonymous No.12521328]

How do I prove that a brownian motion is a martingale???

[Anonymous No.12521336]

>>12521282
The Last Psychiatrist ;_;

[Anonymous No.12521340]

>>12517220
>I forgot the definition of a set again
lost it right at the end. beautiful job

[Anonymous No.12521350]

>>12521328
Markov:
[math]\mathbb{E}[W_t|\mathcal{F}_s]=\mathbb{E}[W_t|W_s][/math]
Increment distribution:
[math]W_t-W_s|W_s \sim \mathcal{N}(0,(t-s)^2) [/math]
Properties of Gaussian:
[math]W_t|W_s \sim \mathcal{N}(W_s,(t-s)^2) [/math]
QED:
[math]\mathbb{E}[W_t|W_s]=W_s[/math]

[Anonymous No.12521375]

>>12521350
I was shitposting, you weren't supposed to answer it seriously.

[Anonymous No.12521397]

>>12521375
I'm a simple man, anon

[Anonymous No.12521490]

>Post-doctoral position in mathematics or mathematical physics for the ERC project "Derived Symplectic Geometry and Applications"
In Montpellier (France) even though the website is Croatian https://www.euraxess.hr/jobs/587374

>>12521277
Please forgive me.

[Anonymous No.12521501]

>>12521198
The exact same proof applies if G is a supergroup of the integers under addition

[Anonymous No.12521589]

>>12521490
She just looks curious there.
At least put one second of effort into it.
https://safebooru.donmai.us/posts?tags=apologizing+

[Anonymous No.12521623]

>>12521589
It's an anime tranny, obviously he has autism and can't distinguish facial expression. I bet you just made him cry.

[Anonymous No.12521666]

>>12521589
It wasn't supposed to be an apologetic face. This would be my expression irl, assuming >>12521623 is correct. I guess apologist would be a good term for an algebraic topologist, the apologist of category theorists.

[Anonymous No.12521686]

what math do i learn in order to impress anime trannies
because i want to have sex with them

[Anonymous No.12521692]

>>12521686
Don't be fooled by the anime pictures. They look cute, but in reality they are disgusting looking men.

[Anonymous No.12521696]

>>12521666
If you aren't actually sorry then don't apologize, faggot.

[Anonymous No.12521734]

>>12521686
For the postgrad one it would be engineering math as "her" man is an engineer.

[Anonymous No.12521740]

>>12521692
bro it's my last chance
i gave up on real females some time ago

[Anonymous No.12521768]

>>12521740
This is what the globalist demoralization campaign on white men looks like

[Anonymous No.12521830]

>>12521768

[Anonymous No.12521849]

>>12521830

>hook up easily

But we're not trying to hook up, I want a gee effay

[Anonymous No.12521886]

>>12521696
Grrrrrrr!

>>12521734
Specialised in nuclear plants and an arch enemy of Rosatom.

[Anonymous No.12521888]

>>12521849
It took me so fucking long to figure out what a «gee effay» was.

[Anonymous No.12521894]

>>12521849
you were pretty explicit about the tranneis

[Anonymous No.12521998]

>>12521830
very true
i'm more of a "i'm inadequate" type
>>12521849
bro... good luck...

[Anonymous No.12522027]

>>12521894

[Anonymous No.12522067]

>>12521830
bullshit argument. low IQ

[Anonymous No.12522180]

Today I discovered a (to me) perplexing relation:

[math]X:=\left(
\begin{array}{cc}
x & 0 \\
0 & x_0 \\
\end{array}
\right)[/math],

[math]H:=\left(
\begin{array}{cc}
0 & h \\
0 & 0 \\
\end{array}
\right)[/math]

then, at least for analytic f,

[math]f(X + H) = f(X) + \dfrac{f(x) - f(x_0)}{x - x_0} H[/math]

(* Some Mathematica: X = {{x, 0}, {0, x0}}; H = {{0, h}, {0, 0}}; X . H - H . X == (x - x0) H // Simplify; MatrixPower[X + H, m] == MatrixPower[X, m] + H (x0^m - x^m) / (x0 - x) // Simplify; *)

In existing language, this family (in, [math]x, x_0, h[/math]) are possible bases of 2x2 representation of the affine algebra (the smallest Lie-algebra), as evidenced by

[math][X, H] = (x - x_0)\, H[/math]

For [math]x_0=-x[/math] that's also a subalgebra of [math]sl(2)[/math] (note that my X is often called H in that context.)

