Advanced Mathematics Question/Answer thread

No Pre-Calculus, Calculus I-III, Differential Equations or elementary Linear Algebra questions allowed.

Other urls found in this thread:ödels-incompleteness-theorems-and-the-axiomatization-of-euclidean-geometry

What's the number that looks like an upside-down 6?

>When you're still in school and divide math by the damn courses you took

High schoolers just call every class "math." I've noticed a lot of people in college who don't take a lot of math do the same.

How do I calculate slope given two points?

This is pretty ad vanced I think.

What does it mean to differentiate something? What are eigenvalues? What is a Delta function? What are polar coordinates? What is Taylor series used for?

Under what conditions does Lie algebra cohomology coincide with other cohomology theories?

Consider the manifold M x^2+y^2+4z^2=4. I have calculated the Gauss map to [math]\eta = 1/\sqrt{3z^2+1} (x/2, y/2, z)[/math]. Is this correct? Then the volume form should be [math]dM = z/c dx \wedge dy - y/(2c) dx \wedge dz + x/(2c) dy \wedge dz[/math]. Correct?

so i have a characteristic zero field F with closure C(F) and the absolute galois group [math] \Gamma_k = \text{ Gal }( \frac{C(F)}{F} ) [/math]
then X is a geometrically connected variety over F, and a geometric point y in X(C(F))
then [math] \pi_1(X) = \pi_1(X,y) [/math] is the etale fundamental group of X
next, let [math] Z = X~\times_k~ C(F) [/math] and let [math] \pi_1(Z) = \pi_1(Z,y) [/math] be the etale fundamental group of Z
thanks to grothi we have [math] 0 \rightarrow \pi_1(Z) \rightarrow \pi_1(X) \rightarrow \Gamma_k \rightarrow 0 [/math]
and because [math] \pi_1 [/math] is a functor, if a point in F is in X, then that sequence has a section
i want to prove the above statement is true when X is a hyperbolic curve over a number field
how do i go about this?


srs question
Do people actually find math through calculus difficult. Literally just may attention and don't even need a strong foundation bc of the internet.

I've never encountered a branch of math that was more complicated than
>Memorize notation and vocabulary
>Keep track of variables
>Memorize some useful lemmas to save time

how is high school treating you, friend?

Think harder about what I wrote.
Hint: Not all variables are numeric.

How much information in bits is a base 10 decimal number in binary? The decimal number (182)_10 = (10110110)_2 so how many bits s that?

how can i show that
[math] \mathfrac{sl}(4, \mathbb{C}) = (1 eigenspace) \oplus (-1 eigenspace) [/math]

or rather
[math] \mathfrak{sl}(4, \mathbb{C}) = (1-eigenspace) \oplus (-1-eigenspace) [/math]

I take it you don't have high reading comprehension?
I can post a test to check your knowledge of scientific principles and you can share the link of your results, timestamped of course.

You have offered zero counter-point, zero counter evidence.
Therefore I see no reason to continue with you if only I have something to intellectually contribute.

Your denialism is fallacious.

let's say i want to generate my own probability distribution -- something akin to the chi-square or normal distribution. what techniques/technologies will i need at my disposal?

a P underlined.

Nah, I'll admit you win. I'll just go fuck off and try to figure out how to wrap my little brain around homomorphisms without using notation or varying anything.

Look, if you're still going to troll or act retarded, that's fine.
- Swear
- Ad hominem; Call people names
- Don't provide counter-arguments
- Reject realism and the scientific consensus
That's ok.
Just don't loop.
Looping is cancer.

Personal incredulity and the argument from ignorance are fallacies. You're ignorant.
You imply you have no knowledge of the other kinds, therefore they don't exist.
That is wrong irrational.

Why set theory seems fascinating at a glance, but is boring af in detail.

set theory isn't really that big in itself, most of the interesting things about pure set theory arise from autists arguing over axioms

if you're comfortable with your set theory skills you should try topology, it's a little bit more fun

or you could just abandon set theory and go for category theory, it's pretty neat

nice try grothendieck

Stable homotopy theory


How prove [math]{H_k}\left( {X;\mathbb{F}} \right) \cong {H_k}\left( {X;\mathbb{Z}} \right) \otimes \mathbb{F}[/math] ?

[math]\mathbb{F}[/math] being a field.

