/mg/ = /math/ general

The problem is about the maximum possible area of a [math]p[/math]-gon for a given [math] p [/math], fixed.
But to answer you question, if [math] p = n-1 [/math], yes, that is the inscribed [math]p[/math]-gon with maximal area (and the one with the greatest area for all values of [math]p[/math] but that's trivial). Also, [math]what[/math] is that area (as a function of the area of the [math]n[/math]-gon)?
Prove it and generalise it for other values of [math]p[/math]. You're going in the right direction.

90% of the content in this infographic is pseudoscience.

[math]A_n[/math] is a formal series Lie algebra generated by indeterminants [math]X^{ij}[/math] modulo some relations, and [math]P_n[/math] is the pure braid group, i.e. the group of actions that braids n particles together such that the initial and final configurations the same. [math]\theta[/math] uses the KZ monodromy [math]\omega[/math] (which gives rise to a conformal field theory) to define a representation of [math]P_n[/math] with values in [math]A_n[/math], and this is useful for finding holonomies of the field theory with connection [math]\nabla = d + \omega[/math] (just take the exp).
Holonomies are important because it tells us the particle statistics of the theory, namely how our wavefunction transforms [math]\psi \rightarrow e^{-i\int_\gamma A} \psi[/math] under [math]P_n[/math], and this is directly linked to the Aharonov-Bohm effect. It is also important in constructing a gauge theory equivalent to the CFT due to the equivalence shown here arxiv.org/abs/1504.02401.
The symplectic form obtained from the holonomy [math]\exp\left(-i\int_\gamma \theta\right) = \exp\left(-i\int_B \Omega\right)[/math] where [math]\partial B = \gamma[/math] can also tell us quantization conditions that govern the CFT via the first integrality condition.
The rest are just calculations based on these.

So this is the mathematics behind how particles interact with each other? Are these equations even set in stone? If so how do we know that they are?

fuck, I hope I will get good at differential geometry one day. what is the non-meme book where I can learn this stuff?

is it A*(1 - 4 (1 - p/n) sin^2(180/n))?, where A is the are of the regular n-gon?

The mathematics itself is set in stone, as shown by Artin who characterized the pure braid group [math]B_n = \pi_1(\mathcal{Q}_n(\mathbb{R}^2))[/math] and other mathematicians that followed who generalized it to higher dimensions and other manifolds. The connection of this concept to particle statistics in physics arises from the observation of the quantum Hall effect and the Laughlin variational wavefunction that describes it, which is a wavefunction that transforms exactly like how I described above under [math]P_n[/math]. (((Jacak))) has a good article on this arxiv.org/abs/cond-mat/0308530 if you'd want to check it out.
If you're more interested about the more preliminary stuff regarding CFTs and geometric quantization I recommend the books by Kohno and Woodhouse, respectively. The results established there are my starting points; all I'm doing here is taking concepts from one place and apply it to another to see if there's anything new popping up in the context of physics.

Then as a well versed scientist, how would you advise someone young to become very cute and feminine?

Is there any new studies or equations being found that apply to physics????
How do we know if the content being found inside mathematics is applicable or if it even works???

>How do we know if the content being found inside mathematics is applicable or if it even works???
How do we know if it does or doesn't if we don't try?!?!?!?!?!

If it doesn't apply, then what's the point of creating the mathematics in the first place?

I don't know ask the mathematicians rofl

On problem (5), it should be [math] M^TM = I_m [/math]. Also, [math] m \leq n [/math].
Nope.

You start in an area that interests you. There is no guideline to reading original papers. You're on your own.

zbmath.org/

Lee's Manifolds.

I would avoid Lee's. If you are looking to get good, try Tu's book for a good introduction (if needed), otherwise if you are already acquainted, try Spivak's volumes to improve.

I'm having a fleeting moment of appreciation of the notation in linear algebra.

