/sci/ - Science & Math

Math is so comfy that I want to focus on it the most but I can't.

Have you ever felt threatened by a math author before?

>>10154644
yes

>>10154644
a physics author called me nigger once.

>>10153853
mochizuki is so fucking based af it just causes a non-removable singularity

>>10154816
Damn, I wish someone called me a nigger.

>>10154644
Math student here
what am I supposed to prove at least once in my life

>>10154819
I never know what he's talking about so I can't tell if he's based or a hack

>tfw to dumb to understand adjoints
I mean I understand the definition but why are they interesting?

>>10155050
Proving stuff. "If the adjoint maps convex sets to convex sets then something something".

Ok I'm trying to prove [math]\mathbb Z[/math] is a Dedekind domain. So far, what I have is:

>[math]\mathbb Z[/math] is Noetherian and of dimension 1, clearly, since it is a PID
>localizing at any prime [math]p[/math] gives a local ring [math]\mathbb Z_{(p)}[/math] which must be Noetherian, and in particular a PID by the correspondence of ideals, so it is Noetherian of dimension 1 again, and in particular, its maximal ideal is principal
>equivalently [math]\mathbb Z_{(p)}[/math], is a discrete valuation ring for every prime [math]p[/math]
>hence, it is a Dedekind domain
is this right?

Suppose I have a solution to a non-linear ODE. How do I find the complementary non-stationary equation (PDE) it also solves? I'm looking at Poisson's equations with non-linear sources with non-integrable singularities and trying to find the stable solutions in case there are infinite for given parameters. I'm already working in cylindrical coordinates, so that I the equation reduced to an ODE.
I know to look for the solutions whose smallest eigenvalue is positive, but I need the PDE first which is solved by the stationary solution (ODE).
Any ideas on how even to google it?

>>10155082
I think this is a shitpost but there's a seed of doubt in my mind

>>10155082
>Noetherian and of dimension 1
>>localizing at any prime [math]p[/math] gives a local ring [math]\mathbb Z_{(p)}[/math] which must be Noetherian, and in particular a PID by the correspondence of ideals, so it is Noetherian of dimension 1 again, and in particular, its maximal ideal is principal
>>equivalently [math]\mathbb Z_{(p)}[/math], is a discrete valuation ring for every prime [math]p[/math]
you don't need any of these

>>10155135
>>10155128
uhh what's your definition of Dedekind domain then? mine is: a noeth domain of dimension one such that is locally a DVR for every prime

>>10155146
https://en.wikipedia.org/wiki/Dedekind_domain

>>10155146
I've never heard anyone use that as the definition. It's equivalent, sure, but it's autistic.
A Dedekind domain is one where (proper) ideals factor into products of prime ideals.

>>10155169
>>10155166
yeah but that seems hard to compute in general

>>10155146
proper ideals factor into prime ideals
notably, any PID is trivially dedekind

>>10155173
In a completely general case, sure. But it means you don't have to trudge through all that bullshit to prove extremely simple rings are Dedekind. Most of the time when you have a PID you either know or it's not too hard to show that you have one.

>>10153853
I had a dream where we were using minions as numbers for some reason. Like the mayan numeric system, but with minions. And they weren't the regular minions, it was like those shitposting jojo minions.

/blog

What did Rudin mean by this?

>>10155288
>What did Rudin mean by this?
What do you not understand?

>>10155293
I just don't get why he actually proved it.

>>10155294
just because you define something to vanish somewhere doesn't mean it actually does.........
obviously there's a reason why he defined it that way, which is why he needs to prove it

>>10155297
>if a function is zero on a list of open sets, it's also zero
>lemme just take half a page to prove that when I've skipped harder stuff
NIGGA WHAT THE FUCK IS THIS DID RUDIN HIDE SOME FUNDAMENTAL SECRET THERE THAT I'M NOT SEEING? HE DIDN'T EVEN MENTION THAT COUNTABLE UNIONS OF OPEN SETS ARE ALSO OPEN TO SHOW THE DEFINITION FITS

>>10155300
>NIGGA WHAT THE FUCK IS THIS DID RUDIN HIDE SOME FUNDAMENTAL SECRET THERE THAT I'M NOT SEEING? HE DIDN'T EVEN MENTION THAT COUNTABLE UNIONS OF OPEN SETS ARE ALSO OPEN TO SHOW THE DEFINITION FITS
Rudin is a meme.

>>10155118
Never mind, I've found a way. Now I can rest assured that I can monkey my way out and find stable solutions if I ever need to model heat transfer in a material that spontaneously combusts. :)

>>10153853
the points where the diagonal touches the circle are the first two points of the triangle, the third is anywhere on the circle, this way I can cover all the possible ratios of the sides of those right triangles. My question is how do I represent visually all the possible ratios of sides of triangles of all possible triangles not just right triangles??

>>10155356
By the sine rule?

Which theorems do /mg/ reject?

>>10155575
axiom of the powerset

>>10155575
The axiom of choice

>>10155592
>>10155629
>T
>H
>E
>O
>R
>E
>M
>S
>>10155575
Zorn's lemma.

>>10155575
Bayes' theorem

>>10155638
>complains about people replying with axioms
>lists something that's equivalent to an axiom
what did he mean by this?

>>10155326
>I need help with something
>never mind I found it :^)
>doesn't mention the solution

>>10155656
>I have a marvelous solution, but the character limit of this reply is too small to contain it

>>10155646
>what did he mean by this?
I'm not a "he".

>>10155663
What did they mean by this?

>>10155663
Don't pretend to be me, it's impolite.
>>10155646
Pay close attention to reply order in that post.

>>10155575
Zorn can piss off.

>>10155575
Anything dependent on AoC.

>>10155748
>no Q-basis for R
what did he mean by this?

>>10155797
Exactly what I said.

I don't think AoC is in any way inconsistent, but it's completely useless except for pedagogical reasons. It's really nice to be able to say that every vector space has a basis, but if you can't, in any way, shape or form specify this basis, then for all intents and purposes, it doesn't exist. Calling it a "true" theorem is meaningless.

