/sci/ - Science & Math

>>11719273
Why do so many people claim Serre is a good writer? He writes like somebody took a regular book and arbitrarily deleted 2/3 of the lines.

>>11719349
He writes concisely and that is good.

>>11719273
is there a book like this but for e&m?

Anyone know the command to format your text like CONTENTS here?

>read serre
The only thing I read is my diary.

>>11720224
You mean apart from typing
>\tableofcontents
in the LaTeX file?
There's a LaTeX StackExchange that surely has your answer. This particular page is fun:

https://tex.stackexchange.com/questions/1319/showcase-of-beautiful-typography-done-in-tex-friends

Incidentally related (just uploaded 15 minutes ago)
https://youtu.be/bk0J7WGwk90

>>11720793
I think he means the funny letters where they're all in caps and the first one is bigger.

>>11719273
am I supposed to understand this

>>11720834
Yeah I donno, as I said look on the SE

https://tex.stackexchange.com/questions/293912/table-of-contents-title-in-uppercase-letters

https://tex.stackexchange.com/questions/37935/table-of-contents-without-title?rq=1

I expect it's a packet in that case.

>>11720839
yes
it's so terse it's basically a skeleton, but it's also page 1 of the review part of the book. it's all really fundamental finite field stuff, any half-decent algebra class would've covered it all

>>11720224
\tableofcontents

>>11719273
>only 119 pages
Something tells me that I'd have to read 1000 pages worth of materal fron other places to understand this

>>11720215
I liked Baez's Gauge Fields, although this was at a much lower level the exposition was just fantastic, and also Smythe's Static and Dynamic Electricity, although that's definitely for practical physicists because the problems are mostly hard shit meant to "weed out the weak" in the author's own words because he hated people who couldn't apply their knowledge

>>11719395
conciseness is certainly preferable to verboseness
but mere few word bare concatenated sentences do not make good math writing

>>11720974
Thats sort of the point. My college taught out of dummit and foote. But if youre just trying to review the fundamentals, this would probably be a good book.

think about this:

for any [math]n \in \mathbb{N}[/math] and any group [math]G[/math], commutative if [math]n\geq 2[/math], there exists a space [math]X[/math] such that
[eqn]\pi_i(X) = \begin{cases}
G & i=n \\
0 & i\neq n
\end{cases}[/eqn]
we can realize groups as blowing up balloons.

HOT OR NOT ??

>>11721292
I dont really understand what you are saying, could you explain?
are you saying that for any commutative group, there is a space such that one and only one permutation group on that space = the group?
and how is that blowing up balloons?

>>11721789
Homotopy group, not permutation group. Read Serre.

Just finished my precalc book. Should I do calculus or linear algebra next? Or something else?

>>11721292
Similarly, if you choose an abelian group [math]A=F\oplus T[/math], where [math]F[/math] is finitely generated & free and [math]T[/math] is torsion, and [math]n\ge 1[/math], there is a space [math]P^n(A)[/math] whose reduced integral cohomology is trivial for all other degrees except [math]H^n (P^n(A)) \cong A[/math]. If you then take the reduced homology of this space, you get [math]H_n(P^n(A)) \cong F, H_{n-1}(P^n(A)) \cong T[/math] and trivial otherwise. Obviously, [math]S^n[/math] is a [math]P^n( \mathbb{Z})[/math], even for n=1, and so [math]\pi_n(X) = [ S^n, X]_* \cong [P^n(\mathbb{Z}), X]_*[/math], so let [math]\pi_n(X; A) = [ P^n(A), X]_*[/math] gives you the nth homotopy group of [math]X[/math] with coefficients in [math]A[/math]. Ordinary homotopy is then the same as the integral homotopy. What do you think?

>>11722003
Oh and notice how you have [math]H^n(X; A) = [X, K(A, n)]_*[/math] so this is sort of cool, ay?

ahhh i just passed my heat transfer final bros. I had to study a shit ton of diff equations solving methods but it was worthed in the end. Fucking bessel functions almost got me but i made it.
t. mech engineer mathlet

>>11722003
very nice butters. very nice

>>11721789
it's not permutation group, it's homotopy group.
homotopy group of a space measures if there are holes by mapping spheres into the space and looking if the spheres can be contracted (if they can't, there's a hole). you can combine two such mappings to get another one, this gives group structure.
read about fundamental group, it's intuitive

>>11722003
is there a group structure on [Pn(A),X] ?? I thinks not

you guys are such nerds

>>11722386
Another nice thing about it is that you can combine it with the fundamental theorem of finitely generated abelian groups. If [math]A[/math] is f.g., then [math]A \cong \mathbb{Z}^n \oplus \sum\limits_{i_1}^m \mathbb{Z}/k_i\mathbb{Z}[/math], and so it suffices to find one of those spaces for each summand and take the wedge of them, as [math]H^n(X \vee Y) \cong H^n(X) \oplus H^n(Y)[/math]. As mentioned before, [math]S^n[/math] can be used as [math]P^n(\mathbb{Z})[/math], so we need to get the [math]P^n(\mathbb{Z}/k_i\mathbb{Z})[/math]s somehow. This happens by taking the degree [math]k_i[/math] map [math]f\colon S^{n-1}\to S^{n-1}[/math], and then its mapping cone gives you [math]H^(C_f) \cong \mathbb{Z}/k_i \mathbb{Z}[/math] and trivial reduced cohomology otherwise. Taking the wedge of these gives you [math]P^n(A)[/math]. Easy to construct.

>>11722871
When [math]n=2[/math], you get just a set, but if [math]n\ge 3[/math], then it is a group, abelian when [math]n\ge 4[/math]. You get the group structure using the comultiplication of co-H spaces, as [math]\Sigma P^n \simeq P^{n+1}[/math] and suspensions are co-H. Then you apply the contravariant functor [math][-, X]_*[/math].

>How exactly is the group operation on an abelian variety generally defined?
>I know it for elliptic curves, but otherwise, for an entirely generic variety, how do I know how to get from two points to a third? What's the definition of the multiplication?

>>11718904
>>11719022
Thanks for the answers. But given that the abelien variety is given by a polynomial expression and appearently there's a group associated with it, why would it be as general?
Isn't it the case that you drop an equation, say the integral or rational are the elements and then have group operation about it? Otherwise, how could you prove general theorems about it, like Mordells theorem?

>>11723279
>ΣPn≃Pn+1
right I didn't realize that
I approve

>>11723572
so you're looking for something like Lie groups "=" matrix groups? then I don't know because I don't know shit about algebraic geometry
but I'll tell you that general theory of Lie groups can be built completely without mentioning any matrices

>>11723626
Not sure why you speak about Lie groups or matrices tbqh

The function [math]\ln{x}\sqrt{sin(x^{1.5})}[/math] for real values of x has the plot shown here. What is the nature of the curve connecting the cusps?

>>11720224
Read the koma script documentation on that topic.
Seriously, documentation is like stack exchange, just infinitely better. In fact 99.9% on there should be answered with RTFM, that would more helpful.

>>11723687
It's where sin(...) = 1, one would suspect.

>"irrational" lengths

>>11723687
just plot sin(x^{1.5}). beyond x=1, the rest of your function only scale that

>>11723687

>imaginary part

oh nonononono

There doesn't seem to be very many examples of categorically different Abelien Varieties.
Either elliptic curves of a construction of differential-forms..

There doesn't seem to be very many examples of essentially different Abelian Varieties.
Either elliptic curves, or a construction of differential-forms...

>>11719349
He's very succinct, but there is never anything missing (although sometimes you kinda have to squint to realize it)

Divisors, how do they work?

What are your /hobbies/, /mg/?

mathlet here, what are some common algebra letters like x y, i j, n m?

>>11724052
Osu, watching anime, shitposting on this board. Not much else beside those.

>>11724057
Currently I spend my days shitposting here, watching anime, and studying math too, but looking for maybe something more productive than anime, maybe something to do before bed

>>11724052

video games
cooming
loafing
loitering
floundering
lying on applications

>>11724052
Lying in bed trying to avoid getting drunk and spending all night shitposting here. Well, almost midnight and here I am... This is starting to make be a bit chubby.

>Smart enough to understand problem statement
>Smart enough to understand its proof
>Not smart enough to ever come up with such a proof by yourself
I feel like shit.

>tfw handwriting is so shitty it sabotages my calculations

>>11724127
You're not shit or any other excrement, anon. You just nee to get used to the tricks of that field.

