/mg/ - Math general

yeah for someone who just started to question how everything works

it's not a bad thing, we all kinda have a phase where we think everything is wrong, eventually you'll get to a point where you realize it's meaningless to try to redefine everything off some new standard if it doesn't even break way to any new discoveries, only to more limitations

go ahead and post in wildberger threads all you want, i'm not going to spend time trying to disprove a claim without evidence given
note: i'm not asking for evidence, just take it to the nearest wildberger thread

>math Chad
no such thing. everyone who's good at math studies or has studied extensively.

Reading Velleman's How to Prove It. Learning a lot coming from a physics background. Recommendations for future reading?

what're you planning on learning?
discrete math and linear algebra are both good for babbies first "with proofs" class

I've been reading Kosinski's Differential Manifolds following an user's suggestion. Going through the introductory chapters right now to dust off some of that differential topology I forgot in the last year I've been out of school.

How're you liking the book?

Pretty well. I like Kosinski's writing so far and I like the material. The exercises can be pretty involved and the book doesn't really hold your hand at all in terms of providing similar proofs or anything, but I haven't gotten too stuck on them. Pretty solid altogether.

That's great, if you need any other book suggestions I can probably recommend some other nice texts.

for all integers, a^2*b^2 = n^2 where n is another integer.
this means you can rewrite it to 7a^2=11b^2
for this to work, 11|7*a, meaning a must be an integer multiple of 11, meaning 77|a. also, 7|11*b, meaning 77|b.

no matter what you will always have a prime factorization of 7*11^a=11*7^b, which through division is equivalent to 11^(a-1)=7^(b-1), and for any integer a and b, they can never be equal since neither side will ever have a prime factorization of both 7 and 11.

rate my proof, i haven't taken a class with proofs yet

starting to read tao's notes on measure theory

terrytao.files.wordpress.com/2011/01/measure-book1.pdf

is this "box" terminology typical? or it k-cell more common term?

>tao had notes on measure theory
didn't know about that
thanks

WHAT DO I DO LADS?!

I've got 2 days to learn Fourier series, Fourier transform and DFT & FFT.
So that I can teach it to a small group of people.

Fourier series is basically just using the orthogonality of sin and cos with respect to the integral inner product from 0 to L (or maybe it's -L to L, it's been a while)

>discrete math and linear algebra are both good
Wrong.

One day is more than enough for that stuff

>a, b, c, d are integers
>I'm trying to derive a contradiction from this
You're pretty much done. Arbitrary integers don't necessarily exist.

Explain

Tao's construction of the Lebesgue measure is godly. It is completely intuitive and very little out of left field proofs. It starts with a completely geometric notion of measure (measure of an interval is it's length, similarly his 'boxes' are n-volumes) and slowly adds in analytic machinery to get the class of Lebesgue measurable sets. It is so much better than the two other authors whose last names start with R that have written widely used measure theory books.

Anyone else feel like Tao's teaching materials are pedagogically really poorly made? It seems like he is trying to show how smart he is and not make people learn. It seems that all his blog posts and course notes are probably really great once you are a fields medalist, but I find them completely inaccessible, and I've taught myself very niche stuff like local field theory.

I can only speak for his intro measure theory book which I used in a grad analysis course. We used Royden in my undergrad measure theory class and Tao in my first semester grad analysis class. Maybe I just was inexperienced when I used Royden but I didn't really get measure theory until my grad class, and I thank Tao's book as well as a great instructor for that. I found every construction was specific and precise, and the exercises perfect complements and extensions of the material. I have never more than glanced at his blog or have read any of his other books.

yeah i started reading Tao's and got annoyed with how much exercise he just starts laying out. but the i went to Rudin and thought: wtf is this? So now im back on the Tao train

Rudin is a meme. Good work not falling for it.

i know that the complex numbers can be constructed as the quotient ring of the polynomial ring of real numbers by the ideal generated by (x^2+1)

is there any interesting structure to quotients of higher order ideals?
and what about polynomial rings of multiple variables?

one thing that comes to mind is [math] \displaystyle \frac{\mathbb{R}[x,y]}{x^3-x-y^2} [/math], or more generally, [math] \displaystyle \frac{\mathbb{R}[x,y]}{x^3+ax+b-y^2} [/math]
which is of particular interest because of the algebraic structures you can define on elliptic curves
what does this "look like"?
i feel like this might be approaching algebraic geometry territory, which i know nothing of
can i get a quick rundown on this?