>>10155300
>HE DIDN'T EVEN MENTION THAT COUNTABLE UNIONS OF OPEN SETS ARE ALSO OPEN TO SHOW THE DEFINITION FITS
I imagine he just assumes that the reader understands that and doesn't have to spell it out. I don't really see the problem here.
>>10155575
The existence of smooth but non analytic real functions really annoys me.

>>10155818
>. It's really nice to be able to say that every vector space has a basis, but if you can't, in any way, shape or form specify this basis, then for all intents and purposes, it doesn't exist.
This, the reality is that most vector spaces do not have a basis unless you assume they have a basis.

I know that [math]\int_{a}^{b} f = \sup \{ \int_{a}^{b} \phi \hspace{0.1cm} : \hspace{0.1cm} 0\leq \phi \leq f \}[/math], where [math]\phi[/math] are simple functions, but would this also work for a general bounded measurable function? I.e., [math]\int_{a}^{b} = \sup\{ \int_{a}^{b} g : g\leq f , \forall x\in[a,b] \} [/math] ?

>>10155897

>>10155899
for a non-simple function though?

>>10155897
Yeah, but it doesn't say anything since f is already the supremum. By the by, [math]d\mu[/math]
That letter is \mu

>>10155908
I don't follow.

>>10155913
Any simple function smaller than some g is by definition smaller than f too, so you might as well just integrate along f.

>>10155916
Suppose I wasn't given the definition in terms of simple functions, rather the latter. How could I show explicitly that the integral can be defined that way?

>>10155928
You couldn't.

>>10155935
it's a question on my assignment

>>10155939
Can I have the explicit statement of the assignment?

>>10155940
Let [math]f:[a,b]\to \mathbb{R}[/math] be a nonnegative, measurable function. Prove that [eqn]\int_{a}^{b}f = \sup \bigg\{ \int_{a}^{b} g : g \hspace{0.1cm} \text{is a bounded, measurable function with} \hspace{0.1cm} h(x)\leq f(x) \hspace{0.1cm} \forall x\in [a,b] \bigg \} [/eqn] .

>>10155948
should be [math]g(x)\leq f(x)[/math] inside the set of course.

>>10155948
Wait, so you can use the normal definition? Trivial.
Let h(x) be any simple function smaller than f(x). Then h(x) is bounded and measurable, and the supremum of the integrals of h(x) is a minimum for the integrals along f(x).
Assume some g(x) has measure larger than any simple h(x). Then there is some simple j(x)<=g(x) which has a measure larger than any h(x), but any function smaller than g(x) is also smaller than f(x), and we're done.

>>10155961
Certainty trivial thank you. But what if I told you we never covered the definition in terms of simple functions in class nor does this abhorrent textbook we're using?

>>10155968
>we need to prove things from a circular definition which just says that the integral is an inner measure
I literally don't have enough axioms to work with.

>>10155972
Well, that is very unfortunate to hear. Guess I'll apply the standard definition and prove it that way then.

>>10155575
Godel incompleteness theorem
Kepler Conjenture
Four colors theorem

>>10155575
classification of finite simple groups

>>10155748
>vector spaces don't necessarily have a basis to them
what did he mean by this?

>>10155575
Existence of Mazur manifolds whose inner product inherited from the extended unitary 4-TQFT vanishes.

every vector space has a basis and anyone who says otherwise is a cock sucking nigger faggot.

>>10155575
Cantor's diagonal argument. Is there a non shitty way of proving it?

>>10156101
>every vector space has a basis and anyone who says otherwise is a cock sucking nigger faggot.
t. freshman

>>10156108
Graduated 10 years ago.

>>10156100
>Mazur manifolds
No such thing.

>>10156113
Based.

As far as I know, x^n = x (mod 2) for every natural numbers x and n (even times even is even, odd times odd is odd), so how are x, x^2, ..., different from each other?

And by the way, where can I learn about finite fields? Going through a linear algebra course but we've never even mentioned them. I'm very intrigued.

>>10156164
Are you retarded?

>>10155575
that relatively simple operations like tensoring or hom'ing are non-exact

>>10156164
What you're talking about is a very small case of Fermat's little theorem [math]a^p \equiv a[/math] mod p. This holds for every prime.

If you are asking how they are different, then unfortunately your high school teacher did not hit you with a yardstick enough times, and you still think of variables as numbers. X is _not_ an element of F_2. It's just a formal symbol.
What you think of as "evaluating" a polynomial at a number is actually a function (technically a ring homomorphism) from F_2[x] to F_2. Just because two polynomials go to the same number, it doesn't mean they're actually the same polynomial.

You can find stuff on polynomial rings and finite fields in most abstract algebra books, although you can do way more cool things in finite fields than most books will let on.

I have a binary form over the complex numbers [math]ax^3+3bx^2y+3cby^2+dy^3[/math] and I want to find its invariants under linear transformations in [math]\text{SL}_2(\mathbb C)[/math]. I've seen that yeah, the discriminant is the only invariant, but how does one derive this fact? How does one find the formula of the discriminant (without hindsight) without autistically looking for combinations of polynomials that just happen to work?

>>10156205
>just because two polynomials go to the same number, it doesn't mean they're actually the same polynomial

Does that mean that polynomials equality is different from function equality? F is equal to G iff their domains and codomains are the same and F(x) = G(x) for every x in the domain.

How do I show [math]M_{2}(n\mathbb{Z}) \triangleleft M_{2}(\mathbb{Z)}[/math] ?

>>10156197
Rejecting Hom functors being non-exact destroys pretty much the entire concept of cohomology.

Every popular cohomology theory arises as an Ext functor in some sense.

>>10156216
In this context, yes, it is different, because the polynomial is not a function. It's just a linear combination of X with coefficients taken from F.
If you would like an example of why it's important to make this seemingly autistic distinction, you can look up how fields of prime power order are constructed. In order to make a field of order 4, you need a polynomial of order 2 over F_2. You can't have one of these if you insist that x^2 = x.

>>10156226
From the definition

>>10156260
but wouldn't [math]BAB^{-1}[/math] contain entries which are not integers on account of the [math]\frac{1}{\det{B}}[/math] floating around?