>>11724223
Maybe you should try learning a nicer handwriting. Also, it is useful to put a little line through the middle line of z to distinguish it from 2 etc. Coloured pens are pretty cool if you need to draw diagrams and stuff.

If you have a topological space X, and a ring R, you can construct the n-th homology module [math]H_n(X, R)[/math] which is an R-module.
But from what i've seen so far, [math]H_n(X, R)[/math] turns out to be R^k for some integer k.
Can you give me examples where this is not the case?

>>11724052
https://www.youtube.com/watch?v=iE6B-R6UXSY

Also volleyball, guitar/singing, vidya, /tg/ stuff and not starting creative projects

>>11724295
oh forgot animu

>>11724287
[math]R=\mathbb Z[/math] and [math]X=\mathbb R\mathrm P^n[/math], or the Klein bottle. I like to imagine torsion subgroups got their name from "torsion" in topological spaces, ie twists and turns in their structure. BTW if you're looking for any other examples, you need to probably consider non-orientable examples, since orientable manifolds (like torus, sphere) don't have torsion (at least at the fundamental class, can't remember about the rest).

>>11724287
Have you tried using a ring like [math]R = Z_2 \oplus Z_3[/math] and computing for [math]X = RP^2[/math] or a similar enough space? Wild guess says [math]H_1 (X, R) = \mathbb{Z}_2[/math]

>>11724287
https://en.wikipedia.org/wiki/Universal_coefficient_theorem
might yield some more examples

>>11724287
Let's be simple and use [math]R = \mathbb{Z}[/math] because I don't want to think about torsion now. Take the degree 2 map [math]S^2\to S^2[/math] and let [math]C[/math] be its mapping cone. Then you have the segment [math]0 \to \mathbb{Z} \xrightarrow{\cdot2} \mathbb{Z} \to H_2(C) \to 0[/math] in the LES, forcing [math]H_2(C)\cong \mathbb{Z}/2\mathbb{Z}[/math].

>>11720224
\textsc
>>11724287
something like this maybe? https://en.wikipedia.org/wiki/Hawaiian_earring#Higher_dimensions

this is so cool

>>11721935
Whatever your heart desires.

>>11724418
Whatever your heart enjoys.

>>11721292
What is your favorite way to make a [math]K(G,n)[/math]? Mine is iterating the bar construction starting with [math]K(G,0)[/math] as the simplicial set

[eqn] G \overset{\overset{\text{id}_G}{\longleftarrow}}{\underset{\text{id}_G}{\longrightarrow}} G \overset{\overset{\text{id}_G}{\longleftarrow}}{\underset{\text{id}_G}{\longrightarrow}} G \overset{\overset{\text{id}_G}{\longleftarrow}}{\underset{\text{id}_G}{\longrightarrow}} ...[/eqn]

Mathlet here, I'm just studying some gas dynamics and as I read up on the Boltzmann-Constant I saw this on the German Wikipedia:

>Is the value of the Boltzmann-Constant written in eV/K the value of the quotient of both the exact numbers 1,380649·10−23 and 1,602176634·10−19 has no finite decimal expansion and therefore has to be written with three dots: 8.617333262...e-5 eV/K.

My question is now, how likely is it that two "random" decimal numbers that aren't terribly long would have a quotient with no finite decimal expansion?
My very basic hunch would be that it's pretty damn unlikely that two unrelated numbers would act this way and that if anything, this would happen if the numbers were very close, which they aren't.

Any explanation on this? Is it common and I just now found out or is this a pretty remarkable coincidence (or perhaps no coincidence at all and the numbers are in fact related)?

>>11724470
That diagram looks autistic. Very based.

Does anyone of you look at the general /sci/ front page anymore?

>>11724471
Firstly, fooks like a sloppy translation on your side.
Secondly, the quotient of two rationals has an eventually periodic pattern, so it's not clear what the claim is here.

>>11724487
>Firstly, fooks like a sloppy translation on your side.
Probably.
>Secondly, the quotient of two rationals has an eventually periodic pattern, so it's not clear what the claim is here.
I'm asking what the rule is on when a quotient of two rational numbers has a finite decimal expansion. 2/5 does not have an infinite expansion, so it can't be every one.

>>11724502
What I'm saying is that, in either case, you can write it down in a finite fashion, since eventually the pattern of a rational repeats. It shouldn't have to be written with "...", at least not in the sense that the full information wouldn't be given.

>>11724517
That's not my question though. I don't really care about the thing with the dots, just wanted to give context.
"I'm asking what the rule is on when a quotient of two rational numbers has a finite decimal expansion."
This is what I actually want to know (specifically so I can understand whether or not it is likely these two numbers are physically related).

>>11724521
have you tried googling?

>>11724521
The reason why yhis holds is that we aee in base 10 asnd those are the only pries dividing 10.

>>11724554
This = >>11724543

I'm trying to compute [math](ij)(12...n)(ji)[/math] but am running into trouble. Obviously computing this permutation is easy, but I don't really know what to do for this general case. Any help?

>>11724543
>>11724554

I tried but my english math vocabulary apparently isn't good enough. Thanks.

>>11720889
Very, very nice.

>>11724560
have you tried seeing what happens to individual elements?

>>11724319
I think I managed to understand the projective space example. This is exactly what I've been looking for.
>>11724333
OK this works too. If I replaced [math]S^2[/math] and [math]H_2[/math] here with [math]S^1[/math] and [math]H_1[/math], the space C would be actually [math]\mathbb{R}P^2[/math], right?

>>11724619
Yeah. The degree n map of [math]S^1[/math] is easy to understand: [math]e^{i\theta} \mapsto e^{in\theta}[/math], so the degree 2 map would identify the antipodal points and give you that. I hope there are no typos.

Good night

>acts on your symplectic manifold
Heh, nothing personal kid.

>>11724681
huh?

>>11724052
undergrad math problems, tutoring, singing
some computer programming

>>11724418
It is ! You can also compute the volume of the n-ball by a similar argument

>>11724418
It is ! You can also prove that the volume of a cone (with an arbitrary base) is 1/3 (area of the base)*(height) using the same technique.
You can also compute the volume of an n-ball using similar ideas

>>11724840

>tutoring

what are your rates to teach me about your cock anon?

>>11724052
Reading books, jujitsu and hopefully soon firefighting.

any current grad students have advice for someone with bad gpa (low 3) , no grad courses (school is anal about granting permission for grad courses ) but good math GRE (90th %ile). Is it possible for me to get into an ma program?

>>11725698
You could probably get into a low tier MA program, unfunded.

>>11725730
is there any way I can step up from that if I do well (into a good PhD program)?

Let's take a triangle. For the sake of the simplicity let it be an equilateral triangle.
Centroid (aka the center of the mass) is the point at which the figure stays in balance. But that does not mean that if we draw any line through centroid then the triangle will be divided into two parts of equal mass (surface area)? It is true if we draw a linie from the vertex )because of median properties) but if we draw a line parallel to the base then we will no longer have two parts of equal weight. In equilateral triangle we clearly have two different weights.
Is there such concept of dividing an object into two parts of equal surface? For symmetric shapes like circle, square or ellipse it is obvious that the any line going through their center will divide it, but what about others?

>>11724681
pls explain on example what does action of torus on symplectic manifold means in mechanics

>>11724052
learning japanese
learning french
learning QFT
drawing
reading manga/literature

>>11719273
hello and good morning every ones, assume am retard with zilch mathematic ability from not applying ones self in academia, where would be a good start with reading, an introduction etc for basics. thank in advance

Is it retarded to start learning Physics II because I became interested in entropy after watching Madoka Magica yesterday?

>>11725698
low 3s is bad? mine's 1.5 kek

>>11726517

>spend four days calculating 3-2
>result: 4
....

>>11726517
Start with the Verbitsky list. Since you seem to be unbound by course work, you should be able to finish it in 2 or 3 years. Unless... well, you know unless what. The limits of human performance need not be mentioned.

>>11726525
It's good to have some motivation for stuff. I applied for maths because of Logicomix, so I personally can't say that would be at least much worse. Just as long as it wasn't the English dub.

>>11726565
I feel you.

>>11726533
low 3 is not what you want for graduate school standards
the formal minimum cutoff for almost all places is 3, and a lot of the places you'd call a "good" school want at least a 3.5

I dunno how 1.5 is even achievable. You must be either severely mentally ill, severely lazy or severely stupid. Or some combination of those.