Let [math]\alpha = (e_1,\dots,e_n)[/math] be an ordered basis (organized in a formal row vector) of some vector space [math]V[/math]. Denote [math](u)_{\alpha}[/math] the column vector of coordinates of [math]u \in V[/math] with respect to [math]\alpha[/math]. Let [math]\beta[/math] be another basis and denote [math](id)_{\beta \alpha}[/math] the transition matrix from [math]\alpha[/math] to [math]\beta[/math]. Considering FORMAL matrix multiplication, all of the following identities and manipulations are valid (notice how the greek letters connect)

[math]u = \alpha \,(u)_{\alpha}[/math]

[math](u)_{\beta} = (id)_{\beta \alpha} \, (u)_{\alpha}[/math]

[math]\beta \, (id)_{\beta \alpha} = \alpha[/math]

[math]u = \alpha \, (u)_{\alpha} = \beta \, (id)_{\beta \alpha} \, (u)_{\alpha} = \beta \, (u)_{\beta} = u[/math]

Furthemore, denote [math]\alpha^* = (f^1,\dots,f^n)^T[/math] the ordered dual basis of [math]\alpha[/math], but organized into a column vector, let's also agree that for a form [math]\omega[/math] we will write [math](\omega)_{\alpha^*}[/math] for the ROW (!!) vector of coordinates with respect to [math]\alpha^*[/math]. Once again considering formal matrix multiplication, we have

[math]\omega(u) = (\omega)_{\alpha^*} \, (u)_{\alpha}[/math]

[math]\alpha^* \, \alpha = E[/math], here [math]f^i . e^j[/math] means evaluation

And of course the "dual" properties showing that [math]^*[/math] is just transposition really

[math]\omega = (\omega)_{\alpha^*} \, \alpha^* [/math]

[math](\omega)_{\beta^*} (id)_{\beta \alpha} = (\omega)_{\alpha^*}[/math]

[math]\beta^* = (id)_{\beta \alpha} \, \alpha^* [/math]

For a linear map [math](\varphi)_{\beta \alpha}[/math] it's similar.

>notice how the greek letters connect
Like how the physicists did back in 1920?
Coordinates in the dual basis can be written as contravariant indices so that [math](u^*)^\beta u_\alpha = \delta_\alpha^\beta[/math]. The bra-ket notation is also another example of good notation: the kets [math] |\psi\rangle \in \mathcal{H}[/math] are elements of a Hilbert space and bras [math]\langle \phi | \in \mathcal{H}^*[/math] is an element in the dual, so that the pairing ca be written as [math]\langle\phi |\psi \rangle = \phi(\psi) \in \mathbb{C}[/math].

Imagine being a fan of anime and a fan of math, but instead of watching anime or studying math, you spend your whole day talking about those two things on an anonymous Bangladeshi child porn sharing site. And you repeat that every, single, God, damn, day. Isn't that just a waste of time?

>two qualifying exams in 30 days
>depression and incompetence reaching maximum
I should have went into the Biomath program instead
No qualifying exams and the research interests are closer

You're obviously out of almost everyone else's league, so why the fuck are you here and not on /r/math or stackexchange/stackoverflow?

How many of you actually are Math PhDs?
Reading this thread I get the feeling it's full of undergrads and CS undergrads with a minor in Math. Not to mention the casuals who "self study".

I'm gonna be a Math PhD one day
I'll make you proud

Neurotypicals cannot be mathematicians because they lack the creativity to visualize the Math inside their brains. All they do is memorize formulas and condition themselves to solve problems.

Is there any simple way to prove that the diophantine equation [math] x^4 + 4y^4 = z^2 [/math] has no all-positive solutions?

H-how do I know if I'm neurotypical?

about to start my PhD. I've got a fresh hot-off-the-presses Bachelor's in math. Got accepted to the same medium-good uni I did my undergrad at. I am satisfied with this.

I'm not the user you responded too, but you will know if you can answer this question:

>What exactly is a vector?

Consider all of these questions:
-back in school days, did other students and teachers consider you as annoying?
-do you feel itching a lot?
-are you somehow sensitive to sounds?
-does hugging, cuddling, touching, etc, disturb you?
-do you spend plenty of time (at least one hour every day or so) fantasizing/daydreaming?
-do you carry any "strange" childhood habits (severely chewing pencils, spinning coins, etc)?
-do you feel as if things should make mathematical sense?
-do you often repeat many things over out of pure habit?
-do you find it difficult to concentrate?
-do you enjoy solving difficult puzzles?
-do you often spend a lot of time staring at things (buildings, the moon etc)?
-do you find it difficult to relate to normal people?
-do you have trouble when trying to socialize?