Anyone have references for any maths concerning deformation, or rather, perturbation from spherical symmetry, of a rotating spheroid due to some impulse or however it would be worded by a mathematician?

I’m currently working on something related with general relativity, but I’m interested to see if there’s any pure mathematics done which speaks on this differently than the physicists do.

>>10156269
The same question could be applied to whether M_2(Z) is a group or not. You might want to calculate that BAB^{-1} determinant.

>>10156270
Depends on what you're doing, but there are results in PDE/functional analysis where if a term like [math]\epsilon S_1[/math] explicitly breaks the symmetry of your action functional [math]S_0[/math] on some Banach space, then for sufficiently small [math]\epsilon[/math] and sufficiently regular [math]S_1[/math] the kernel of the first-variation map for [math]S_0 + \epsilon S_1[/math] is [math]o(\epsilon)[/math]-away from that of [math]S_0[/math]. This means that critical points (if they exist) of [math]S_0 + \epsilon S_1[/math] is close to those of [math]S_0[/math].
Of course you have the standard results like Noether and Goldstone that tell you what happens when the compact Lie group [math]G[/math] of symmetries is broken into a subgroup [math]H[/math], but they do not guarantee the existence and "closeness" of the critical points.

>>10156085
Why do so many old fart professors hate the classification theorem? Are they opposed to mathematical progress?

>>10156269
Bruh if you’re saying it’s normal as a group, then it’s an abelian group so all subgroups are normal. What you want is that it’s an ideal, it’s “normal” in the 2 by 2 matrices as a ring. Show that the product of an element of [math]M_2(n\mathbb{Z})[/math] and an element of [math]M_2(\mathbb{Z})[/math] is still in [math]M_2(n\mathbb{Z})[/math]

how do I visualize/graph these?
[math]z \in \mathbb{C}[/math]
[math]\{z:|z-1|<2\}[/math]
[math]\{z:|z-(i+1))|\geq \frac{1}{2}\}[/math]
[math]\{z:|z-2)|< \sqrt{5}|z-2|\}[/math]

>>10155300
Because vanishing is defined dually. It is trivial that a continuous function that is zero on an open cover of its domain is zero, but here, for a distribution, vanishing on an open set U means vanishing at each test function with support in U.
Just because you vanish at each test function with support in U_i for some family (U_i) does not directly imply that you vanish at each test function with support in the union of the U_i's.
The argument shows that a test function with support in the union of the U_i's is actually a sum of test functions, each of which is supported in one of the U_i's, which is nontrivial and requires a partition of unity.

>>10153853
Is the agamma function related at all to the laplace transform? They seem pretty similar and both operate on the frequency domain.
Also, any good sources on the Laplace and Fourier tranforms? Im learning about them thru a book on engineering (EE student) and I would very much like some text that really explains what's going on in detail.

>>10156232
I know, that's why i hate it

>>10155575
It is not a theorem, but Church thesis and everything is "proved" using that.

kek algebraists btfo

explain this

>>10155575
Mochizuki's proof of ABC.

A line in the plane of a triangle T is called an equaliser if it divides T into two regions having equal area and perimeter. Find positive integers a>b>c with a as small as possible such that there exists a triangle with lengths a,b,c that has exactly two distinct equalisers.

>>10157873
Weierstrass needed to come up with a really weird function to show that continuity didn't imply differentiability.

>>10156586
mathmode isn't working for me, so I'm just reading raw tex, but here's a go
>[math]z \in \mathbb{C}[/math]
the entire complex plane
>[math]\{z:|z-1|<2\}[/math]
a circle of radius 2 about 1+0i
>[math]\{z:|z-(i+1))|\geq \frac{1}{2}\}[/math]
the complement (everything except) of a circle of radius 1/2 centered at 1+i
>[math]\{z:|z-2)|< \sqrt{5}|z-2|\}[/math]
you have unpaired parentheses here, but if you mean {z:|z-2|<sqrt(5)|z-2|}, then it's also the entire complex plane minus 2+0i, since |x|<k|x| is always true if k>1 and |x| does not equal 0.

>>10157982
also
>inb4 disks not circles

>>10157982
You got them all correct. Here's your official TexMaster certificate.

>>10157883
No, bad anon. What did Uncle Mochizuki ever do to you?

>>10158375
Based and Mochipilled

Should I drop out /mg/? I just failed my abstract algebra exam. 3 months behind in topology and exam is 1 month away. I keep making the same mistakes. Telling myself "i'll catch up, I'll catch up, I'll ace the final". Then I work hard as fuck for 2 weeks, but in reality it's too late. I feel I need to be realistic and accept that I fucked up and just drop out for now. Come back later maybe. In general my life is pretty fucked here and I want to leave my country. Lmk what you think.

>>10158575
Similar situation here, abstract algebra was really hard for me. The first half was trivial but the second was a nightmare. And by the way

>work hard as fuck for 2 weeks
Really? 2 weeks is a lot of time, you know? You can get really deep into a subject if you use that time (say, 4~5 hours a day) to self-study

Can't you leave your country for academic reasons? I.e. interchange?

>>10158575
>Fail 2 classes
>Guess I should drop out
Drop out, but only because you obviously don't care about it

https://arxiv.org/pdf/1811.08500.pdf
>Collatz Theorem
>Dagnachew Jenber
>(Submitted on 19 Nov 2018)
>This paper studies the proof of collatz conjecture for some set of sequence of numbers.The extension of this assumed to be the proof of the full conjecture, using the concept of mathematical induction.

Am I being dumb? Is this really that obvious? I can't see how to prove it

>>10158622
>Is this really that obvious? I can't see how to prove it
What have you tried?

>>10158623
well, I know you're shitposting and probably don't know the answer, but I kinda had the idea that somehow the invariants form a 2-dimensional space since the coefficients of a quadratic form only have 2 degrees of freedom, so the identity and one other invariant does it. But I can't see how to formalise the argument

>>10158585
I did all of group theory in 2 weeks, but my understanding was still too weak. I know it's all trivial and I get depressed knowing if I just kept up I would've been fine.