>>11725698
How much money do you have to apply with? Low 3 isn't great, but it's not unworkably bad, and it will be counterbalanced a bit by the very strong GRE score.
You're not getting into MIT or anything, but if you can get some decent rec letters and you can afford to spam like 8 or 10 PhD applications you have a pretty reliable shot of getting into at least 1 or 2 places you'd be happy with.

>>11726565
forgot pic

>>11726582
lol, looking at it now, I made a mistake going from the fourth to the fifth line

i want to say FUCK differential forms
if [math]d\omega=0[/math] it doesnt follow that [math]\omega=0[/math] at all, why would you name this operation "derivative"

>>11726630
df/dx = 0 implies f = 0 ?

>>11726630
you have discovered that geometers are a cancer upon mathematics
there is hope for you now

>>11726630
what?

Seems you mean constant. Even if, why would it follow, given it's about cooridnate free presentations..

>>11724052
taking moocs in deep learning
films
reading rudin
vidya

>>11726640
>>11726643
yes i meant: [math]d\omega = 0[/math] doesnt imply [math]\omega[/math] is constant
>>11726643
how is being coordinate free relevant here

>>11726656
df/dx=0 doesn't imply f=constant either.

>>11726630
There's a reason everyone just visualizes the exterior differential as "Stokes works" and more abstract forms of "Stokes works."

>>11726656
first of all, df/dx = 0 doesn't imply f is constant
second of all "partial derivative of f = 0 doesn't imply f is constant why would you name this operation "derivative""

>>11726656
third of all, what does it mean for a differential form to be "constant" ?

>>11726665
>>11726680
let U be an open connected subset of R^n
let [math]f:U \rightarrow \mathbb{R}[/math] be a differentiable function
if [math]f' = 0[/math] everywhere on U, then f is constant

for comparison, consider the 1-form [math]\omega = \frac{-ydx + xdy}{x^2 + y^2}[/math] defined on [math]\mathbb{R}^2 \setminus /{(0, 0)/}[/math]
then [math]d\omega = 0[/math] but [math]\omega[/math] is not constant in any meaning of this word
please explain to me: how does the behavior of [math]d\omega[/math] relates to [math]\omega[/math] itself

>>11726721
Stokes.

>>11726721
>connected
this is an extra assumption. df/dx = 0 => f is constant holds only when the domain is connected.
for a k-form w, dw = 0 => w is constant (whatever that means) only when the domain k-connected

f is constant means that the value doesn't depend on the point. note that f is 0-form and point is 0-manifold.
w is constant if it "doesn't depend on the k-manifold". here I mean integration and I'm very very imprecise, but the generalization is there.

>>11726721
I see you are going through the traumatising differential geometry course.

>>11726672
>Stokes works
What are these? Or do you mean a mantra like "Stokes works Stokes works Stokes works..."?

>>11726721
>>11726737
I'll add that in your example the domain is not simply connected (1-connected) and so the implication dw = 0 => w = dv doesn't hold

if w = dv holds, then the integral of w over any path depends only on the end points. this is what "constant" means. you can understand it that the integral of this 1-form depends only on something 0-dimensional.
for a function (0-form), it means that the "integral" of it (which is just evaluation at a discrete set) depends on something (-1)-dimensional which is the empty set. so it doesn't depend on anything. and you recover the usual notion of constant function.

>>11726737
>>11726757
thank you this clears up my confusion to some extent
>>11726742
i had my differential geometry course 2 years ago and i never learned differential forms properly
and now i need to know this stuff

>>11726577
>severely lazy
yeah that's my problem
>severely stupid
I'm not, my IQ is average

>>11726630
Differential forms are weird combination of calculus and linear algebra. You’re thinking strictly in Calculus but for differential forms this is true.

>>11726815
To cut open the wounds once closed... The best of luck, anon.

>>11719273
What is a hamiltonian walk?

>>11724052
programming
watching movies

>>11726815
>and now i need to know this stuff
Is it something autistic like equivariant de Rham theory or did you somehow dodge learning Chern-Weil theory for two years?

>>11726871
What's a good book on those topics?

>about to finish my master's
>going into a job in finance
>i will no longer have any motivation to study maths any further
feels absolutely awful
i've only just started with topics like arithmetic geometry and i wish i had the time to take them further

>>11726907
What area of finance?

>>11726921
quantitative trading
i mean, i love the job, but it's just sad that this is essentially the end of my mathematical journey (unless i can drum up the motivation to continue learning in my spare time but i doubt that)

>>11726871
stupid tranny

AAAAAAAAAAAAAAAAAAAAAAAAAAA
I WANT TO BE GOOD AT MATHS AAAAAAAAAAAAAAAAA

>>11726577
Would it improve my chances of getting into a good PhD program if I did an MA first ( assuming I do well in it)?

>>11726960
You know what to do.

>>11726971
Is it to find a cure for ageing so I never become 24 years old?

>>11726969
Yes.

bros...

>>11726656
>how is being coordinate free relevant here
there's no "w(x)", so what do you expect from a derivative on a coordinate free object. What would the equation, with zero derivative, look like so you wouldn't complain the way you did, is what I'm asking

>>11726969
Yes. This is not a totally uncommon thing to do for students who got interested in graduate school too late to be competitive for the PhD they want. So if you really want to push for a top-tier school it's not a terrible idea.
Although I should point out that what the first poster said is terrible advice. You should not even consider accepting an unfunded offer anywhere. In mathematics (and in most non-professional graduate degrees) unfunded means "we don't want you, but if you're stupid enough to pay us for something you should be getting paid for we'll take your money".

>>11727010
>You should not even consider accepting an unfunded offer anywhere.
not true at all, in my country universities are free and the only ones who get paid for doing a PhD are the geniuses who got only 9s and 10s during their majors

Why does .ps format exist?

>>11726902
>equivariant de Rham theory
No idea. I only know about it because of Audin's Torus Actions book.
>Chern-Weil theory
Tu's Differential Geometry, I suppose. Someone else probably has a more advanced recommendation.

>>11727028
The dude is talking about GRE scores and writes his GPA on a 4.0 scale, he's clearly American.

>.psst kid

>>11726997
NO!!!!!!!!!!!!! YOU ARE NOT 24 YET!!!!!!!!

>>11727047
What?

>>11727051
W-why, what happens when you turn 24?

>>11727095
>he doesn't know
OUCH

funfact of the day
https://en.wikipedia.org/wiki/Tennenbaum%27s_theorem

>>11726871
>tfw im about to start going through Tu's book on equivariant cohomology
>tfw im autistic

>>11727095
Some of us have crossed that line, some of us haven't. This leads to a problem in communicating this clearly, for I have seen the other side and you have not. It is beyond belief what kind of alterations your mind and body go through when you go through the veil of 24 years. It is not something your baby mind could comprehend if I were to use words to describe it. It is something you will have to experience yourself. It is the point of no return. Have you ever wondered why so many musicians die at the age of 27? Usually it involves substance abuse, as they can't handle the pressure brought by crossing the line. Speaking of that, I managed to resist the bottles and surprised my liver with water today. /blogpost

>>11727107
I really tip my imaginary hat to people who can prove those things. I remember trying to prove some very very very elementary stuff in the logic course back in the days and it was really surprisingly hard. I don't think it was even anything as "big" as the commutativity of addition.

>>11727125
what book?

>>11727107
Hey, he published that result when he was 32, there's still time.

>hey Dmitry, remember that time me and Liouville collaborated on the Arnold-Liouville theorem?
>best collaborator I ever had, absolutely peerless to work with

>>11727134
>tfw 24 in 10 days
I'm in a good grad school, so I'll make it bros...right?

>>11727141
Respect the good grad school by spending those 10 days writing a high quality letter of resignation. I'm sorry, anon. That is all that is left to be done anymore.

>>11726565
I got it down to the precedessor of 2!

>>11727141
>tfw 23 in 6 days
>don't even have a PhD
I'm....I'm not gonna make it bros..........

>lost my one chance for true love
>lost my one chance for a success in academics and doing something with my life
>24 in few months
bros...

>>11727134
I actually don't like this sort of theorem. Or not even the field, really.

But, as often is the case, some things become interesting (for a moment) when it's the resolution to a thing you were wondering about yourself, or when you ask yourself a question. There's this interesting thing where something you'd not have given much attention becomes memorable simple because you discovered it yourself or because it helped you. This is a good argument for actually doing things yourself and now skimming over a text and simply taking in the theorems and tactics.