The formula generalizes the standard matrix computation of its exponential map, [math]f=\exp[/math], but from the standard representation (which has [math]x_0=0[/math]), it's usually not clear that a finite difference quotient shows up there.
In the limit [math]x\to x_0[/math] it's somewhat clear that

[math]f(X + H) = f(X) + f'(x) H[/math]

because H is nilpotent, but it's baffling me that the expression also holds on the finite difference level.

It essentially mirrors the idea

"[math]\dfrac{f(x + h) - f(x)}{h} = \dfrac{f(x) - f(x_0)}{x - x_0}[/math]"

but holds exactly.
Has anybody seen this before?

[Anonymous No.12522232]

inverse systems, I tell you hwat

[Anonymous No.12522237]

>>12522027
>Nationwide Transgender Equality Campaign Debuts on FOX News During Republican Convention
lol

[Anonymous No.12522296]

Any anons can recommend literature covering KdV, infinite toda lattice (using Lax representations and inverse spectral method) and the AKS scheme?

[Anonymous No.12522320]

>>12522180
Yes people have seen it but I'm not allowed to say. See if it holds for powers of two.

[Anonymous No.12522366]

Are there any open problems left concerning single variable degree-2 polynomials over [math]\mathbb{C}[/math]?

[Anonymous No.12522373]

>>12522366
Outside of algorithm efficiency, no.

[Anonymous No.12522430]

Starting Spivak's Calculus. Any tips??
:)

[Anonymous No.12522477]

>>12522430
Use a pen and paper. No pencils and conversely, no electronic devices.

[Anonymous No.12522484]

>>12522477
This, because writing keeps your visual cortex involved in the process -> everything will be easier to visualise.

[Anonymous No.12522501]

>>12522366
i think it's fair to say that mandelbrot set and related topics are about "single variable quadratic polynomials over C" and I suspect you could find some open questions there

[Anonymous No.12522548]

>Proof: The diagram in fig X commutes

[Anonymous No.12522595]

>>12522548
> Proof: by inspection

[Anonymous No.12522623]

>>12522501
Iterated functions don't really count, but here's an open mandelbrot question:
>Can you find a set of four points [math]a,b,c,d[/math] exterior to the mandelbrot set, arranged as the vertices of a square, for which the number of iterations each of [math]a,b,c,d[/math] takes to have a magnitude greater than or equal to 2 (also known as the escape time) is any quadruple of natural numbers?

[Anonymous No.12522644]

>>12522623
Honestly if you allow modulo 2^32-1 instead of all natural numbers you have the question of can you have any 2*2 combination of pixel colors in some depiction of the set colored by escape time. You can then extend this to any m*n rectangle of lattice points to get the broader question: do all computer-displayable images exist in escape time-colorings of the mandelbrot set?

[Anonymous No.12522666]

>>12522595
The patrician way to do diagram chases is proof by moving fingers. You just put your right index finger on the diagram and move it around. When an element lifts to something, you use your left index finger to track this new element. The canonical choice of hand usage is justified by the similarity of left and lift.

[Anonymous No.12522689]

>>12522666
So this is what math majors choose to do when they find out they can't do integrals? Damn that's rough.

[Anonymous No.12522720]

Can I get recommendations for books on math history?

[Anonymous No.12522727]

>>12522720
yes

[Anonymous No.12522739]

>>12522689
If you do it on a blackboard, it's a good morning stretching exercise.

>>12522720
https://www.amazon.com/History-Mathematics-Carl-Benjamin-Boyer/dp/0691023913 is good, if I remember correctly.

[Anonymous No.12522786]

>>12517202

The theorem is correct. Its a stupid theorem because it doesn’t even claim the negation of the statement (that if you pick a number [math] z \neq x [/math] then [math] b + z \neq = b [/math]).

[Anonymous No.12522789]

>>12522366
Never mind quadratics, are there any open problems concerning systems of single variable linear equations over [math]\mathbb{C}[/math] or is that topic solved?