Originally my plan was to do this year then go on exchange, but I don't think this is a good option anymore.

>>10158597
There's more to it than that.

>>10158616
>the current state of non-western mathematics
fucking hell, the paper is literally copy pasted (the words are all the same over and over), and that's not even considering that he's using the "concept of mathematical induction" as a keyword

>tfw abc is true in Japan
>tfw Collatz is true in Ethiopia
>tfw RH is true in Nigeria
>tfw P=NP is true in India
>tfw Navier-stokes is true in St. Petersburg (probably tho)

Is there a collection of solutions for the problems in Richard Stanleys EC1 book that lack a solution in the book itself?

Poetry

god being dumb is the greatest curse ... I would give up my ability to walk for a workable IQ

>>10158622
>determinantal

>>10154856
the riemann hypothesis

>some girl tried to convince me today that the existence of an inverse does not imply bijectivity
>tried to prove her point with f(x) = x^2, claiming that the square root was the inverse
And of course she defined neither domain nor range of her function...

>math education major in number theory class asks why the remainder upon division by a number is less than the divisor

>>10150884
>>10150957
I just realized that I can archive all of my notes and answers to exercises and use that as an opportunity to learn latex. Thanks.

>>10160011
>iestening to roasties
Kek

Differential geometry gurus: Frankel, Nakahara or something else for a comprehensive overview at that level?

How can they be diffeomorphic if the map from one to the other is not injective?

>>10160075
the second space is probably supposed to be a helix (in R^3)

>>10160085
I think not because the author uses this notation.

>>10160088
How should he have drawn the spiral, then?

>>10160091
Using the usual Knot Theory notation with overpass and underpass like he did in the image I posted above.

Hey friends, undergrad idiot here
Currently taking abstract algebra, linear 2, probability.
What are some self research ideas that I could grasp? Or is that something reserved for the graduate level?

>>10160106
Problems in combinatorics can be taught to children, but otherwise you'll have to go for some application.

>>10160088
then they are not diffeomorphic

PRIMS rejected my Riemann paper in about 10 days. That makes a lot of sense because PRIMS does not publish notes. It's about 14 days on my sine and cosine paper. I hope they publish me.

Is it safe to say that most of Hilbert's problems were not actually that important in the scheme of things? (with notable exceptions such as CH and RH)

>>10156082
One of these is not like the others

What's /mg/'s level of formal education? I'm wondering if the reason I feel like a plebeian here is because the population is skewed towards wizards.

https://www.strawpoll.me/16898749

>>10160404
I'm studying money-science and maths on the side. Stuff's fun.
By the by, lurkers who don't post will ruin the poll results.
>>10160281
Nah.

>>10160179
No Tooker, nobody is going to publish you, you are unpublishable

>>10160413
The sheer number of lurkers always spooks me. The rule of thumb is 1% of users are creators, 9% are contributors, and 90% are lurkers. With this thread having 51 posters so far, that means there should be about 500 lurkers!

>>10160056
Frankel for math Nakahara for physics.

>>10160433
An effective condition of unpublishability seems like the most likely scenario

I am a math undergrad at ucsd. I am currently taking my first upper division course in upper level math called intro to proofs. I am getting a B in this class. My dreams are to study mathematics and get a phd and work in academia. I have been used to getting all A's in my lower division courses, but this class has hit me like a brick. What is the possibility of me getting into UCSD grad school for a phd in pure mathematics given this information. I suppose the better question would be how good does my GPA have to be in order for me to have a chance. I am kind of pissed off that the class average on the midterms was 65%. I know I should just study harder but I need the assurance of people on a Polynesian wood carving forum because I am a weak minded simpleton. I apologize for off topic.

>>10160455
>intro to proofs
>upper division
The U.S. was a mistake. Here in monkey-land we see proofs in high school. Fucking monkey-land.

>>10160075
Diffeomorphisms need not be embeddings. Self-crossings are allowed.
>>10160088
Now this is a diffeomorphism AND an embedding. No self-crossing here.

>>10160455
>course in upper level math called intro to proofs
Confirmed school for brainlets.

>>10160455
>My dreams are to study mathematics and get a phd and work in academia
kek

>>10160455
With math, there are always some classes that are just harder at a given level. Don't fret too much about not getting a perfect grade. Just know that the material you're learning in that class is bedrock for the rest of your math career, so effort now will pay dividends in years to come.

>>10160477
thanks. I also do not go to office hours, work in groups etc. I will also buy my math textbooks early and try to learn the material before class.

>>10160460
You know, I see you using weird definitions every now and then. Where you from?

>>10160480
Don't look at office hours as a necessity. They're a tool; For some people discussing things with a prof can be really helpful, but other people are just fine cramming a textbook in. It sounds like you don't really need that. If anything that's good, since you don't have to worry about that becoming a crutch, since profs won't be there forever. Same for groups.

>>10160484
thanks. I suppose I just need to study a bit better. I have this hunch that I am capable of getting into a phd program in pure maths, but at the same time I also have just as big of a hunch that I am just genetically not cut out for it. I suppose it is a very foolish mindset, but I hate uncertainty and I am pretty insecure of my abilities.

>>10160483
I think you're confusing the embedded image into [math]\mathbb{R}^2[/math] with its representation where there is meant to be no ambient space. Again, the image having self-intersections in an embedding into some Euclidean space does not imply that the spaces are not diffeomorphic. E.g. the cylinder is diffeomorphic to the [math]2n[/math]-twisted Mobius band even though there are a bunch of self-intersections in the embedding of the latter in [math]\mathbb{R}^3[/math].

>>10160493
>[math]\mathbb{R}^3[/math]
Meant [math]\mathbb{R}^2[/math], obviously.

>>10160484
Can an American explain to me what "office hours" are? Is that the exercise classes/tutorials?

>>10160487
>I am just genetically not cut out for it
Fuck that /pol/ shit. Number one factor in you going all the way is the ability to stay motivated, and that largely depends on you staying interested. Don't just crunch through your classes; You have to actually be interested yourself!