>What EXACTLY is a vector?
I really fucking hate these kinds of questions with a passion, because if I say something stupid like "an element of a vector space" or whatever, I'm sure to get it really fucking wrong, and I've probably already proven that I'm neurotypical or something, so that's just another data point to back the at this point undeniable conclusion that I'm fucking worthless.

I didn't want to have a good night anyway.

Because I love you guys.

Not the user you responded to, but this is /spectral analysis general/ now.

>-back in school days, did other students and teachers consider you as annoying?
Possibly.
>-do you feel itching a lot?
I used to.
>-are you somehow sensitive to sounds?
No.
>-does hugging, cuddling, touching, etc, disturb you?
No.
>-do you spend plenty of time (at least one hour every day or so) fantasizing/daydreaming?
I used to.
>-do you carry any "strange" childhood habits (severely chewing pencils, spinning coins, etc)?
No.
>-do you feel as if things should make mathematical sense?
Yes.
>-do you often repeat many things over out of pure habit?
Maybe? Not sure what you mean exaclty.
>-do you find it difficult to concentrate?
No.
>-do you enjoy solving difficult puzzles?
Yes.
>-do you often spend a lot of time staring at things (buildings, the moon etc)?
I used to.
>-do you find it difficult to relate to normal people?
Somewhat.
>-do you have trouble when trying to socialize?
Yes.

Element of a vector space.

>-back in school days, did other students and teachers consider you as annoying?
No.
>-do you feel itching a lot?
I don't know what constitutes "a lot", but from what I've seen from other people, not especially, no.
>-are you somehow sensitive to sounds?
Yes
>-does hugging, cuddling, touching, etc, disturb you?
Yes.
>-do you spend plenty of time (at least one hour every day or so) fantasizing/daydreaming?
Definite yes.
>-do you carry any "strange" childhood habits (severely chewing pencils, spinning coins, etc)?
Yes.
>-do you feel as if things should make mathematical sense?
Yes.
>-do you often repeat many things over out of pure habit?
Yes.
>-do you find it difficult to concentrate?
Yes.
>-do you enjoy solving difficult puzzles?
No, because I know I won't be able to solve them, so I usually don't try in the first place, and when I do, I fail.
>-do you often spend a lot of time staring at things (buildings, the moon etc)?
Yes.
>-do you find it difficult to relate to normal people?
I'm not sure.
>-do you have trouble when trying to socialize?
Yes, but I always pull through, and people always think I'm friendly and nice (which disturbs me).

So what's the verdict doctor?

>-back in school days, did other students and teachers consider you as annoying?
Yes
>-do you feel itching a lot?
No
>-are you somehow sensitive to sounds?
Extremely
>-does hugging, cuddling, touching, etc, disturb you?
Only if someone else initiates it
>-do you spend plenty of time (at least one hour every day or so) fantasizing/daydreaming?
Yes, though it usually involves math/physics
>-do you carry any "strange" childhood habits (severely chewing pencils, spinning coins, etc)?
Yes, I used to take the green twist ties from the supermarket and I'd make soldiers, vehicles, dragons, boats, I never had a reason, I just did. I also used to do this thing where if there were tiles I could only step from one to another by going three units up and one to the left or the right, basically to how a knight in chess moves, always just seemed fun. I also used to divide up any shapes by their axis of symmetry.
>-do you feel as if things should make mathematical sense?
Yes
>-do you often repeat many things over out of pure habit?
Yes
>-do you find it difficult to concentrate?
No
>-do you enjoy solving difficult puzzles?
Yes
>-do you often spend a lot of time staring at things (buildings, the moon etc)?
Yes
>-do you find it difficult to relate to normal people?
Sometimes, for the most part I can guess why someone does something, but I never really "get" it, some behavior/social convention that's supposed to be normal just seems a bit weird/useless.
>-do you have trouble when trying to socialize?
Yes

Neurotypicals, all of you.