[Anonymous No.12522809]

>>12522789
If you are only studying linear equations, [math]\mathbb{R}^2[/math] is all you need.

[Anonymous No.12523028]

anybody knows how to determine the normal vector that points ''outwards'' of a surface? say i define a cube with a set of vertex and edges, and say every 4-loop is a face, i'd calculate the normal vector by doing a cross product, but how do i know which direction points to the ''outside'' of the cube? it's easy to see which direction points outwards for this one cube, but for a giant mesh in general, how does one determine which normal vector to take?

[Anonymous No.12523081]

holy shit my parents thought i was studying applied math/statistics this whole time
they just found out im studying pure math and their flipping their shit
what do i do

[Anonymous No.12523083]

>>12518368
Most infinitist proofs can be demonstrated in formal verification systems. Of course they are incomputable proofs but that doesn't mean they can't be verified.

[Anonymous No.12523178]

>>12523081
Tell them that with a math PhD you get 300k to start minimum and any job that you want.

[Anonymous No.12523219]

>>12523178
Stfu

[Anonymous No.12523325]

>>12523081
tell them there is no difference

[Anonymous No.12523346]

>>12516743
>>12517285
fuck this shit board

[Anonymous No.12523351]

>>12517410
>click on the nlab link
found your problem

[Anonymous No.12523372]

>>12517744
(If f is a real nowhere differentiable function and x is a real number the for which x is
x^f differentiable.)
This is a problem in baby Rubin, and he provides an elegant 2 line solution.

[Anonymous No.12523392]

>>12522180
>l0ok at me i goo oon stak exxhange and post i t to soi/

[Anonymous No.12523402]

Reminder that you wasted years of your life studying "higher mathematics" and "category theory (not math)" are wasted and any high schooler who studies elementary math in depth will be able to prove more interesting results.

[Anonymous No.12523467]

Springer is giving me a 30 Euro discount on a book since I published with them but idk what to use it on.
Any recommendations? I love reading in most fields except statistics so any classics in your fields you can recommend would be great.

[Anonymous No.12523869]

>>12523346
This is the moment when you should tell us that you will never ever like really never ever in the literal meaning of never ever even open /sci/ again, and then we will reply "see you tomorrow, darling".

>>12523351
They are sometimes one of the few quick sources for things, and I'm not talking about some HCT stuff. Additionally, they have references mentioned. It's like doing a presentation for school. Wikipedia itself was not accepted as a source, but its reference lists allowed me to skip finding info on my own.

>>12523467
>rotman's book on algebraic topology
>whitehead's book on homotopy
>fuchs-fomenko if that version also has the illustrations
>pierce's book on associative algebras
>brown's book on group cohomology

[Anonymous No.12523898]

>>12523467
Book about fractals with lots of colorful pictures.

[Anonymous No.12523909]

>>12523402
true and dangerously based

[Anonymous No.12524252]

If there are different sizes of infinity are there also different sizes of infinitesimals?

[Anonymous No.12524262]

>>12524252
Yes.

[Anonymous No.12524506]

Just learned about this formula. I think it's interesting.

[eqn]
\sum_{k=1}^{n}(k) = \frac{n(n+1)}{2}
[/eqn]

[Anonymous No.12524531]

>>12524506
that's very admirable, but I don't think a 10-year-old should browse 4chan

[Anonymous No.12524581]

>>12524506
Now find a formula for
[math]\sum_{k=1}^n k^m[/math] for general m.

[Anonymous No.12524596]

>>12520436
I actually loled at this one

[Anonymous No.12524601]

Mathbros... what's a good personal project that can a CSlet do that has some practical applications but is kinda related to pure maths?

[Anonymous No.12524637]

What is an ordinal number of set of natural numbers with lexicographic order? (0<1<10<100<1000<10000<...<11<110<1100<11000<...<12<120<1200<12000<...)

[Anonymous No.12524645]

Tips and tricks to give proofs by diagram chasing? I just follow the maps and a given element till I have a solution. Is it just this?