It's called a coordinate ring, try skimming en.wikipedia.org/wiki/Affine_variety

Probability theory

And some algorithms textbook

Progressing slowly through Mendelson's Introduction to Topology while I live the wageslave life. I was an applied math major who didn't even take Real Analysis, so I probably should have done Baby Rudin or something first, but there is enough coverage of continuity and a little section that hammers epsilon/delta so we'll see how this goes.

Can you define the dot product for "continuous matrices" as as [eqn](A \cdot B)(x, y) = \int\limits_{-\infty}^{\infty}A(x,s)B(s,y)\ \text{d}s[/eqn]

>I was an applied math major
How can you major in something nonexistent?

>i know that the complex numbers can be constructed
Not really.

What do you mean?

here's a better proof: 7 is not a square
QED

Currently going through Aluffi Algebra Chapter 0, killed almost every section but I'm onto modules and fuck Im kind of plateauing. Any tips?

what math books have u actually read?

does anyone here go on irc?

depends on ur goals

sure, assuming appropriate regularity conditions

...

Is it possible to be bad at calculus but good at more advanced math? My math logic class comes really easily to me but calc 3 doesn't? What gives?

What exactly are you having trouble with?

In general, a ring R can be thought of as a ring of functions of the affine scheme Spec(R).

The schemes associated to the rings you listed are not particularly nice because the reals aren't algebraically closed.

For an alg. closed field k, and a prime ideal p of k[x1,...,xn]. The ring k[x1,...,xn]/p is isomorphic to the ring of regular functions on the variety V(p).

ex. If p = < y^2 - x^3 > then k[x,y]/p is the ring of regular functions on the affine curve y^2=x^3

>Is it possible to be bad at calculus but good at more advanced math?
I don't know anything about calculus, so probably.

I don't think it's terribly uncommon for that to be the case. I had a hard time with calc 3 because I'm pretty lazy with algebra so I had negative sign accumulation. I got an A, but I definitely worked for it.

There was a girl in our math department who failed Calc 3 but got a 40 on the Putnam exam one year and was allowed to teach graph theory as a senior.

If we accept the assumption that you're actually good at math the problem is that you're missing foundations.
In a logic course you start from scratch, but if there's concepts you didn't understand in the first two sections of calculus calc 3 is going to be difficult.

If you struggle with calc 3 while actually understanding calc 1+2 you are unfortunately a brainlet, there's nothing in that course that's more than a minor variation on stuff you already know.

>2 days
>2
>learn Fourier series
>learn DFT
>learn FFT
>learn it enough to teach people

I pity you and the people who will be confused and have such a beautiful topic ruined by an obvious mental retard.

Why are linear algebra and discrete mathematics (like combinatorics, set theory, graph theory, etc) all taught after calculus? there is no calculus prerequisite for any of this

are there more branches of math like this, where there aren't really prerequisites beyond geometry and algebra?

oh, probablility and statistics too

>are there more branches of math like this
Yes, basically anything even remotely interesting.

You actually use some calc in probability and stats such as applying the geometric telescoping series to certain problems.

How can math explain non numerical concepts like gravity?

I know you can calculate gravity with math but how does one use math to explain gravity?

WHY DO THEY CALL IT FAST FOURIER TRANSFORM IF I CAN'T LEARN IT IN LESS THAN 30 MINUTES

like????
???
>?
????

yeah but a lot of you don't use anything at all

a subfield known as physics

by giving physical things a symbolic representation, we can extend math into the real world
e.g. let s(x) be the displacement between two points in real space, s'(x)=v(x), or velocity

right now we're still in a math-like realm, so no real major assumptions have been made, this isn't true for most things higher than this. Pretty much everything is based on observations, or just fitting some math to the observations.

when it comes to gravity, almost all of it is based on observations (see Kepler's laws, all are based on observations)
all newton did was showed that F~m1m2/r^2, with there being a constant G for that, all based on data he had. this is good enough for every entry physics course and is pretty good for general purposes. Obviously this doesn't 100% hold true, otherwise general relativity wouldn't be a thing. There's also some discrepancies about how Mercury's path is like 50 arcseconds per century off or something.

In fact, Newton even tried to let F=Gm1m2/r^2 + Bm1m2/r^3 at one point.

so basically you just give axioms based on what you observe and hope they're right enough.

>yeah but a lot of you don't use anything at all
are you fucking serious?

Could a kind user present link.springer.com/chapter/10.1007/978-3-642-02094-0_7

it's hard to say but it's the only section where i've had trouble almost every exercise

Brushing up on my Galois theory.
I'm trying to take a combinatorial approach.