>>11727139
What are you getting at?

>>11727150
If you're spending 4 days, why not use a tool to guide you.
Like coding up this
https://en.wikipedia.org/wiki/Church_encoding#Translation_with_other_representations
and then redoing the whole thing to print out the intermediates. This actually sounds like a nice project - simulating this execution in a way that you can runtime inspect it.

At one time I wrote a tiny universal class for General Recursive Function execution, that you functionally pass the incremental step. I passed it the "3n+1 else n/2" task and then took this Turing-machine like execution and dovetail it,
https://en.wikipedia.org/wiki/Dovetailing_(computer_science)
to explain recursive sets
https://youtu.be/Ox0tD58DTG0
That was really fun, tqbh.

>>11727183
I think I'm 33.
bros...

>>11727222
>I think I'm 33.
How do you think you're 33, shouldn't you know that?

November 86, I can make the calculation if you I have to. If I just say a number it might be a year off

>>11727236
You must be younger than 24 if you've never had to stop and think to remember how old you are
Once you get past your early 20s, the combination of it no longer having any relevance to anything and time starting to rapidly accelerate means the exact number stops being an automatic reflex in your brain

>>11727222
>What are you getting at?
I find this entire thing funny.

>>11727253
Sometimes, when I strive through the library I stop at that old unsorted shelf with 60's and 70's mathematical physics textbooks and the stuff of Arnold and co always makes me depressed. This is because they already use all the ideas and language that's also currently hip and I feel that most things have already been done and things just move too slow.

I wonder if Thiel & co. are right and there's stagnation in the sciences (including math), or if it's just so much that we can't - at any point - see all the cool stuff that's emerging at that moment.

Arnold is also interesting for his aggressive rhetoric.
I actually read my fair bit of epistemology and have edgy views myself, but it's hard to talk about those things in a concise, organized manner, because it's a huge lot of work to find a starting point. And if you don't address potential counterpoints from every sperg from Hume to Dennett, you end up visible in discussions that make it seem like your take was a naive one. So too much work.

>>11727222
>I actually don't like this sort of theorem. Or not even the field, really.
Neither do I, and the same holds for analysis. Yet I must still say I feel a lot of respect to people like Lebesgue for doing the stuff I would never imagine being able to do (or interested in, but that is irrelevant).
>where something you'd not have given much attention becomes memorable simple because you discovered it yourself or because it helped you
Yep. It is the act of getting your hands dirty that really teaches the stuff, I think. A really simple example would be when I needed to get a map [math]B(\mathbb{Z}/n\mathbb{Z})\to \mathbb{C}P^\infty[/math]. I'm fairly sure I would have had to spend a lot of time on that if I hadn't discovered [math]\mathbb{C}P^\infty \simeq BS^1[/math] by playing around with stuff. This is also why I always tell people to prove the identities when they complain about logarithms or trigonometric functions.

>>11727236
>>11727251
Yep, and when you are no longer a happy-go-lucky youngling (under 24) you try to forget your age to protect yourself from pain.

>feel anxiety every time someone posts symbols I don't understand on /mg/
how do i stop this...

>>11727371
have you tried not being a gigantic faggot

sisters...

>>11727371
You could just ask. The touhou poster sees all and seems quite capable of answering pretty much everything. The more symbols you know, the less there are scary unknown symbols.

>>11727413
Wanna be bitten, ay? I'll smash yer foockin' 'ead in if ya keep this up.

>>11727417
>Wanna be bitten, ay? I'll smash yer foockin' 'ead in if ya keep this up.
Not her, but yes

>>11727417
>and seems quite capable of answering pretty much everything
I have never seen him/her/it answer anything.
Usually he goes on a tangent about whatever interests him/her/it wants to talk about, completely ignoring the question or giving an answer the person clearly is unable to understand.
I hope he isn't teaching anybody, or if he is, he is not doing the same shit he is doing on 4chan.

>>11727417
>The touhou poster sees all and seems quite capable of answering pretty much everything.
I'm out.
See you lads in July.

>>11727438
I have never seen him/her/it answer anything.
Usually him/her/it goes on a tangent about whatever interests him/her/it wants to talk about, completely ignoring the question or giving an answer the person clearly is unable to understand.
I hope he/she/it isn't teaching anybody, or if he/she/it is, he/she/it is not doing the same shit he/she/it is doing on 4chan.

>>11727427
One bite per every year of age. That will be at most 23, right?

>>11727438
>>11727442
I don't know. I don't understand most of the questions, so maybe yes maybe not.

>>11727439
Now that you mentioned July, does it not bother anyone else that the 9th-12th months are called 7th-10th? They just had to ruin everything by putting Julius and Augustus before them.

>>11727438
sort of agree

>>11727439
what are you doing in the meantime?

>>11727417
>>11727427
what do you faux lesbos have going there?

>>11727222
I have no idea how to code it up, considering church numerals are both functions and numbers, and a programming language won't know when to treat it as a function and when as number
pic related, it obviously gives an error since "add" has 4 parameters but only 2 are given (the other 2 are irrelevant when adding 0 and 0, but the interpreter doesn't know that)

Find the next 4 terms of this sequence and describe it:
2,6,15,35,77,143...

>>11727460
Just trying to scare people into success. That's some heavy duty mouth breathing in the background btw.

>>11727508
N-not doing your homework.

>>11727529
>N-not doing your homework.
It's not a homework problem, I just kind of pulled it out of my ass and found it interesting because it's not on OEIS

>>11727480
"clearly" the issue is not that
>a programming language won't know when to treat it as a function and when as number
but an issue is how you display the return value if it's a function.

Regarding your 4 vs. 2 problem, I'd think the argument in func of your add-definition should be a 2-ary lambda function (the arguments being n and m) and it's return value is a lambda term taking f and x

>>11727508
Google gives me that task as Quora question.

But I have a problem.
Bros.
How to pronounce this theory of the Aczel in pic related?

>>11727556
>Google gives me that task as Quora question.
Their sequence is not the same

>>11727508
1*2
2*3
3*5
5*7
7*11
11*13
13*17

DAB

>>11727558
??

>>11727556
Also, another not regaridng the quote statement, I don't think you ever "have to treat them as numbers", just treat them as functions through and through

>>11727565
I used startpage so I must have looked at a different question

>>11727562
Yep

>>11727534
What I tried to say is that it is too hard for...

>>11727562
I mean, obviously. If you didn't come up with this solution in less than 5 seconds, I have some bad news for you.

>>11727427
Akko is for STRAIGHT only. Stop posting this OOC garbage.

>>11727569
>I mean, obviously. If you didn't come up with this solution in less than 5 seconds, I have some bad news for you.

>>11727576
Don't worry, I didn't get it either.

>>11726933
Care to blogpost about how you got the job and what the process was like? Current PhD student here, halfway through my degree and considering non-academic career options. Is it true that there's a growing demand for functional programming in finance? Were they interested in any particular courses/research you had done, or were they just happy that you had a math grad degree?

>>11719273
Noob question, but what are some good (free) resources to learn differential geometry?

>>11727556
>Regarding your 4 vs. 2 problem, I'd think the argument in func of your add-definition should be a 2-ary lambda function (the arguments being n and m) and it's return value is a lambda term taking f and x
No. Church numerals are functions with 2 inputs, like a->b->c in lambda calculus.
If you want a function which transforms 2 numbers into another one, like add, you need a function with 4 inputs, like summand1->summand2->x->y->z. Otherwise the resulting function would not have 2 inputs and therefore not be a number.
You can check on the page you found yourself, by the way, addition is defined as a function of 4 variables here too
https://en.wikipedia.org/wiki/Church_encoding#Translation_with_other_representations

>>11727635
Why do you write "No" here?

I think you didn't understand what I wrote there.
Your add function will be 2-ary, each of it's argument being a lambda term. What the add function returns is another lambda term (another function which takes arguments)

>summand1->summand2->x->y->z
Yes, your add will take summand1, summand2 and return a function of type x->y->z.
The return value (the number "summand1 plus summand2") is another python function, call it A. As such, the add function doesn't need 4 arguments. The add function doesn't need the argument slots that A would have.

All inputs and return values are always functions.

>>11726815
all jokes aside, differential forms are neat as fuck once it clicks. you never want to go back

>>11727457
~20 bites then!~

>>11727757
This anon is too scary to be bitten. Who knows what kind of evil machinations would await me?