[Anonymous No.12524648]

>>12524637
w^w

[Anonymous No.12524767]

>>12524506

why would you TeX a trivial result like this? just makes us more prone to think CS kids are doing baby math in university

[Anonymous No.12524791]

>>12524601
Literally nothing, if you don't have a foundation in pure math.

[Anonymous No.12524804]

Can't believe I fell for the meme and spent my life doing mathematics, guys please stop and do something life affirming if u still have time
> PhD in math
> any bridge I want
> neverending depression

[Anonymous No.12524821]

>>12524506
For few embarassing seconds I thought that [math](k)[/math] has some special meaning.

[Anonymous No.12524826]

>>12524601
>>12524791
>foundation
Now that you mention it, a database of formalized proofs might be suitable, if done in the "type-theoretic" paradigm (Curry-Howard-Lambek, Brouwer-Heyting-Kolmogorov, or whatever they're calling it these days).

[Anonymous No.12524896]

>>12523392
If you're implying you found the relation on SE, please share the link.

I got at it by trying to recast an unsolved number theoretic problem into an algebra context.
Namely there's only two solutions known to
[math] \dfrac{x^m - 1}{x-1}=\dfrac{y^n - 1}{y - 1} [/math]

e.g.
[math] \dfrac{5^3-1}{5-1}=\dfrac{2^5-1}{2-1}=31 [/math]

and a second one, and the conjecture is that those are all of them, see
https://en.wikipedia.org/wiki/Goormaghtigh_conjecture

I recast the condition as one of a rep of sl(2), where this says that the following difference is diagonal
[math]\left(
\begin{array}{cc}
x & h \\
0 & 1 \\
\end{array}
\right)^m-\left(
\begin{array}{cc}
y & h \\
0 & 1 \\
\end{array}
\right)^n[/math]
(the condition is saying that the top right component vanishes for any h)

>>12524601
How pure are we talking?
And do you want it to be actually actually usable for you later on (something you end up using)?

I bought this 100$ drone (pic related) that some Indian guy on youtube will do a basic course on and I plan to overcomplicate by implementing some routines in some-close-to-the-math algorithms for it's mechanics and dynamics.

You're welcome to join my leisure effort. One might indeed also port the stuff to a more mathematical language (I'm interested in Idris)

[Anonymous No.12525076]

>>12524896
> Nolting
Physishit or methhead that took the arnold pill and now studies physics?

[Anonymous No.12525102]

>>12524804
If you are planning to do something stupid, then please choose a bridge such that the arch shape is the hypperbolic cosine. That way you will get to end your life mathematically as well.

[Anonymous No.12525121]

>>12525076
Why is he so sexy bros

[Anonymous No.12525148]

what are you smoking there anon

[Anonymous No.12525158]

>>12525121

that's a man dude

[Anonymous No.12525224]

>>12524821
For a few embarrassing seconds I thought I cared.

[Anonymous No.12525225]

I'm getting filtered by modular forms atm, is pure math just a pyramid scheme?
Mathematicians get paid by institutions that only do math and are disconnected from the economy, the only money that flows into the system is tuition and book sales.

[Anonymous No.12525232]

>>12525148
learn english motherfucker

[Anonymous No.12525249]

>>12525225
>is pure math just a pyramid scheme
yes

[Anonymous No.12525251]

>>12525225
>institutions that only do math and are disconnected from the economy
god I wish that was the case

[Anonymous No.12525255]

>>12525121
idk bros I'm more into quirky frenchies

[Anonymous No.12525262]

>>12525225
>Mathematicians get paid by institutions that only do math and are disconnected from the economy
lol

[Anonymous No.12525536]

Any number theorists here?
>>12525523

[Anonymous No.12525539]

I don't care if you study in MIT or Cambridge or whatever new hot shit is topping the QS Rankings, my Uni is still comfier than yours. It's in the middle of a forest ffs how can any other even compete?

[Anonymous No.12525544]

>>12525539

>Brazil

they compete by being located in a real country

[Anonymous No.12525545]

>>12525539
That does look comfy, but my Uni is a literal castle, so I won.

[Anonymous No.12525556]

The thread has been slow and garbage, so I'll post some problems from Grillet.
This particular batch is an immense pain in the ass.

[Anonymous No.12525569]

Just 11.