Cool, guess I'm not the autismo I thought I was, this is great news, time to sell out and become a chad

The easiest way to tell if someone is autistic is by asking the person directly, if you have ever thought to yourself "I have autism" you most certainly don't. Autists are too oblivious to care about such matters, those who make claims like I have autism are usually just attention whores.

can anyone gave me a problem for a math brainlet like myself

What is 2+2?

((2)+(2))

fish XD

What criteria did you use?

You're not autistic, just an attention whore.

I never said I was autistic, I've never been diagnosed and autodiagnosis is retarded as fuck.

Honestly, I hate this kind of "so far above that tedious shit" attitude in math. I had one fellow math friend in my grad program who wasn't like that but have lost contact, feels bad.

I don't know where you're at so it's hard to guess what problem might fit you. Probably you're in the wrong place though, more due to the attitudes than the content though.

What is this type of problem called?
I'm studying for the accuplacer for College. I have always done poorly in math and basically needed to learn arithmetic and Algebra again through self study but I can't study this because I have no idea what the fuck I'm trying to do.
I know this is very basic to all of you so I figured I would ask

>Honestly, I hate this kind of "so far above that tedious shit" attitude in math.
What are you talking about exactly, user?

It's basic arithmetic, here's a hint, distribute the sqrt(50) to each of the terms and try working it out form there.

How long did it take you to get to your level of knowledge? How many hours a week do you study, and since what age do you do that? Did you get a Math Ba or was it self taught? I'm sorry for being overly curious, but I guess it would help having someone as a compass.

You have one meter of rope. Someone asks you to form the rope into a right isosceles triangle. What's the area of the triangle in square meters?

For the record, if you can't answer this question you are a turbo-brainlet who even brainlets look down upon.

Why even bother? They were probably acoustic about maths and physics their whole life, were born a genius of some kind and now are just reaping the benefits of superior genes.

1- I didn't ask you.
2- If a Finnish 2hu poster can without trying then so should I.
3- Even if not, you clearly don't understand what a compass is.

>How long did it take you to get to your level of knowledge?
24 years since I'm almost 24 now.
>How many hours a week do you study, and since what age do you do that?
I don't have a set schedule. Since I don't have data on my phone whenever I commute I bring a couple of texts/articles with me to read on the train. That'd probably amount to 4 hours per day back and forth. I've been doing this since sophomore year when I was 18.
>Did you get a Math Ba or was it self taught?
My bachelors was in mathematical physics. Most of the topics here are self-taught but it does help me understand theories in physics better, and vice versa.
Also I'm not Finnish.

You forgot the single most important and universally relevant question for this type of inquiry

What is your family's academic background and area of profession?

inb4 phDs, professors, did math with me as a child

>commute
>4 hours per day
What third world country are you from?

I can sense you got really flustered. How so, user?

Dad was an insurance broker and mom was a trader for an international corporation.
I live two towns away from where I work/studied.

Element of a vector space is the correct response. The mark of a true mathematician, congrats.

I asked what country you live, not the distance between your place of residence and place where you studied/worked measured in human settlements.

>4 hour daily commute
Damn, sounds rough, though I guess if you can get quality reading time in then it's not so bad. Did ya already finish your PhD or are you still a student?

I've just finished up a MSc in physics. I'm heading elsewhere for a PhD (also in physics) soon.
Doesn't mean I'll stop self-studying math though.

What country are you from?

Was your master's thesis also on cft or maybe some other topic (I know in the US you do a thesis for your masters, don't know about elsewhere). What math? Non-commutative geometry, knot theory, algebraic geometry, K-theory...

I did do a thesis, it was in Canada. It was about graphene; it's a very archetypal "physics" thesis. Don't dox me btw.
>What math?
Mostly what you see here. CFT, TQFT, geometric quantization, knot theory, etc.

Can you please tell me what country you are from?

I was born in Taiwan.

Your racial group is Asian?