>>11727757
>>11727869
stop shitposting

>>11727508
212, 226,76,-412

>>11727883
I will. Good night.

>>11726721
>please explain to me: how does the behavior of dωdω relates to ωω itself
Stokes theorem. Stop bitching now.

>>11727869

bros.... how do i start doing maths again....

>>11726933
London or NYC/Chicago?

>>11723777
>use ruler to draw 1 cm length
>rotate 90 degrees
>draw 1 cm length
>hypotenuse is sqrt(2) cm

>>11726525
It's never retarded to better yourself because something you love inspired it!

>>11728068
just do it. i got my bsc, did algebra up to algebraic geometry, graduated, became alcoholic, now studying Rudin to refresh. will refresh Galois theory after.

Books for getting into differential geometry?
Besides analysis on manifolds by spivak.
Books to read after that would be nice too.

>>11727609
i applied for an internship while halfway through my degree and got the job offer off the back of my performance in that. i think that's how the majority of traders are hired, though there are some grads that jump straight in without doing an internship. not sure about finance in general but i know my company is big on functional programming and i wouldn't be surprised if it's taking off elsewhere as well.

honestly if you're a maths student i'd fully recommend at least looking into it as a career path. it's a very mathematically minded job that's pretty intellectually engaging with good working conditons and great pay.

most companies, at least in my experience, won't be looking for any kind of research or courses or economic background when hiring. hiring is largely skills based, so interviews will consist of difficult probability or game theory problems to gauge how you think, not what you know. the key things they're looking for are mathematical ability, composure under pressure and an ability to assess one's confidence (i.e. not only should you crucially not be overconfident, but to also be able to quantitatively state how confident you are in a belief). so e.g. in an interview you may get given x minutes to try and find an optimal strategy for a game and estimate the expected value under the optimal strategy, then state how likely it is you've found the optimal strategy. if that sounds like your sort of thing then definitely look into it.

>>11728139
london

>>11728376
Differential geometry by spivak

>>11727508
0,0,0,0 obviously.
It's the polynomial in pic related.

>Cats climb entails mammals move: preserving hyponymy in compositional distributional semantics
Nice title. Thanks arxiv mailing list.
https://arxiv.org/pdf/2005.14134.pdf

>>11727480
>and a programming language won't know when to treat it as a function and when as number
It would, if you told it to.

For the love of god, don't use python to do this, if you were using Haskell, or any language with a very strong type system and good Functional programming support I think all your Problems would vanish instantly.

Also, your fucking code does not make any fucking sense.
The line alone "return self.f(*args)" is non-sensical. You give __call__ multiple arguments, so when you do "return self.f(*args)" you call self.f(arg[0],arg[1],...), but arg[1], is a string, so what the fuck?
Seriously, you need a type checker.

>>11729042
kek

>tfw too stupid

>>11729132
My HS maths teacher would say you have a weak butt. Better train those sitting muscles.

>tfw no HS maths teacher that was supportive and gave you some kind of advice

>>11729145
He also told us to make our bodies accustomed to arsenic by starting with small doses and then taking more, or how he doesn't smoke because he ate cigarette stumps as a kid and vomited. He'd call everyone by their surnames until the third year when only the ones who had survived the harder version of maths were left. Then he'd start using first names and for some reason it felt really good to hear him use it. Great guy.

Could you beat him, /mg/?

>>11719273
no clue if this belongs here but I couldn't think of anywhere else to ask.

in 6th grade my teacher taught us a way to approximate square roots. I don't know if he read it somewhere or if he made it himself, I remember him as a pretty smart guy whose favorite topic was math when teaching. anyways I remember it working something like this, with the number 180 as an example. the nearest perfect squares are 13^2 = 169 and 14^2 = 196. I remember the difference between the two nearest perfect squares being important too somehow, in this case 27. after this it became a matter of finding which square was closer to the number, in this case 169 is closer to 180 than 196.

anyways after some arithmetic, the part that I forgot, he could arrive at any square root accurate up to 3 decimal places, verified by a calculator. it was only a couple steps and didn't require recursion like Newton's method, it was accurate after one shot. a while back I looked for him on social media to ask him about it but I couldn't find him. sure there are faster, more accurate, or even easier methods, but I remember liking it because it could be done without any complicated math and the way it worked 'made sense' to me somehow. anyone know what I'm talking about?

>>11729193
In a fist fight?
Probably.

>>11729244
Probably a variation of this:
https://en.wikipedia.org/wiki/Bisection_method

>>11729193
he may know his logic but i'm pretty sure he would shit the bed when given any IMO problem

>>11729193
I would let him challenge me to a duel. Then I could choose what the duel is about and maximise my chances of victory. I would also shame him for not challenging me which leads to 3 possible outcomes:
(1) If he has no intention to duel me, neither of us loses.
(2) If he wants to duel me, but has no courage to challenge me, I win the duel of wills.
(3) If he wants to duel me and does so, I can try to organise things so that my victory is almost assured.
I would say it is unlikely that I would lose to him.

>>11729193
no... im too stupid...

>>11729261
>IMO problem
not science or math

>>11729320
that's actually math
logic and set theory is not math

I hope hijacking this for my own idiotics is, for a brief moment, acceptable. Existential crisis related to studies in mathematics.
Previously, I spent a year studying mathematics, but being immature and unable to understand what it means to study something, I would spend about a week before each test going through the course material. Top grades, but no deep appreciation for the material itself. Drifting through life, I ended up studying architecture instead. Not entirely at random, but certainly not as a result of lengthy consideration.
Now, some years later at age 25 I find myself with a bachelor's and master's degree in architecture, a published thesis/papers and opportunities most can only dream of, among which is a position at what would be considered the absolute top workplace in the architectural world, while still being able to appear as guest critic in academia.
Yet, I absolutely miss mathematics. The average architect is prone to arriving at illogical conclusions, which I and my autistic self can't handle. And especially not for the course of an entire life-time. Sure, there's is plenty computation, engineering and AI in the field, but only at surface level. Except for your occasional transdisciplinaries, but these are a rarity. Given my previous studies I could quite easily manage to, during a two-year period of studies on the side while also working, prepare everything necessary to begin studying for a master's degree (Europe) in mathematics, taking up an additional two years once started. Graduating at age 30-31, depending on potential fuck-ups. Am I being an absolute idiot? Beautifying things? Is starting a PhD at 31 too late? Is going out into the world at 31 too late?
Unsure whether I have, or possible is about to, fuck up life. Any insights, similar experiences, call-outs or general dread-reducing comments are very welcome. Thanks boys.

I hope hijacking this for my own idiotics is, for a brief moment, acceptable. Existential crisis related to studies in mathematics.

Previously, I spent a year studying mathematics, but being immature and unable to understand what it means to study something, I would spend about a week before each test going through the course material. Top grades, but no deep appreciation for the material itself. Drifting through life, I ended up studying architecture instead. Not entirely at random, but certainly not as a result of lengthy consideration.

Now, some years later at age 25 I find myself with a bachelor's and master's degree in architecture, a published thesis/papers and opportunities most can only dream of, among which is a position at what would be considered the absolute top workplace in the architectural world, while still being able to appear as guest critic in academia.
Yet, I absolutely miss mathematics. The average architect is prone to arriving at illogical conclusions, which I and my autistic self can't handle. And especially not for the course of an entire life-time. Sure, there's is plenty computation, engineering and AI in the field, but only at surface level. Except for your occasional transdisciplinaries, but these are a rarity. Given my previous studies I could quite easily manage to, during a two-year period of studies on the side while also working, prepare everything necessary to begin studying for a master's degree (Europe) in mathematics, taking up an additional two years once started. Graduating at age 30-31, depending on potential fuck-ups. Am I being an absolute idiot? Beautifying things? Is starting a PhD at 31 too late? Is going out into the world at 31 too late?

Unsure whether I have, or possible is about to, fuck up life. Any insights, similar experiences, call-outs or general dread-reducing comments are very welcome. Thanks boys.

>>11729442
>miss mathematics
that would be any of our trannies

>>11729442
>Is starting a PhD at 31 too late?
anything past 24 is too late

>>11729447
If these are complimentary by virtue of mentioning them, I will have one.

>>11729449
>anything past 24 is too late
Other than suicide, I guess. Here we go.