[Anonymous No.12525579]

[Anonymous No.12525584]

Alright, I'm done.

[Anonymous No.12525586]

>>12525569
max order of elements

[Anonymous No.12525593]

>>12525586
I don't follow, elaborate.
[math]\mathbb{Z}_6[/math] has an element whose order equals the entire group's order, but it still splits as [math]\mathbb{Z}_2 \oplus \mathbb{Z}_3[/math].

[Anonymous No.12525613]

>>12525593
Jesus, man... Just think a little more...

[Anonymous No.12525645]

>>12525613
I have a better idea:
What if you actually posted your solution?

[Anonymous No.12525652]

>>12525645
You do realise your counter-example is unrelated because 6 is not a power of any prime?

[Anonymous No.12525655]

>>12525593
what is the

[Anonymous No.12525659]

>>12525652
My counter-example is unrelated to his solution because he didn't post anything even distantly resembling a solution, hence I had to guess he meant something like "order of element too large hence doesn't split".

[Anonymous No.12525665]

>>12525569
Lagrange theorem. If it were decomposable A+B, A would have order p^l for some l<k. Then every element would have order at most max(l, k-l)

[Anonymous No.12525677]

>>12525665
That's acceptable, good job.

[Anonymous No.12525689]

>>12525677
I don't understand why you compliment me when doing this problem was such a triviality. I was not showing off my skill, but trying to help you.

[Anonymous No.12525695]

>>12525584
trivial since "s" stands for "simple"

[Anonymous No.12525697]

>>12525689
...help with what? I know how to solve the problems.
Well, other than >>12525584 . Don't really remember or want to suffer through that one.

[Anonymous No.12526137]

>>12525697
why call it wew.png when the answer is obvious
ugh.png too

[Anonymous No.12526600]

>>12525225
>I'm getting filtered by modular forms atm
what's wrong with modular forms? i love that shit

[Anonymous No.12526624]

>>12526137
Usually I just name files whatever so Windows doesn't scream at me for saving more than one Capture.png
You should have been able to tell from >>12525556's text that I was legitimately just posting random shit to move the thread.

[Anonymous No.12527498]

>>12516628
https://www.youtube.com/watch?v=rbi213MKbww
Pattern of Prime Numbers

[Anonymous No.12527694]

>>12525556
i hope he does do the proof by contraposition

[Anonymous No.12527696]

>>12527694
>does
not

[Anonymous No.12527816]

>>12525544
As someone who is located in a total nowhere I would love to be in Brazil

[Anonymous No.12527907]

Anyone have tips for prolonging the life of Springer hardcovers? The bindings (at least on the longer books) always get destroyed after a few weeks.

[Anonymous No.12527979]

>>12527907
The old ones used to be really good. No idea what the fuck happened to so called German quality. They've outsourced most of their shit to India too.

[Anonymous No.12528022]

>>12527979
>They've outsourced most of their shit to India too.
Hmm, I never checked before but it looks like the one I just picked up (Lee smooth manifolds) was printed in the US. Fingers crossed it’s of better quality.

[Anonymous No.12528183]

>>12525544
Nice redditspacing

[Anonymous No.12528257]

>>12525556
3. Associativity OK, so we need an identity and inverses. Fix any [math]a\in S[/math] and let [math]r_a, l_a\in S[/math] be such that [math]ar_a = a = l_aa[/math]. Given any [math]b\in S[/math], there exist [math]x, y\in S[/math] such that [math]ax = b = ya[/math], so now [math]l_ab = l_a(ax) = (l_aa)x = ax = b[/math] and, similarly, [math]br_a = b[/math]. Thus, those work as left and right identities, and this gives [math]r_a = l_ar_a = l_a[/math], from which it follows that [math]e := l_a[/math] is a mutual-sided identity. Next, consider the equations [math]ab = e, ca = e[/math]. These make sense by assumption. We have [math]b = eb = (ca)b = c(ab) = ce = c[/math], so any one-sided inverse is a mutual-sided inverse.
M.O.T.