>>10160497
Almost all profs will schedule one or two hours a week where they will be in their office and students can come and one-on-one ask questions about the class.

>>10160503
first of all im a poltard and I agree with a good amount of things they say. That aside you are correct and I need to be interested in the math aside from drudging through classes. At the end of the day as much as i like to say that we are all capable of the same depth of thought (assuming you dont have down syndrome etc) I really can't ignore the nagging feeling that I am wrong. I know this sounds probably very foolish but I this class has made me quite insecure of my ability. Now that I think about it, Im probably just being a little bitch.

>>10160514
Not gonna lie, I go there too. Who knows, you might be right; Genetics are powerful, and I don't even know you. All I know is that if given the choice between continuing to work at it, or giving up early because you feel sorta bad, I'd choose the former every time. You don't have a good life doubting yourself that much and giving up that easy.

Tips on getting better at problem solving? I'm able to follow the logic of the text, but when it comes to the exercises I give up really quickly and look at the answers. When I see the answer I can understand why it's true, but I hate doing this and feel I need to stand on my own two feet more. It's like reading a newspaper in a foreign language and having to look up words you don't know in the dictionary every two seconds. Really frustrating.

>>10160526
You are right. Thanks for reinvigorating me. Sorry op i didnt mean to turn this thread into a brainlet cope. That aside you are right, I would forever kick myself for not learning about what I love. I realize that nothing is more comfy than math.

>>10158597
this

>>10160075
>>10160088
>>10160460
Just saying, if you consider the second manifold as (+structure) the topological subspace of R^2 in the picture, then that stuff is not even a topological manifold (topology 101, show that a cross is not a segment). The first answer @10160085 is trivially correct.

>>10160543
>if you consider the second manifold as (+structure) the topological subspace of R^2 in the picture
That's what an embedding is, dear.

>>10160531
Try doing the problems backwards and if that doesn’t make sure start with easy problems and then move up to more complex ones that’ll ensure that you know all the required techniques (no-shortcut way of getting better)

>>10160548
Of course substructure = embedding as usual, but you have to give a structure before talking about maps. Self crossing are not allowed.

>>10160531
>I'm able to follow the logic of the text, but when it comes to the exercises I give up really quickly and look at the answers.
Wow I used to be exactly like this! It's funny. I realized if I want to pass my quals that I have to be better at solving problems and also fix my attitude.

First, attitude. The problems/exercises are not impossible. Unlike research problems, they are solvable within a reasonable amount of time. Go with the attitude that you can solve them Maybe it sounds like bs, but it helped me. This way you won't give up 30 seconds in. Mull over the questions.

Second, get a problem book (these are available for fundamental subjects like real/complex analysis, algebra, topology) and practice solving a few problems every day. And check the solutions to see if there is another way of solving a particular problem. Afterwards, you start seeing patterns like learning tricks of how the problems were "made." Other problems become relatively easier.

>>10160552
>you have to give a structure before talking about maps
The structures necessary to talk about maps don't need you to embed the space into anything. Again, you can still talk about diffeomorphisms in the absence of an ambient space.

>>10160556
Then any function is a diffeomorphism, just give the structure to the codomain such that it is so. gg.

>>10160562
That's wrong though.

>>10160566
Bijection, don't let me state the obvious.

>>10160569
That's wrong though. Are you conflating a smooth structure with an embedding? Because I don't think that's a good idea.

>>10160573
>That's wrong though
Let A be a manifold, f:A->B a bijection, define a structure on B as follows:
- for every open U in A, f(A) is open in B (and those are the only open subsets),
- for every chart (U, x) on A, (f(U), x . f_{|U}^{-1}) is a chart on B.
Rhen f is a diffeomorphism.

>Are you conflating a smooth structure with an embedding?
No, I'm saying that considering the manifold as submanifold of R^2 from the picture makes no sense, hence that the first answer is the only reasonable.

>sheaf of sets
Make it rings at least.

>>10160596
>tautological construction
I mean [math]B[/math] is then determined from [math]A[/math] and [math]f[/math] and that's not very interesting now is it?
>I'm saying that considering the manifold as submanifold of R^2 from the picture makes no sense
Then we agree. If you [math]have[/math] to think about manifolds by embedding it in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math] (smoothly or otherwise) then you won't get anything nice; but the point is that you don't.

Thanks for the tips anons.

>>10160551
I only do the assigned problems atm, but I think yeah it's a good idea to go back and do easier unassigned problems if I'm struggling.

>>10160554
It's an attitude problem in part for sure. I honestly kind of dread doing the problems and see them as something I need to get through.

My prof recommended me a book he liked a lot and said it was pretty much only problems. Of course I avoided it at the time, but now I see how it could be helpful.

>you start seeing patterns like learning tricks of how the problems were "made."

This is really where I want to get. To be able to recognize a type of problem and immediately have a certain argument or approach come to mind. This sometimes happens to me and it's a nice feeling.

>>10160606
>that's not very interesting now is it?
That, if you don't say what you are talking about then it might as well be true by bananas.
>manifolds embedded in [math]\mathbb{R}^n[/math]
The first contact with manifolds. There is no shame in using manifoldic subsets of R^n; you can then study the "abstract" manifolds, or even more abstract geometric objects. Differential geometry is ugly anyway.

>>10160632
>The first contact with manifolds.
Well yeah but that's not really the point of contention.
>subsets of [math]\mathbb{R}^n[/math]
Well you can embed any smooth manifold in [math]\mathbb{R}^n[/math] for sufficiently large enough [math]n[/math] but that won't help with visualizing the manifold on a page of a book now would it? The point is that how manifolds are represented on a page does not fully capture its smooth/topological content.
>Differential geometry is ugly anyway.
At least diff geo is local and allows much more freedom in what you're studying. Besides, we have gadgets like spin-structures and principal bundles etc. that are much more powerful than just pretending we're doing vector calc on local charts.

>>10160628
To add to what the other anon said, some textbooks want you to use previous problems (usually in the same section) to solve the problems that follow (which may be the ones you are assigned). So it's good to at least look over the preceding unassigned ones from the same section before attempting others.