>>11728376
Depends on your experience with analysis, I'd say reading (or skimming) a book on curves and surfaces leading to Gauss-Bonnet is a good start (maybe Baby DoCarmo or similar).
Otherwise (or after) straight into Lee or DoCarmo. Also really depends what area of differential geometry you want to get into. Sharpe is also very nice but not so introductory.

>>11729442
You mentioned Europe and you are awake at this hour, so I would assume you are one of us. Would you have to pay for your studies? If not, then give it a go if you think you have passion for it. If you get to the point where you have your master's degree and would like to get a PhD, then feel free to try. Yes, it may lead to a total disaster on your behalf and you would then take someone else's place, and this someone else could have used the position for something extraordinary etc., but here's the catch: you don't actually make the choice yourself. There's the spray-pray-gay triad that characterises the process: first you spray the whole world with your applications, then you pray for someone to accept you, and then you are gay (in the original sense) that someone picked you up and then you accept the position. The first and last choices are yours to make, but the big one is not. If the people in charge think you are what they want, they will choose you. Even... even if you are... I dare not say it, sorry. But yeah, give it a go and be happy.

>>11729513
stop giving him false hope
he's 25, it's over

>>11729523
Your words of ageism sting my ancient heart. Euthanise me. I am like the pet dog whose legs have been amputated and replaced with wheels because its body can't stand the age. Be merciful.

>>11729442
What would be the point?
Is your ultimate goal becoming a research mathematician?
Why?
> Is going out into the world at 31 too late?
Going out into the world to do what?

>The average architect is prone to arriving at illogical conclusions
Every human does, rationality is usually a loosing strategy.

>>11728441
Oh right, I know the firm then. How are you guys doing in this market?

>>11728441
Is it possible to prepare for interviews like that or do you have to be naturally good at arithmetic/puzzles? How did you go about it?

bros... i studied maths today...

>>11729640
did it hurt?

amigos...

>>11729068
>arxiv mailing list
Is it one with some sort of filter?
How do you get the interesting stuff?

>>11729640
Very good, anon. You will grow big and strong if you keep it up.

>>11729817
You pick the genres and then you get a mail like:
>Submissions to:
>Commutative Algebra
>Algebraic Topology
>Category Theory
>Group Theory
>K-Theory and Homology
>Rings and Algebras
>Representation Theory
>received from Wed 27 May 20 18:00:00 GMT to Thu 28 May 20 18:00:00 GMT
And then it will contain titles, abstracts and links for all the submissions in those thingies during that time period.
https://arxiv.org/help/subscribe
>genres
What are they? They are not genres. How can I lose a word like this? Well, you get the point anyway.

>having trouble with exercise
>google it
>every proof depends on Zorn's lemma
maths was a mistake

>>11729920
thx

>>11729934
which subject?

>>11729938
algebraic geometry

>>11729920
this seems like a great way to have your email flooded with terrible chink papers

>>11729934
Maybe I'm a filthy algebraist but Zorn's lemma has always been the form of AoC I was least uncomfortable using.
For some reason situations where Zorn is the natural form to use almost always feel intuitively true. You rarely get full-retard results out of Zorn like you do the AoC.

>>11729964
One post a day is hardly anything compared to the uni propaganda mail. It's just a list of all the submissions. Besides, you may find something nice there.

>>11729973
>the form of AoC I was least uncomfortable using
Now that you mentioned that, I can't think of personally using https://en.wikipedia.org/wiki/Well-ordering_theorem for anything other than to show you can invert surjective functions because it's nicer to pick the first element instead of any element. Have you used this a lot?

>>11729938
You're welcome.

Speaking of axiom of choice, well ordering and such things, here's an exercise:
Prove that if CH holds, then there exists a function [math]f:[0,1]^2 \rightarrow \mathbb{R}[/math] such that
[math]\integral_0^1 f(x, y)dx = 0[/math] for all [math]y \in [0, 1][/math], and
[math]\integral_0^1 f(x, y)dy = 0[/math] for all [math]x \in [0, 1][/math].

>>11729973
Well, most of the arguments I've seen that use Zorn only use a very simplistic method, ie taking unions or intersections over inclusions, which is not particularly sophisticated, thus making its results quite intuitive, even if Zorn itself isn't particularly intuitive.

>>11729985
I've seen it used in a theorem about global dimension by Auslander in noncommutative ring theory, but it was a pretty niche application where he ordered elements so as to get chains of submodules

Here's an exercise related to well-ordering etc:
Assuming Continuum Hypothesis holds, prove that there exists a function [math]f:[0,1]^2 \rightarrow \mathbb{R}[/math] such that
[math]\int_0^1 f(x, y)dx = 0[/math] for all [math]y \in [0, 1][/math], and
[math]\int_0^1 f(x, y)dy = 1[/math] for all [math]x \in [0, 1][/math].

>>11730007
wasted quads

>>11730009
yeah ik

>>11730007
E-easy. Left to the asker as an exercise.

>>11730001
Yep. Dugundji derives AC from it quite nicely. Order your sets well and choose the first elements.

>>11729985
>Have you used this a lot?
Hardly at all. But I think this is mostly because it's such an unappealing way to think about things that it's almost never the most natural way to do something unless you're balls-deep in autistic set theory.
IIRC you _can_ do things like Tychonoff or Hausdorff's maximal chain theorem using the well-ordering theorem instead, but the only thought in your head the entire time will be "god I wish I was just using Zorn for this". I think that accounts for 99% of places where you could apply it.

>>11730029
Yep. It's sort of a cute result, but for most of u it just exists and hardly anything more.

>>11730007
how?

>>11729590
i personally didn't do any preparation so not sure the best way to go about it. but two of my friends did quite a bit of prep when they were interviewing; one got a slew of job offers, while the other got none. so preparation should help but it;s not a replacement for natural ability and intuition.

>>11730009
that's not quads.

>>11730123
I think to a large extent interviewer in those kind of interviews don't want you to prepare much for them.
The point is to watch you think under pressure; getting the questions right isn't the important part. If they give you a puzzle and you start reciting the answer you remember from a book of interview questions, you're not showing them anything they want to see.

>>11730007
Proof: Think

>>11730134
Do you guys just scalp the bid ask spread or are there much more complex strategies?

>>11730007
Is that a Riemann integral?

What's the best textbooks on the following topics for someone who has dipped their toe in these subjects, but never taken a formal class on them, and wants to learn these things extensively (but starting from an introductory level)

>Abstract vector spaces and operator theory
>Abstract algebra
>Topology
>Functional analysis

>>11730516
my diary

>>11729442
Why not try to combine your interest for math with the knowledge you actually have? Why not try to leverage your interest in math and applying it to architecture? I'm sure spacex or blue origin would be interested in the architecture of space habitats or something like that, idk (I really don't know anything about architecture I'll be honest)

>>11730477
Lebesgue integral.

Quick anon! What is a logical equivalent of OR, TTTF?

You guys think I'd have a good shot at a PhD in USA/Europe/Asia with a masters at impa? Especially USA or Japan. I'm looking for something more on the applied side because I don't want to live off research.

>>11730630
academic research I mean

>>11730612
SPS or SL+ if you conjugate

>>11730612
Yes.

I need to learn differential geometry, almost from scratch.

I am looking for books on the level of Theodore Frankel's Geometry of Physics and Nakahara's famous books.

And recommendations?

>>11730664
I was blind and didn't see de morgan's law. Thanks tho.

>>11721292
It is hot anon

https://www.youtube.com/watch?v=2BclyVYo0Tc

>>11730755
Do Carmo

>>11730630
>Especially USA or Japan
Impossible to really say. There are a fuckload of schools in the USA. You certainly have a good shot at _some_ of them, but which ones depends on your personal circumstances.
Japan depends even more on personal circumstances. At the graduate you level you need to provide a solid answer to the question "why here instead of somewhere else" and "I'm king weeb I ruv grorious nippon so much" isn't going to cut it. If you have some research-related reason that justifies you wanting to fly around the entire globe to work at that particular department, then your chances may be decent.
> because I don't want to live off research.
Why would you ever do a PhD if you don't want a research career? That's literally all a PhD prepares you for. Even if you don't get a professorship and do academic research, the only jobs you need a PhD for are industry research jobs.

>>11730859
>Why would you ever do a PhD if you don't want a research career?
You can get one for fun.

>>11730918
I think the intersection of "doesn't want to do research for a living" and "has fun doing 4 years of obsessive research about useless autism" is a null set

>>11730938
Good point.