[Anonymous No.12528321]

>>12525556
4. Fix any [math]x \in S[/math] and define [math]f_x \colon S\to S[/math] by f_x(y) = xy[/math]. If now [math]f_x(y) = xy = xz = f_x(z)[/math] for some [math]z\in S[/math], then [math]y=z[/math], by assumption, so [math]f_x[/math] is an injective endomorphism of a finite set. This shows that [math]f_x[/math] is bijective, so there exists [math]r_x\in S[/math] f_x(r_x) = xr_x = x[/math]. Similarly, one obtains an element [math]l_x\in S[/math] such that math]l_xx = x[/math]. Given any [math]y \in S[/math], now [math]y = f_x(z)[/math], for some [math]z\in S[/math], by bijectivity, and so [math]l_xy = l_xf_x(z) = l_x(xz) = (l_xx)z = xz = y[/math]. Similarly, [math]yr_x = y[/math], so [math]e = l_x[/math] is a mutual-sided identity. Since [math] e = f_x(y)[/math] for some [math]y\in S[/math], and similarly [math]zx = e[/math] for some [math]z\in S[/math], now [math]y = ey = (zx)y = z(xy) = ze = z[/math], showing that one-sided inverses are mutual-sided. Since [math]x[/math] was arbitrary, we are done.
[math][ ^o\smile^o][/math]
Can't come up with an example in 15 seconds, so skip that.

[Anonymous No.12528330]

>>12528321
Oops these typos and forgetting [math] (like really WTF?). Shamefur dispray!

[Anonymous No.12528341]

>>12528330
dumb tranny

[Anonymous No.12528391]

Is probability theory by Renyi a good book? I was considering getting the Dover edition of it

[Anonymous No.12528483]

>>12528391
Never looked at Renyi. Consider Çinlar for a more modern introduction.

[Anonymous No.12528573]

>>12528483
Thanks anon. I always heard of durrett as the book to get for probability theory. How would you say cinlar compares?

[Anonymous No.12528596]

>>12528321
>he got stuck in the single easiest part of the entire problem
The cancellation laws hold in the free semigroup on one variable (vurgarly known as the natural numbers) but it's not a group.

[Anonymous No.12528607]

>>12528596
Ah, yes, they actually do. Very true. Anyway, off I go, so have a nice a new year /mg/!

[Anonymous No.12528904]

Did any of you teach yourselves high school math? If so how long did it take you to master it?

[Anonymous No.12529055]

>32 statistics exercises to do
>25 differential equations exercises
>20 analysis exercises
>19 abstract algebra exercises
I need to do all that in 6~7 days at most. Give it to me straight, is this humanely possible or did I just fuck up my entire semester? Which of them should I start by?

[Anonymous No.12529057]

>>12527816
>I would love to be in Brazil


How wealthy are you? We've no working public services or resemblance of welfare and rule of law, to enjoy a dignified life here you will have to buy everything in the private sector, all very expensive, whilst paying one of the heaviest tax burdens in the world for those public utilities you don't use. Also, we don't produce anything internally beyond food and prostitution so any useful product will be imported and cost 2000% because of our braindead taxation polices, weak currency and overall inefficient economy.

[Anonymous No.12529067]

>>12529055
you should just start instead of complaining. do the real analysis right now, once you get stuck switch to algebra. while you're stuck on both do the shit for the other classes. switch between all 4 until done.

[Anonymous No.12529069]

>>12529067
I wasn't complaining.

[Anonymous No.12529077]

>>12529055

Stop taking so much credits for a degree you're not getting a job with anyway

[Anonymous No.12529101]

>>12529069
sure i just meant stop posting on 4chan and start. why are you still responding to me?

[Anonymous No.12529119]

>>12528904
I suspect that was never a problem for people here
In any case, depending on your starting point it shouldn't take too long

[Anonymous No.12529122]

>>12528904

you can solve highschool with the Gelfand circuit

[Anonymous No.12529132]

>>12528904
When I was in high school I practically taught myself with some more advanced books, took me around six months to go through them all (7 volumes), but I barely did the exercises. Doing all the exercises would probably take two or three more months, maybe less depending on the time available.