If you've ever wondered how many irreducible monic polynomials of degree n there are over a finite field, here you go.

>yeah but a lot of you don't use anything at all
probability theory = measure theory with a bounded measure (and some weird-ass notation). integrals all over the place. if not, you're doing something wrong.

You could describe the 1/r^2 part pretty easily.

The surface area of a sphere is proportional to r^2.

If you are a distance r away from a point mass of mass m, you can imagine a sphere of radius r centered at the point mass (your position will be on the surface of the sphere). Now imagine the mass is radiating gravitational energy equally in all directions in proportion to m. If you calculate the flux density of this radiation at your point on the sphere, you will find it to be proportional to m/r^2.

what about fields of the form Z/p^n ?

>what about fields of the form Z/p^n ?
Those aren't fields (unless n=1).

I have an integral test coming, so can you guys post your nastiest integrals within the material of calc. 1 and 2.

>so that i can teach it to a small group of people
why are you teaching it if you dont even know it?

i guess he meant [math]\mathbb F_{p^n}[/math]

I think he thinks that [math] F_{p^n} [/math] is [math] \mathbb{Z} / p^n \mathbb{Z} [/math] .

what does this have to do with galois theory? pretty much all you need is the fact that the polynomial ring is a ufd. still a nice argument.

I had my integral test last week, turns out I diverge.

Get me into maths anons, I hate it.

So 10 is equal to 0.2 ?

>Proof of this theorem has been left as an exercise to the student

Then you prove it like you were solving any other problem, but no! It's the redditfrog retard who needs to have solutions in the back of the book instead of being able to proofread its (yes, it doesn't even deserve to be treated like a human bean) own proofs. Just give up, you will never become anything.

pretty sure facebookfrog lad hasn't taken a proofs class (or any class involving proofs) yet, so it's kind of justified.

almost everyone struggles at first with proofs when you haven't done lots of practice with them before

>tumblr spacing

>everyone struggles with proofs when you haven't done lots of practice
>yet he complains about authors forcing him to practice
wew

>someone struggles with practice problems because they haven't done a lot of practice
yes it's a straight forward conclusion

i'm saying it's not like it's all hopeless because he struggled to prove a theorem, that it's just through lack of practice

the point is he's complaining about a book guiding him through the practice he needs, and that's fucking retarded. he needs to suck it up and do it

[eqn]\int_0^1 \log (x)\log(1-x) dx[/eqn]

refer to Sorry about the shit notation.
I'm pretty sure the counting argument still works if you replace p by p^N everywhere.

Counting the isomorphisms that fix the base field will be fun
If you start with a field F, you work with polynomials over F, namely, F[x].

To do a field extension you pick an irreducible polynomial p(x) in F[x].

Then you mod out by the ideal generated by p(x), . is just all polynomials having p(x) as a factor. You could write as (F[x])*p(x).

You get F[x]/ is a field extension of F.
Addition works like you think it should (it inherits the structure from F). It is pretty much vectors with components in F.
Multiplication is done like in F[x] but then you reduce the result mod p(x)

To do iterated extensions, you let K=F[x]/ and look at polynomials over K, namely, K[y].

You can pretty much represent the whole theory as multivariate polynomials satisfying the modularity rules defined when "modding out" by the polynomial ideals.

Where can a math major undergrad intern in a shitty country

[math]\int \sqrt{\tan{x}} \mathbb{d}x[/math]

have fun

[math]\int e^{e^x}dx[/math]

[math]\int e^{-x^2}dx[/math]

which country

pretty sure that you need to have studied integrals in more than one variable to solve it in a reasonable manner

New Zealand

[math] \int e^{e^x}dx [/math]. Let [math] u = e^x [/math]. Then [math] du = e^x dx = udx \implies dx = \frac{du}{u} [/math]. So the integral becomes [math] \int \frac{e^{u}}{u}du = Ei(u) + C = Ei(e^x) + C [/math]

Heh, that was trivial. You are like a little baby compared to me.

Integrate this
*unzips integral*
[math]e^{e^{e^{.....^{e^{e^{x}}}}}}[/math]

I am afraid not even the exponential integral can do against this BEAST. However, after quickly applying Tai's method I noticed it was trivial and therefore I leave it as an exercise to the reader.

quick how do I prove that the intersection of two subgroups of a group forms a subgroup

I can't see where to start with proving closure

en.wikipedia.org/wiki/Subgroup_test

u=x^2, du=(dx)2x=(dx)2u^(1/2)
Then know the gamma function.