I know view on this, and the circumstances people are in, differ quite a bit, but I like the student life and I'd recommend a PhD pretty much in any case. Preferably one where you read a lot, also philosophy.

I'm finally out of academia and currently own 3 times as much a month as I spend (including rent), and while it's good to not having to worry about money, I have the uncomfortable suspicion that I'm wasting my life (by not reading more).

I know views on this, and the circumstances people are in, differ quite a bit, but I like the student life and I'd recommend a PhD pretty much in any case. Preferably one where you read a lot, also philosophy.

I'm finally out of academia and currently earn 3 times as much a month as I spend (including rent), and while it's good to not having to worry about money, I have the uncomfortable suspicion that I'm wasting my life (by not reading more).

>>11730938
Actually doing research for a living involves a lot of struggle to get funding one way or another, while a clever PhD student can leave all of that to his advisor.

>>11729442
I feel you on the existential crisis. I had one myself when I attempted a Ph.D. at a certain
public Ivy college. I am okay, somewhat, but my
petit crisis sits far in the background. Staring.
Waiting for me. Wanting to play with me again.

What was harder?
>Poincare conjecture for n>4 (Smale)
>Poincare conjecture for n=4 (Freedman)
>Poincare conjecture for n=3 (Perelman)
>Poincare conjecture for n=2 (Poincare)

>>11731013
>still no proof for n=1

>>11730852
ok boomer

>>11731034
I think that's just the classification of 1-manifolds.

>>11730532
>>11730007
Ok so how do you solve it?

>>11731100
And I was making a low effort bait to get some funny responses, but nay. I'm losing my edge. I've also been tormented by this question for hours now: are the genres of maths called subfields or what? I've completely lost that word and it's irritating me.

>>11731108
Step by step.

>>11731121
Branches

>>11731126
THAT'S IT! Thank you, anon. <3

Is it possible to construct a logical statement from any arbitrary truth table?

>>11731158
I guess the easiest way would be to use conjunctions to get truth on precisely one row and then take a disjunction of those. There could be fancier ways to do it, but that would at least give you what you want. For example, if you wanted FTFT, then you'd take [math](p\land\neg q)\lor (\neg p\land \neg q)[/math]. The values would obviously be like:
p q
T T
T F
F T
F F

>>11730007
>>11731108
Here's the solution:
1) Let [math]\omega_1[/math] denote the first uncountable ordinal. For any [math]a \in \omega_1[/math], consider the set [math]\{b \in \omega_1 : b < a\}[/math] - this set is countable,because it is an ordinal smaller than [math]\omega_1[/math].
2) The set [math]\omega_1[/math] has cardinality [math]\aleph_1[/math]; assuming continuum hypothesis, [math]\aleph_1 = 2^{\aleph_0} = \textrm{card } \mathbb{R}[/math], so we can find a bijection [math]i: [0,1] \rightarrow \omega_1[/math].
3) Finally, let [math]f(x,y) = 1[/math] if [math]i(x) < i(y)[/math], and [math]f(x,y) = 0[/math] otherwise.
For any [math]y_0[/math], we have [math]f(x,y_0) = 0[/math] for all [math]x[/math], except countably many.
For any [math]x_0[/math], we have [math]f(x_0, y) = 1[/math] for all [math]y[/math], except countably many.

>>11731182
Neat, thanks.

>>11731210
Pretty cool. Did you come up with this proof yourself?

>>11731243
No problemo.

>>11731130
You’re welcome ESL-kun.

Unrelated:
I'm reading this
https://arxiv.org/pdf/1712.01815.pdf

>>11731210
Thanks.

I'm a bit confused when applying the same argument to the successor [math] S\omega_0 := \omega_1 \cup \{ \omega_1 \} [/math], that has the same cardinality
There should also be such [math] j : [0,1] \to S\omega_0 [/math], such that we can freely choose a [math] w \in [0,1] [/math] and set [math] j(w) = \omega_1 [/math] . We set [math] f(x,y) = j(x) < j(y) [/math] like before but with j. The thing doesn't work anymore, since [math]f(x, w)=1[/math] except [math]f(w, w)=0[/math].

And yeah, I suppose since we can say nothing concrete about the order on [0,1], this is worsely pierced than the characteristic function of Q in R and we won't Riemann integrate it.

>so we can find a bijection
>We choose the axiom |A|=|B|, so we can "find" a bijection
this language should be discourages tbqh.

>>11731439
Book covers are so full of text nowadays that I thought your image was one. Especially when it was combined with
>unrelated
>i'm reading this

>>11731409
You have no idea how dumb it felt to go through options like subfield, subdiscipline, subject etc. without finding the right one. You gave me tranquility.

>>11731291
No, I learned in my measure theory course.
I think it's cool because many people say CH is "intuitively true", and here it leads to something weird.
I wish I knew some equally weird consequence of assuming "not CH".

>>11731439
In general (with the standard definition of ordinals-as-sets), for any ordinals [math]\alpha < \beta[/math], one has [math]\{x \in \beta : x < \alpha \} = \alpha[/math].
So the set [math]\{x \in S\omega_1 : x < \omega_1 \} = \omega_1[/math] is uncountable and the original argument no longer works.

>>11731580
>I wish I knew some equally weird consequence of assuming "not CH".
Sounds a bit forced, don't you think?

>>11723739
documentation is like 99.9% garbage its written by experts for themselves not people learning it from zero.

>>11731595
>Sounds a bit forced, don't you think?
Maybe you're right, but set theory is a world where anything can happen.
Assume there's a cardinal [math]\aleph_0 < \kappa < \textrm{card } \mathbb{R}[/math], and let your imagination run wild.

>>11731729
https://en.wikipedia.org/wiki/Forcing_(mathematics)

>>11731747
If you are making a joke on the word "forcing" here, I am sorry for not getting it.
If forcing is actually relevant here, I have no clue because I know nothing about it.

>>11731580
>I think it's cool because many people say CH is "intuitively true"
I think that depends on the formulation. It's "infinitely true" in the "there's no set between, in size, N and R" and that's because people can't imagine well the there object that actually have no tangible counting functions. I.e. indeed people can't even properly "see" the subsets of N. In this formulation, CH is about a "weird not before seen inbetween" objects between the set N they know and between the set R they mistakenly think they know. Because of this mistake, people have the intuition that R is the smallest uncountable set.

But if you abstractly define ordinals, then |omega_1| suddenly becomes the abstractly defined smallest uncountable set. That is, the intuition people tie to R is realized in formal sense. And now CH is actually the claim that R is small, namely as small as just omega_1 .
So in that formulation, to me at least, there's no clear reason why CH is true.

Can we make parts the example more explicit?
Looks like we can choose to have [math] i(q) \in \omega_0 [/math] for [math] q \in { \mathbb Q } \cap [0,1] [/math]. That is to say, we can use all finite ordinals to cover the dense rational subset of [0,1] . At this point we make a first limit ordinal jump in induction, jumping beyond [math] \omega_0 [/math] .
Of course, we can choose some field extension of Q, say by [math] \sqrt{2} [/math], count it and have [math] \omega_0 + d [/math] map onto those. With this new objects we get to [math] \omega_0 \cdot 2[/math]. And so on.
Adopting CH says that the remaining subset of [0,1] beyond the rational, algebraic etc. is actually "so sparse" that the first jump beyond countable ordinals exhausts it.
R is uncountable by Cantors diagonalization.
When it's possible that an eventual jump to [math] \omega_1 [/math] doesn't exhaust the reals in [0,1], then at [math] S\omega_1 [/math] there's still reals left, and this then has to be an uncountable subset of [0,1].

>>11731790
My mercy is upon you. You have been forgiven. It is somewhat related to your idea of a "non-CH isn't obvious" example, as it is used in proving the independence of CH from ZFC. Or independence of... my English is breaking down. Imagine a hug.

>exercise II.2.4 of Hartshorne has a 3 page proof in some other book
make this stop

Nighty night, /mg/.

>>11731985
Fall not into despair!

>>11731798
I agree. For a long time, I had the misconception you mentioned in your first paragraph.

Another problem related to the topic "cardinalities between [math]\aleph_0[/math] and [math]|\mathbb{R}|[/math]" came to my mind:
Is it true that every Borel subset of [math]\mathbb{R}[/math] has cardinality [math]\leq \aleph_0[/math], or equal to [math]|\mathbb{R}|[/math]?
I can see this is true for open and closed subsets. I'm going to think more and when that fails, google it tomorrow.