>I honestly kind of dread doing the problems and see them as something I need to get through.
It is striking; I had the same exact attitude. I used to think why waste time on (masterfully convoluted) problems rather than speeding through the text and work on research. But on the other hand, these problems help you get acquainted with the theory just introduced. It's like practicing what you have learned in a language lesson; they are really crucial. I wish I realized sooner. haha

On the flip side, as you progress through your training, you will have to deal with such things as research problems are themselves, in some way, a test to whether you really understand the theory or not.

good luck, anon!

>>10160640
deal with such things less and less*

>>10160636
>The point is that how manifolds are represented on a page does not fully capture its smooth/topological content.
The whole field and language of differential geometry is built on the concept of differential manifold, no meaningful escape if not rejecting it all.
>spin-structures
Am I getting /physics vibes?
>gadgets
Am I getting /analysis vibes?
>vector calc on local charts
High school tier is cringe, I agree.

>>10160645
Like it or not tools inspired by physics is being used to solve problems in pure topology (c.f. monopole Floer cohomology & Seiberg-Witten theory).

>>10160640
I really do think the language analogy is a good one. In my home country I can write down a list of 20 new words, learn them, and forget them the next day. When I'm abroad and am learning words then immediately using them in daily life it's really surprising how well they stick.

In this way I can see the exercises somehow tell one's brain that this is relevant information it should retain. Also yeah I often think I understand the material better than I actually do, and the exercises sober me up and make me realize I'm missing something important.

Thanks for the wish anon, hopefully I can tackle some research problems like you one day.

>>10154644
Halmos?

>>10160487
>hate uncertainty
>wants to do research
hmmm

Just curious. What are you trying to show with this intrusive anime demonstration?
Like "hurr durr I am wannabe mathematician (i.e. pretentious basement dweller w/o a single publication) hurr durr let's get these potatoes pissed with muh anime pics in every fukcing single post"

>>10155592
Return to """reddit""" """/u/sleeps_with_crazy""" """please"""

Does anyone have a proof that for (finite)sets [math]A, B[/math], [math]|A \cup B| = |A| + |B| - |A \cap B|[/math]

>>10160875
>complaining about anime on a Nigerian rutabaga peeling forum

>>10161160
you simply remove what you've double counted - that is, what's in the intersection

>>10161160
A\(AnB) is disjoint to B, so the cardinality of their union (A\(AnB))uB is just the sum of the cardinalities. But the union is equal to AuB, and the cardinality of A\AnB is |A|-|AnB|.
Qed

>>10161172
Thats not a formal proof.
>>10161183
Is it true that the cardinality of the union of disjoint sets is the sum of cardinalities?

>>10161199
Nigga are you retarded?

>>10160875
>Like "hurr durr I am wannabe mathematician (i.e. pretentious basement dweller w/o a single publication) hurr durr let's get these potatoes pissed with muh anime pics in every fukcing single post"
What did he/she mean by this?

>>10161207
>potatoes pissed with muh anime pics
Do you think *he pisses regularly on potatoes?

asdas

>>10161217
sdadsdsadsfd

>>10155219
Kek

>>10161217
>>10161236
Quality posts

>>10160804
Correct, the set theory one.

>>10160864
Ok obviously I dont mind not knowing, What I mind is not knowing if Im capable of knowing. I hate the fact that I convince myself I am not capable of upper level math.

>>10160106

Probably been asked a lot before but is there a no bs maths curriculum for self studying?

Real analysis classmate argued that 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ... converges to 1/2. I'm fucking baffled. It's like he hasn't paid attention to the last four chapters of the textbook.

>>10161780
https://en.m.wikipedia.org/wiki/Grandi%27s_series

Do you believe that math competition type of problems are worth doing, other than perhaps for fun?

>>10162012
>perhaps for fun
Yes, the tricks for solving them are always neat.

>>10160493
>that backtracking

>>10161734
>Probably been asked a lot before but is there a no bs maths curriculum for self studying?
High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

>>10162387
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

>>10162389
Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

>>10162390
Sophomore (cont):
• Reductive groups, root systems, representations of semisimple groups, weights, Killing form. Groups generated by reflections, their classification. Cohomology of Lie algebras. Computing cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and the cohomology of its algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "The Classical Groups: Their Invariants and Representations"). Constructions of special Lie groups. Hopf algebras. Quantum groups (definition).

Junior:
• K-theory as a cohomology functor, Bott periodicity, Clifford algebras. Spinors (Atiyah's book "K-Theory" or AS Mishchenko "Vector bundles and their applications"). Spectra. Eilenberg-MacLane Spaces. Infinite loop spaces (according to the book of Switzer or the yellow book of Adams or Adams "Lectures on generalized cohomology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. The Green's operator and applications to the Hodge theory on Riemannian manifolds. Quantum mechanics. (R. Wells's book on analysis or Mishchenko "Vector bundles and their application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch formula. The zeta function of an operator with a discrete spectrum and its asymptotics.
• Homological algebra (Gel'fand-Manin, all chapters except the last chapter). Cohomology of sheaves, derived categories, triangulated categories, derived functor, spectral sequence of a double complex. The composition of triangulated functors and the corresponding spectral sequence. Verdier's duality. The formalism of the six functors and the perverse sheaves.

>>10162392
Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

>>10162395
Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

>>10162398
Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

>>10162400
Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 2).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

>>10161780
Look up Cesaro summation

>>10162387
>high school
nice meme
if you didn't come out of middle school knowing this then you're not going to make it

>>10158616
>paper shows how you can count steps
>have to know the number of steps for a bunch of other numbers, those numbers could be greater than your starting one so induction won't work
>paper says nothing about how you can prove those other series terminate in finite steps
>papers premise is doomed from the start anyway, since determining whether the series applied to an arbitrary number will converge to 1 is the same as solving the halting problem

>>10162387
>>10162389
>>10162390
>>10162392
>>10162395
>>10162398
>>10162400
based curriculum poster

>>10162497
His reasoning was some algebraic bs, not Cesaro summation.