Are memes calculable?

i dont know shit but cant you use
since the torsion with a field is 0?

can somebody tell me what the shit a free vector space is please

do you at least have a definition of it?
its probably analogous to every other free algebraic construction

is there a general way to count the unique combinations of prime factors of a number

so like, 12 = 2*2*3, so |{2,3,4,6}| = 4 ways

to specify, ways that are smaller than the original number (so all combinations up to n-1 of n factors)

i wrote this down wrong, i was copying from three pages of notes
[math] \gamma_k [/math] should be [math] \gamma_F [/math]
similarly [math] \times_k [/math] should be [math] \times_F [/math]
it should have been obvious from the context but i just wanted to fix it

It's all numbers in the interval (1,n) that have a common factor with n.

It's the same thing as n-φ(n)-1, where φ() is the euler totient function.

What's the point of Series, Polar Coordinates and Parametric curves?

As a mechEng student (going to take Physics and Calculus 3 in the next semester), will I ever find a use for them in the future of my academic life?

A vector space not oppressed by the white man.

"A Survey of Modern Algebra" and "Algebra" by Birkhoff and Mac Lane.
What is the difference?

Go contribute some advanced math paradox to please. I need more.

>no calc 1-3 questions
>asks calc 2 question

Most likely, the most useful thing to come out of them for you will be learning how to set up and solve problems and critical thinking skills.

>Engineer student
>academic life
nice try, you tradesman

Every vector space is free. Free just means it has a basis. Commonly used when talking about Modules (which a vector space is a special case of) because Modules don't always have a basis.

*(ignoring infinite dimensional spaces)

Assuming ZF+C, every vector space has a basis and, hence, is free, or am I missing something?

no you are correct. this is one of the main applications of choice.

Usually you'll see free vector space over some prescribed basis elements, and it just means take all formal linear combinations of the elements.

this is exactly why i was asking

the book i'm using seems to make a distinction between "vector space" and "free vector space", but because it's a higher-level book, it never bothered to define either term

i tried googling it but results were too varied or inconclusive to be of any help

pretty annoyed desu

polar coords are fun!

can Veeky Forums help me with part e?

nvm got it

thanks user.



> series
Taylor approximations. Used often in control theory
> polar coordinates
Integrating over pipes
Its easier with polar coordinates
> parametric curves
Umm any 2D mechanics problem?
If two dimensions are functions of time... The easiest cases can be written as parametric curves of time

If a form has coefficients in complex numbers, does a q series whose coefficients are the absolute value of the forms coefficients constitute a form as well?


How would the laplace transform be expressed as a linear map between spaces?

I mean we have [math] f\left( t \right) \mapsto \mathop {\lim }\limits_{x \to \infty } \int\limits_0^x {{e^{ - st}}f\left( t \right)\operatorname{dt} } [/math]

So would we write something like: [math] \mathcal{L}:{L^1}\left( {{\mathbb{R}_{ \geqslant 0}}} \right) \to {L^1}\left( \mathbb{C} \right)[/math] ?

Do Godel's incompleteness theorems apply to Euclidian Geometry?

How do I prove that any scheme is a Z-scheme? It's really easy for an affine scheme, using the fact that Z is initial in CRing, but I can't help myself in the general case.

>Calculus I-III, Differential Equations or elementary Linear Algebra questions allowed
>advance math

>posting algebra 1 homework
>advanced math

Assuming you don't want to formally specify with respect to what you speak of "linear" (you'd have to specficy the module/vector space for the function space), it will be something like this. Except I guess you can extend the domain to C too (not sure), and btw. there is this whole theory of Laplace transforms for measures, of which this will be a special case, I guess.ödels-incompleteness-theorems-and-the-axiomatization-of-euclidean-geometry

Take an affine cover and glue the morphisms down to Z.

in the context of category theory and higher level mathematics, what exactly is equality?
specifically, equality of (large) categories
i know equality of sets is defined by axiom of extensionality, but when working in high-level stuff like category theory, you don't necessarily work with sets
often, you're not even working with proper classes, but rather a faint notion of "collection of objects" that isn't really rigorously defined
i tried googling it first, of course, but only found a bunch of autists arguing over some other irrelevant foundations shit

equality isn't given by axiom of extensionality, that' just an equivalence to equality that is given for sets.

equality is a primitive or logic

>not having heard of Schauder bases

I don't know enough functional analysis to get into specifics of infinite dimensional spaces. So I just excluded them all together.

How do I get a gf mathematically?

>Schauder bases
A Schauder basis is not a basis if your Banach space is infinite dimensional. You can prove this by first showing that every infinite dimensional Banach space contains [math]l_2[/math], which shows that its dimension is uncountable. A Schauder basis is a countable subset of your Banach space with the property that the closure of its span equals is the Banach space.