[Anonymous No.12529369]

>>12522180
Look up Kac-Moody affine Lie alg's and affine Manin triples.
>>12522296
Guest's book has a nice take from the quantum cohomo perspective.
>>12522720
Katz.
>>12523028
A (trivializable) normal bundle determines your orientation, hence which normal direction to take is up to you. However once you've picked one for a segment of your mesh, you can ensure orientability by making sure it does not change sign as you move across the segments.

[Anonymous No.12529441]

test

[Anonymous No.12529473]

>>12529441
test failed

[Anonymous No.12529522]

>>12529369
>Kac-Moody affine Lie alg's and affine Manin triples
Okay. That's a broad direction, do you have any particlar formulas or properties in mind?

[Anonymous No.12529606]

I've learned about sines, cosines, and tangents and have applied this knowledge in video game programming.

Is there a logical "next step" to this? I like being able to do fancy things with angles using these pre-made formulas. Like, I love how a tangent between 1 and -1 is like a 90degree cone around the player and I didn't have to jerryrig any batshit code to make that happen when comparing the enemy's position to the player's position.

[Anonymous No.12529607]

>>12525539
The acclaimed and spoiled place where no one knows what is an ordinal.

[Anonymous No.12529744]

>>12529606
It's not clear what kind of answer to this you're searching for or what

>I didn't have to jerryrig any batshit code to make that happen when comparing the enemy's position to the player's position.
??

[Anonymous No.12529745]

>>12529607
?

[Anonymous No.12529757]

>>12529441
???

[Anonymous No.12529787]

I'd like to understand godels incompleteness proofs. It looks like logic textbooks span from babby ven diagrams and intro to proofs stuff to books that I think assume you've already learned set theory and algebra at a moderate level. What's a good logic book for a comp sci code monkey?

[Anonymous No.12529789]

>>12529744
In the absence of "Rise divided by Run = Tangent" knowledge, I can write a script that can effective work the same way but way more fucking complicated. I love these little tricks that I would never have thought up on my own.

[Anonymous No.12529803]

>>12529789
I think you would if you would study it. Sounds like you just want to learn Euclidean geometry

[Anonymous No.12529814]

>>12529606
>>12529789
the next logical step would be a text on arithmetic geometry. i recommend the lecture notes of peter scholze.

[Anonymous No.12529899]

>>12529814
>peter scholze.
He's a meme. Couldn't even understand IUTT.

[Anonymous No.12529987]

would anyone here have links or something about the polynomial algorithm about pascal triangles? i have it as an assignment for college, but the library is closed indefinitely so i'm really struggling with finding sources about it.

[Anonymous No.12530012]

>>12529987
This video is extremely basic, but I find it beautiful and am sad that they don't do this anymore

https://youtu.be/Di-jAhrAXOY?t=510
(especially from 8:35 onwards)

Similarly, this
https://youtu.be/dBH-Id8VC3U

[Anonymous No.12530026]

>>12530012
i've only skimmed through the videos, but i'll definitely be able to find something in them that will help with the assignment as i also need to write about its implementation, history and such. those videos are really comfy, thanks.

[Anonymous No.12530085]

I'm sorry if my question is dumb for some of you, but what does [math]\mathbb{Z}/n \mathbb{Z}[/math] mean?

[Anonymous No.12530108]

>>12530085
Set of fractions [math]a/nb[/math] for [math]a, b \in \mathbb{Z}[/math].

[Anonymous No.12530132]

>>12530108
I thought it was that, but I got confused because of pic related. Can you explain to me why there's a direct sum in that group and why it says that that group needs to be finite?

[Anonymous No.12530183]

>>12530132
Ok, I'll stop trolling.
It's the group of residues module [math]n[/math]. Each integer leaves a remainder (nonnegative integer smaller than [math]n[/math]) when divided by [math]n[/math], having the same remainder gives an equivalence relation between integers, and as a set, [math]\mathbb{Z} / n\mathbb{Z}[/math] is the set of equivalence classes under this relation. You add two classes (this "direct sum" symbol is the addition being defined) by taking the class of a sum of two representants of each class (you have to understand what it means to be and to prove that this operation is well defined). This gives a finite group with its elements being the residue classes of [math]0, 1, \ldots, n-1[/math].
Later on you'll see that this is the quotient group made from the additive group [math]\mathbb{Z}[/math] by its subgroup of integers multiples of [math]n[/math]. That is why we use this notation, it's the notation used for quotients. More than that, integers form a ring and this subgroup is an ideal, making this quotient a quotient ring.