>>10162696
Then he's a retard

>>10161160
Let E={x_1,....,x_n} and F={y_1,...,y_p} let E and F be disjoint
Consider:
Φ: {1,...,n+p} → E∪F
k → x_k if k<=n, y_k-n if k>=n+1

Φ is a bijection, I'll let you show it. The result will give you that cardEUF=CardE+CardF, this is a lemma for what you're trying to prove. After this you'll have to show that CardE\F=CardE-CardF and then with this you'll be able to prove what you wanted to prove. You'll have to show that |E\(E∩F)|+|F|=|EUF|

>>10157985
Well they aren't!

Does transfinite induction show up anywhere other than balls-deep in autistic pure set theory?

>>10160460
>Diffeomorphisms need not be embeddings. Self-crossings are allowed.
Maybe it IS an embedding of that non-manifold, meaning they're not diffeomorphic.

>>10160453
No, you just can't do rigorous mathematics.
You're a crank.

>>10160493
A picture of a point and a line, with the word diffeomorphic beneath would make perfect sense to you, then, because they're both really images of R^2 ?

>>10163308
Try learning to read. It will do you well in the future.

>>10155050
same problem here
afaik, there's a lot of properties of self-adjoint and normal operators, and spotting whether the corresponding operator of a matrix is self-adjoint/normal or not is quite quite easy

but I have zero intuition behind what an adjoint is

>>10157901
how come the absolute value function didn't suffice for this

like just let y=|x|

or are you not aloud to do that because it's not "elementary" or somethign like that

What are the best resources to start learning maths?

>>10163716
Because whoever you replied to is a fucking idiot. The point of the Weierstrass function is that differentiability can fail [math]everywhere[/math] even if the function is continuous

>>10163716
Absolute value function is differentiable and continuous except at the origin where it is neither IIRC

>>10163726
I lied, it's continuous everywhere and differentiable everywhere except the origin

>>10163724
Can differentiability fail everyhwhere if the function is uniformly continuous? If the answer is yes, what about lipschitz continuous?

>trying to learn a proof based course from a book
>do the exercises
>there's no solutions for it, no way to tell if my proof is accurate or if it's as shitty as "uh suppose it's false, then we've reached a contradiction because you said it's true"
how do you lot do it straight from a book without a solution manual or chegg, how can you tell you're on the right course?

There's only one (finite-dimensional) vector space structure for a given abelian group and field, right?

>>10163797
Ask a professor? Also chegg sucks

>>10163797
>suppose the Riemann Hypothesis is false
>it isn't because Riemann said so
Where's my million?

>>10163810
"Structure" isn't notation for anything, speak properly.

>>10163862
A vector space structure is a (non-trivial) action of a field F on an abelian group V, ie, a (non-zero) ring homomorphism F -> End(V).

>>10163882
Now that's notation I've never seen before.
Let A be a vector space over F with a base A' with n elements, and B over F with base B' with n elements. We define φ such that it maps A' on B' bijectively, and φ is also linear. Showing φ is a bijection is trivial.

>>10156955
>I learned about Turing machines one time.

I bet you have no coherent reason to reject it you turd.

>>10153853

Back in highschool I found physics far more interesting than mathematics but now in university they have flipped. I am contemplating doing my masters in operations research or applied math, does anyone have experience in those fields?

I have never felt more brainlet than the moment I got stumped breaking down 13/32 into 1/2 1/4 1/8 1/16 1/32 for IEEE 754.
Please can some big brain mathsbois bring me back to 13 year old tier syllabus and help me nail this sort of stuff. Things like 9/64 are pretty straightforward but 13/32 just utterly stumped me.

>>10163988
1/2=16/32>13/32 => no 1/2
1/4=8/32<13/32 => one 1/4
subtract 8/32 from 13/32 to get 5/32 remaining
1/8=4/32<5/32 => one 1/8
subtract 4/32 from 5/32 to get 1/32 remaining
1/16=2/32>1/32 => no 1/16
1/32=1/32 => one 1/32

so 13/32=1/4+1/8+1/32

>>10161199
Are you retarded? I'll do it formally for you, just because when I started learning this things I asked myself the same questions.

Let A and B be disjoint sets. Let |A| = n and |B| = m. By definition this means that there is a bijection [math]f[/math] between A and {1, .., n} and a bijection [math]g[/math] between B and {1, ..., m}. There's obviously a bijection between B and {n+1, ..., n+m}. This gives us a natural bijection between AuB and {1, ..., n, n+1, ..., n+m} that will be presented as follows.

This bijection is defined by (sorry don't know how to do piecewise things here)
if [math]x \in A[/math]
[math]x \mapsto f(a)[/math]
if [math]x \in B[/math]
[math]x \mapsto f(b)[/math]

Since the sets are disjoint the function is well defined and completely determined. Proving that it's a bijection will just be separating into cases and using that [math]f[/math] and [math]g[/math] are bijections.

Are you retarded? I'll do it formally for you, just because when I started learning this things I asked myself the same questions.

Let A and B be disjoint sets. Let |A| = n and |B| = m. By definition this means that there is a bijection f between A and {1, .., n} and a bijection g between B and {1, ..., m}. There's obviously a bijection between B and {n+1, ..., n+m}. This gives us a natural bijection between AuB and {1, ..., n, n+1, ..., n+m} that will be presented as follows.

This bijection is defined by (sorry don't know how to do piecewise things here)
if x∈A
xf(x)
if x∈B
xg(x)

Since the sets are disjoint the function is well defined and completely determined. Proving that it's a bijection will just be separating into cases and using that f and g are bijections.

>>10163994
That explanation is brilliant
Thank you so much anon.
Would hug you if I could.

>>10154856
the snAKE LEMMA

>>10164008
>HOMOlogical algebra

>tfw you finally come up with an elegant proof

>>10164073
>and they still tell you to fuck off because they like the pathological sadist pedo faggot heretic better

>a bijective function can have a non bijective derivative
[math] (x^2)' = 2x [/math]
>a non bijective function can have a bijective derivative
[math] (2x)' = 2[/math]

What does math mean by this?

>>10164107
>x^2
>bijective
>2x
>non-bijective
>2
>bijective
what did he/she mean by this?