>>11731835
>>11732009
Thank you and good night.

>>11732012
Yes, it is proved by showing that every Borel set has the perfect set property.

>>11732009
nnmg

>>11732012
donno

gn

>>11732012
>>11732030
Maybe I should say a little more, it is also provable within ZFC that analytic sets, those continuous images of Borel sets also have the perfect set property. It is independent of ZFC whether co-analytic sets have the PSP, in fact is equivalent to [math]\omega_1[/math] being inaccessible to reals.

>>11732012
>>11732030
just found this which seems to discuss contemporary views quite nicely
http://logic.harvard.edu/EFI_CH.pdf

I also point it out since it has 2 pages on the three variants of it's definition
(and especially talks about questions of definablility)

Which also reminds me that David Forster Wallace (who, if I remember correctly, wrote a bachelor thesis on modal logic), at one point at the end of his career (at the end of his life, but I don't want to be rude) wrote a book about Infinity in mathematics.
Turns out it was a disaster in various regards and he got brutally raped by reviewers. Pic related is an except, talks about his section about CH too.
The short review is the first link here
https://en.wikipedia.org/wiki/Everything_and_More_(book)

That said, I like that interview
https://youtu.be/iGLzWdT7vGc
What's even better is this cut:

https://youtu.be/pqmIAbHXr0Y

Tonight, I need some help. Don't know who else to ask. This is the first in a series of issues. I got some of these wrong, don't know which. Any help would be stellar.

>>11724418
Apostol is neat.

>>11732369
Got this one figured out, updating with new one. At least one of the following is wrong, also, at least one is still wrong if I change the answer in 'a' from T to F.

>>11719273
which book is this? I'm just starting Galois Theory and algebra seems interesting as a field for further studies, but I have no other authors except Fraleigh and Aluffi

>>11721935
Calculus. When you finish Single-variable Calc pick up Linear Algebra w/a good multivariable calculus book like Hubbard&Hubbard or Shifrin's Multivariable mathematics. After that you should think about learning analysis, probability, algebra and diff eq

>>11731603
>documentation is like 99.9% garbage its written by experts
Yes. It is written by the people who wrote the package, instead of ranndom people with dangerous half-knowledge of stack overflow.

>not people learning it from zero
I never told him to learn LaTeX through the Kona script documentation. But it is enormously helpful for things like toc manipulation, because it lists literally everything you can do with it.

>>11733152
>Aluffi
Really advising against reading Aluffi before you've gone through a proper undergrad or even entry-level grad algebra course

>>11731182
>>11731158
This is exactly what you do when you are constructing circuits in computer engineering.
There are some methods to simplify the resulting expressions, but if they get large, better let a computer handle that.

>>11721935
Is this for fun? You learn both simultaneously usually I think
>>11733160
You know, I realized I never see Apostol II recommended for multivariable.

>>11719273
she's gonna grade it... right...?

>>11733166
I know. I usually stick to Fraleigh, but sometimes I read Aluffi since the guy is just comfy. Something like a broader picture of stuff.

>>11733186
g-g-guys?

someone make a new thread. I wanna ask a question but this thread is dead

WHATTHEFUCKWHATTHEFUCKWHATTHEFUCK SHEDIDN'TGRADEMYTESTSHEDIDN'TGRADEMYTESTSHEDIDN'TGRADEMYTESTIMGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELFGONNAKILLMYSELF

/gmmg/

>>11733170
Yup. That came to my mind when I was about to post, and that is why I put the truth values like that. I've seen engineers do it upside down.

>>11733306
What's the hurry?

>>11733375
it's been weeks and I need this grade to get into summer classes at a top uni

>>11733450
Have you considered asking her to grade it? You have your reasons why you would need it.

bros... i did maths today... second day in a row

>>11733533
I wouldn't want to bother her. Women have it very difficult in stem as it is.

>>11733537
Well done, anon! I declare this to be a self improvement general.

>>11733538
True, but duty is duty. Besides you would just be notifying her that the grade you are missing is important for actual reasons and not just so that you don't have to worry if you passed or not after binge studying the last night before the test or anything like that. 'm not saying you should write a message like "yo bitch where my fuckin' grade yo"

>>11732030
>>11732045
>>11732112
>>11732159
>>11732161
Thanks. I'm not really a fan of set theory, but it's interesting whenever it overlaps with "normal math".
A thought on "not CH":
For a cardinal [math]\aleph_0 < \kappa \leq |\mathbb{R}|[/math], we can define a measure [math]\mu_{\kappa}[/math] on all subsets of [math]\mathbb{R}[/math] by the condition:
[math]\mu_{\kappa}(A) = +\infty[/math] if [math]|A| \geq \kappa[/math], [math]\mu_{\kappa}(A) = 0[/math] otherwise.
Assume we have some cardinals [math]\kappa_1 \neq \kappa_2[/math] as above.
Then the following observation might be somewhat interesting: [math]\mu_{\kappa_1}(A) = \mu_{\kappa_2}(A)[/math] for any Borel set A, but the measures aren't equal.

>>11733375
good morning

>>11733603
>it's interesting whenever it overlaps with "normal math"
This stuff kind of stuff can pop up in surprising places. https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry#The_Lefschetz_principle

>>11733603
Is this telling me that the Borel sets can't be categorized/ramified/distinguished in a sense, w.r.t. sets between N and R?

As opposed to the other guy, I feel of R as sort of uncontrolled and big.
So if my reading of that is correct, then this sort of says that measure theory doesn't care much for those set theoretical questions?

I'm not a big fan of set theory either, especially when "set theory" means stuff of cardinals beyond [math] V_{\omega + \omega} [/math]. Countable ordinals are rich enough.
Moreover, judgements and inductive types and pattern matching are imho "right" objects in math to have - from that perspective set theory feels blind in an unnecessary way.

>>11733543
>Well done, anon! I declare this to be a self improvement general.
I feel already accomplished today, since I sent out an email regarding bathroom reparations that I should have sent out weeks ago.

>>11733543
>True, but duty is duty.
I'm going to have to stop you right there. Lines like that have been used to keep women down for decades. Check your privilege and shut the fuck up.

>>11733622
i've seen the proof of ax-grothendieck theorem in a model theory course
other than that, i don't really have a clue about algebraic geometry
>>11733624
i think the main takeaway is: stick to Borel sets, and then you dodge all the set-theoretic problems.

>>11733624
>I feel already accomplished today, since I sent out an email regarding bathroom reparations that I should have sent out weeks ago.
Very good, very good. It's the small things that are the hardest to do. I remember the struggle against myself as an undergrad when a toilet stall on the campus had a note saying it was closed. I almost wrote "does not imply not-open" on it.

>>11733641
Then we will stop. I respect your feelings.

>>11733646
>i don't really have a clue about algebraic geometry
Same. Goes way over my head.

>>11733663
Same as you
As a topologist, I also shit on closed toilet seats, given being closed doesn't imply they aren't open.

Empirically, at my own home, I ended up finding shitting everywhere ends up making a lot of cleanup work. But if you want to become a homotopy terrorist, you can't give up.
I still shit everywhere in restaurant bathrooms.
I'm a shitter man, like you.

>>11733690
I can see the plans of world dominance there. You do realise territory is marked using pee and not poo right, though?

>>11733712
I leave lasting cusps.

>>11733538
>>11733641
Cuck alert.
You are a failure. I am glad people like you get walked all over.

>>11733603
Every Borel set has card equal to |R| or countable, so any Borel set with have measure +inf or 0. CH doesn't really matter here. A question that does interest set theorists is sort of a reverse mathematics for CH. Suppose in olden days that a proof in analysis assumed CH. There are many other cardinals that can be defined that are stronger than CH, in the sense that, if CH is true all these cardinals have size equal to R. If not CH though, they might be different. The idea is that when you assume CH, what is it that you actually are using? For instance, if not CH, and |N|<k<|R|, is the union of k many Lebesgue null sets still null?
>>11733646
Yes, Borel sets have almost all regularity properties you would want. But, the projective sets are very nicely defined sets, and regularity properties fail at very low levels. I used to not really care about large cardinals, like >>11733624, but as soon as I started learning descriptive set theory, you reallize that they are intimately tied with nice properties of sets of reals.

>>11733663
>Then we will stop. I respect your feelings.
BASED

final post

>>11734238
Yes, that's me.

>>11741159
You sure?

counter

>>11741187
Riposte!