By the way, are you the anon from the comfy uni >>12525539 ? Good luck!

[Anonymous No.12530205]

>>12530183
Oh, ok, I think I understand it now, thank you and yes, that is me.

[Anonymous No.12530363]

Good materials for being able to do IMO level number theory problems? Able to do my countries olympiad problems but I'm struggling with IMO level.

[Anonymous No.12530419]

Any books you would recommend going through on vacation? I have already taken calculus 1 and linear algebra 1

[Anonymous No.12530437]

>>12530363
want a honest opinion? Stop wasting your time solving olympiad problems, they are totally counter intuitive when it comes to learning actual mathematics, same goes for college competitions such as the Putnam.

[Anonymous No.12530441]

>>12530419

James stewart's book on calc 2

[Anonymous No.12530448]

>>12530441

stewart? 2020?

[Anonymous No.12530456]

>>12530448
I don't know about the newer editions, however the older ones surely make the job.

[Anonymous No.12530461]

>>12530441
Funny you say that, I know a guy selling that one. Guess it'll do

[Anonymous No.12530462]

>>12530437
Surely getting better at problem solving is good though right? If not what are some better number theory resources since I'm just interested in number theory.

[Anonymous No.12530468]

>>12530462
Anyways, since you've asked for it, here it is

https://artofproblemsolving.com/wiki/index.php/Category:Olympiad_Number_Theory_Problems

[Anonymous No.12530482]

>>12530468
ty fren

[Anonymous No.12530495]

>>12530363
Google it, there are tons of material and comments about it on the internet. In particular, you'll find plenty on that AoPS website.
I didn't train specifically for IMO, so I can't say anything more specific.

This anon >>12530437 is wrong, it is definitely good for mathematics. But don't ever think that it's enough for becoming a good researcher, it's not the same and there is more to that. Olympics who want to become mathematicians should gradually put those problems aside until virtually not spending more time on them before becoming graduate students.

[Anonymous No.12530497]

/mg/ doesn't know anything.

[Anonymous No.12530513]

>>12530497
Prove it

[Anonymous No.12530518]

∀

[Anonymous No.12530532]

>>12530437
Care to expand on that opinion, pal? The problem with this site is that people are so entitled that they think they can make outrageous claims like that one without providing any argumentation to sustain it.

[Anonymous No.12530551]

>>12530495
Thanks fren, I'm off to uni next year so I'm doing this rather than getting ahead as I'll be just relearning if i do :(

[Anonymous No.12530560]

>>12530551
You're not as intelligent as you think you are. You're just an entitled little high school kid, know your place! Can't wait for college to filter you.

[Anonymous No.12530572]

>>12530560
Yikes

[Anonymous No.12530582]

>>12530363
is your country good at imo? you know everything you need for imo number theory? some relatively advanced stuff off the top of my head:
quadratic residues, lagrange symbol, the (p/q)(q/p) theorem, all kinds of things connected to mod p arithmetic, the (p-1)!=-1 thing, generators and their existence mod p^k or 2p^k, pell equation
maybe gaussian integers
other than that, you just need to train a lot, the resources are almost infinite - you can go through all available IMO problems, or go to chinese/usa olympiad problems - these are slightly easier than IMO usually. Generally, older problems (say pre-2000) are usually easier. So you can probably find something from those that suits your level. By this I mean, you should be able to solve 1/3 - 1/2 of the problems you attempt (this is just my opinion)
There are also IMO shortlist problems
the AoPS forums mentioned is a good place

>>12530551
enjoy your olympiads as long as you are in high school

[Anonymous No.12530866]

Happy new years math general
I am entering the heart of darkness NYC
Nothing can be more blessed than the new light at the beginning of the new year, the sunrise of freedom

[Anonymous No.12531221]

>>12530560
Shut up dude

[Anonymous No.12531281]

>>12523028
for a convex object, just see if the line in the direction of the normal intersects the object elsewhere, if it does, it must have been inward.
as the other poster noted, however, this doesn't matter at all.