Wait, Canada? Huh, that's kind of funny, of the three guys I work for one was a prof at waterloo and the other got his master's there. I actually wanted to do a masters (or PhD) at perimeter (if I went to waterloo it'd likely be in pure math with an eye towards theoretical physics, I know of few guys in the pure math department that do that sort of thing) though from what I've heard it's difficult as all hell to get in

Some time ago there was someone who declared themselves to be latino, and now there is a self declared Asian. I used to think the worst that could happen to this thread were Americans.

>though from what I've heard it's difficult as all hell to get in
That's correct, and it's extremely competitive. Your CV and recs have to be stellar in order to get into Perimeter, since they'd only take in students that they're willing to give substantial fundings for. I've tried when I was looking for masters but to no avail.

Worst comes to worst I apply and get reject but thanks to the fact that they've uploaded the video lectures for their courses it makes a great supplement to learn a lot of these topics independently so it won't be all bad. Would be nice if they uploaded problem sets, though I guess at that point asking for more would just be greedy.

Yeah that can't be helped. Good luck user.

>unable to find a professor with a research position
>scared of failing my qualifiers
>considering dropping out
>repeat this sequence of thoughts over and over for the last 3 months
>receive an email a few hours ago from a professor with a possible research project in something I'm not experienced in

Is it a sign?

How far out of depth is it? Is it a tangentially related field?

I haven't got the details yet, but it's a project in PDEs requiring experience in numerical linear algebra and finite differences. The later two I'm well versed, but I won't even take the first course in graduate PDEs until this coming semester.

Thanks man, that's all I needed.

I can suggest a few grad level PDE books you can check out during the summer. If you already have a decent amount of knowledge it couldn't hurt to contact the prof and see if you can handle it.

>what are you studying this summer?
The amount of alcohol and opium/opioids that can be combined without croaking.

2. is easy, assuming we know the following things:
>module categories satisfy two topos axioms but not necessarily the subobject classifier thingy (lawvere-tierney)
>in a topos an arrow whose codomain is an initial object is an isomorphism
>trivial modules are both initial and terminal
>there are non-trivial modules over unitary rings
It follows that there can be no subobject classifiers for these module categories, in particular the category of abelian groups.

If you're so concerned with where everyone is from why don't you start off by telling us where [math] you [/math] are from, ass-hat.
>Don't dox me btw.
Too late for that. Gorilla has been so talkative that I already know who both of you are.

The fact that his posts often go un-replied is his fault actually, since most of what he's posting, when it is correct, are elementary constructions ("it follows from the definitions" stuff), and of the rest quite a few are inane symbol salads.

I'm just here to post this (real, a professor at my school put this on their social media) image because it's relevant to the thread theme.

I mean if the prof emailed you that's probably a good thing? Don't let the impostor syndrome get to you and keep at it.

Help I can't see the mistake, I suspect is something related to the fact that you may get [math\frac{h}{h}= \frac{0}{0}[/math] but I'm not quite sure

This captures the essence of the proof, but there is a technical problem. You know as h -> 0 that g(x+h) - g(x) -> 0 as well, so you'd like to say that f'(g(x)) is equal to lim_{h -> 0} (f(g(x+h))-f(g(x)))/(g(x+h)-g(x)). But this isn't actually the definition of the derivative, and you would need to show formally that you get the same limit. Doing this properly really requires messing around with epsilons and using the definition of the derivative.

let g(x) be a constant function.

>Not to mention the casuals who "self study".
Just like Software Engineers program for fun so should Mathematicans self-study for fun. Don't you think it's academics who self-study rather than "casuals"?

user please...
>module categories satisfy two topos axioms but not necessarily the subobject classifier thingy
This is "proof" by assertion/begging the question. What you have to prove is exactly that they don't satisfy it.
>in a topos an arrow whose codomain is an initial object is an isomorphism
This is trivially false. Were it true, every set would have cardinality [math] 0 [/math]. ([math] \textbf{Set} [/math] is a topos.)
>trivial modules are both initial and terminal
True and potentially pertinent to the proof, depending on how you'd choose to approach it.
>there are non-trivial modules over unitary rings
True but not actually relevant.

This user was going in a good direction. (I hadn't noticed his post yesterday. And by the way, I posted a link with it that you might want to read. Section 5 is particularly useful.)

Damn, all of them