>>10163999
Checked.
>>10164107
Das because if f(x)=f-(x), then Df(x)=-Df(-x), and if f(x)=-f(-x), then Df(x)=Df(-x), fool.
It's easy to picture if you think in terms of the integrals rather than the derivatives.

https://thewalrus.ca/the-greatest-mathematician-youve-never-heard-of/

>>10163810
No. For example, R and R^2 are isomorphic as Q-vector spaces (because they both have the same uncountable cardinality). In particular, they are isomorphic as abelian groups, hence the underlying abelian group can be endowed with two essentially different structures of R-vector space.

>>10164151
>The most basic—yet still painfully inaccessible—explanation of Langlands’s calculations is that he related Galois groups, which are number field extensions

>>10164107
Are you actually retarded?

WHY ARE UNIQUENESS RESULTS SO HARD

>>10164218
just prove the definitions are equivalent nigga

>>10164255
I’M NOT A NIGGER

Where can I learn about lipschitz, hölder, uniform, absolutely continuous functions and stuff like that? Would love a book that touches those subjects.

>tfw trying to prove the free group on two generators is a coproduct of Z in the category Grp without Yoneda Lemma
fuck

>>10164470
Literally just using definitions

>>10164470
Literally just use the universal property

What's the best book to teach myself topology?

>>10163924
Actually, I learned about recursive functions one time (as undergraduate student in math).
You are right on the second one. I hate when people use an unclear hypotesis to prove things, even though they can avoid making use of it; I hate when they feel proud and say with a loud voice "HENCE BY CHURCH THESIS.." over and over, and they think they are oh so smart. They look like Hegel fan-boys. It is silly but that's all there is to it.

>>10164107
you swapped them, pal

ODE question

how to get from
X''(x) + λX(x) = 0
X(0) = X(L) = 0

to
X_n(x) = sin(n pi x/ L)

if that's even correct

>>10165197
λ_n = (n pi / L )^2, n natural

Find du, d^2u, if given u = f(xyz, xy^2, xz^2).
How to proceed? I don't even understand the task. Am I asked to find the solution for general case?

>>10165328
Yes, the task is to find the differential of the composite function
(x,y,z) -> (xyz, xy^2, xz^2) -> f(xyz, xy^2, xz^2)
think you can do that? The d^2 will also be zero if .., easy. And it is for the ""general"" case, yes.

>>10165413
OK, I'll try to. Is it viable to substitute a = xyz, b = xy^2, c = xz, and then proceed as f(a, b, c)?

>>10165420
Read the subsubsection about the differential of the composition.

huh

How do you solve this limit:

lim x->0+ of (log(senx) - logx) / (1 - cosx)

>>10163295
Measure theory to prove the Borel sets of the real numbers have cardinality c. I've seen other problems where you could use transfinite induction if you really wanted to (mostly dealing with sigma algebras), but there are much better ways of solving them.

>>10153853
does anybody know where i can find the solutions manual of pic related? i specifically need the fifth edition please im going crazy over here, i also cant buy it right now.

test

>>10166240
Stop cheating on your homework brainlet

>>10164836
anyone? are the /sci/ wiki ones good?

>>10163408
There's the physishit approach for intuition: complex conjugate transpose.

>>10164836
Pretty much every undergrad topology course uses Munkres. I think it's okay for self-study, but there's probably better ones out there.

>>10164836
>>10166379
Munkres is a good book but I don't think I would want to self-study it. Unless you really know what you're doing, generally speaking you want to pick thin books to self-study things.

I would suggest that instead of reading "topology" you find a specific area of topology you want to learn. Manifolds, or algebraic topology, or knot theory, or whatever tickles your pickle.
I am of the fairly strong opinion that you should not learn general topology until you absolutely need it for something.
Point-set topology for its own sake is autistic nonsense; you have to be coming from the context of why it _needs_ to exist in order to take anything useful away.

Same phenomenon as the faglords who used to be common here a year or so ago who were shitting up everywhere about their category theory when they barely knew any algebra. You need experience in order to understand categories. Without the background it's just fart sniffing.

Pls help me I am a noob and I don’t understand how they got this result

>>10164499
>>10164518
No I know, you just use universal property of free groups, there exists a set function g: {x, y} --> F({x, y}) s.t. for all group G and for all set functions f:{x, y} --> G, there's a unique group homomorphism φ: F({x, y}) --> G s.t.
φog = f
And you apply this to make the inclusion group homomorphisms i: Z --> F({x, y}) (two of them I'm assuming) to be initial in the Grp^(Z, Z) category

I'm stuck because I know I have set function g, but I don't know if it's unique in order to force F({x, y}) --> G to be unique for any g
And I don't know what the inclusion homomorphism i: Z --> F({x, y}) even looks like because the book constructed free groups to be the group of all the reduced words on the set: R({x, y}) with concatenation as the operator

Then again F({x, y}) is only unique up to isomorphism, but what would the equivalent Z*Z isomorphic to R({x, y}) be? I know it's the disjoint union ZUZ, do I just have have n in Z be mapped to x^n?

Filling in the blanks is what's stopping me
Also
>inb4 reddit spacing

>>10164836
Bourbaki is my favorite. I know people are going to say it is a meme, but Munkres doesn't even use filters and barely treats any ordinal arithmetic and uncountable anything. When it does it makes problems that should be easier harder because it doesn't want to develop any big machinery. Bourbaki actually is fairly gentle and I think easier than Engelking or Willard.

Big problem is Bourbaki delays metric spaces and in my mind function spaces should be towards the beginning. But it is still the best in terms of a textbook.

>>10166448
I guess my underlying confusion is: I thought natural morphisms of a category (natural projection and inclusion) come from the category itself, like for Set they were provided, but I'm not sure if they are constructed beforehand and then simply applied, or if they're defined along the category's definition
I thought Grp's natural morphisms would inherited from Set, with the additional property of being homomorphic, but I thought that would only be the case for the domain being a subset of the codomain

>>10166443
Solve for X.
X <= 4
X =/= -1 since that would make your denominator 0.