/mg/ math general - Empirical Mathematics Edition

Empirical Mathematics Edition [TQFT, String Theory, AQFT over exotic spacetimes, etc.]

NO Grothendieck or Serre.

What have you been studying, /mg/?

Other urls found in this thread:

math.stackexchange.com/questions/1878298/spivak-calculus-chapter-1-question-4-6
etoix.wordpress.com/category/calculus-by-spivak/page/2/
Veeky
en.wikipedia.org/wiki/Cantor's_diagonal_argument
math.stackexchange.com/questions/580397/spivak-calculus-chapter-1-problem-1v
terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
calnewport.com/blog/2008/11/14/how-to-ace-calculus-the-art-of-doing-well-in-technical-courses/
ncatlab.org/nlab/show/coherence law
twitter.com/NSFWRedditImage

where the fuck do i start with number theory? elementary, analytic or algebraic? getting sick of the attacks on /mg/

>What have you been studying, /mg/?
I've been studying French to start reading Grothendieck's ETQFT.

undergrad math major here. I can understand how to "do problems" but I'm having trouble with what it all means. I can't help but feel that there's always more out there that I can't grasp, and that I'm only scratching the surface of the surface of math. Anyone else relate?

In Spivak's Calculus, chapter 1 problem 4 (v-viii), perhaps even more, how does the author expect you to prove inequalities such as:

>(v)[math] x^2-2x+2 > 0 [/math]
>(vi) [math] x^2+x+1 > 2 [/math]
>(vii) [math] x^2 -x + 10 >16 [/math]

I understand these are "completing the square" and quadratic equation problems, but how am I supposed to derives this myself given only the properties in pic related?

The only solutions I've seen involve so much creativity it seems infeasible:
>math.stackexchange.com/questions/1878298/spivak-calculus-chapter-1-question-4-6

Here is the top answer from said link, regarding question vi (I'm praying to god the tex works out):
[math]
x^2+x+1&>2 & \text{Given}\\
x^2+x+1+0&>2+0 & \text{By Addition}\\
x^2+x+1+0&>2 & \text{By P2}\\
x^2+x+0+1&>2 & \text{By P4}\\
x^2+x+\left( \frac{1}{2} \right)^2+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P3}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P9}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+ (-1)\left( \frac{1}{4} \right) + 1 &> 2 & \text{By Multiplication}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right) &> \left( \frac{5}{4} \right) & \text{By Addition, P3, and P2}\\
[/math]

In this example, it seems infeasible to be expected to see the possibility of [math]-1(\frac{1}{2})^2 + \frac{1}{2}[/math]

I understand these questions are intended to be difficult, but I'd like to leave no stone unturned throughout this book.

Tex did not work out, here's a lazy screenshot:


The answer ends up being [math] x > \frac{-1 + sqr{5}}{2} /math] or [math] x < \frac{-1 + sqr{5}}{2} /math]

>[math] x > \frac{-1 + sqrt{5}}{2} /math] or [math] x < \frac{-1 + sqrt{5}}{2} [/math]

which fucking thread is the real one. fucking autists

>(vii) x2−x+10>16

This should've been
>[math] x^2 +x + 1 > 0 [/math]

well this one is the only one with math so far

Nevermind, I'm getting it. I never really understood what completing the square really was, but finding a geometric example was insanely helpful. Turns out the 1/2^2 I complained about wasn't expected to be found by an arbitrary stroke of luck, but comes straight out as a result of completing the square

When solving an equation or inequality, at any given point you have plenty of options, each of which may lead to a new assortment of options.

Are there any heuristics involved to determine which options are generally more productive?

One example might be above, the inequality [math] x^2 + x + 1 > 2 [/math], and it's partial solution, here: .

My first reaction upon trying this problem was to simply manipulate it into something like this:
>[math] x^2 + x > 1 [/math]
>[math] x(x+ 1) > 1, x \neq -1 [/math]
And from here, it seems I've manipulated into a much more troublesome problem than what it was when I started. Often it seems the slickest solution is the path down a binary-tree-like structure, and I often find myself lost amid the branches, unsure which node is most promising. Do these kinds of issues lessen with experience? Clearly, a more experienced mathematician would instantly recognize that the above was a square looking to be completed, so it that the basis of my inadequacies? Of course, practice always helps, but I'd still be any related material or heuristics for improving my problem solving of this nature,

>the slickest solution is the path down a binary-tree-like structure
I meant to say it's a select set of the possible paths down said structure, hopefully the analogy and intended meaning was clear.

I think analytic is the best place to start. Algebraic number theory tends to use rather heavy machinery to do anything interesting, and elementary number theory is either stupid toy problems or special cases of respectable theorems done in ad hoc ways.
Analytic number theory lets you get to pretty cool stuff (like the PNT or Dirichlet's theorem) pretty quickly with only some undergrad tools.
Just don't go too deep or you'll become an unsalvageable epsilon-muncher

>Analytic number theory lets you get to pretty cool stuff (like the PNT or Dirichlet's theorem) pretty quickly with only some undergrad tools.
So in other words one shouldn't waste any time on it?

Thanks, I was leaning towards analytic anyway since Apostol wrote a book on it. Do I need to visit elementary at all if I've already worked through a couple discrete math/combinatorics/proof text that covered (what I assume to be) similar problems? Or will I understand elementary as I work through analytic/algebraic? And need I visit algebraic after analyitc?

>[math] x^2 - 2x + 2 > 0 [/math]
My solution
>[math] (x-1)^2 > -1} [/math]
Because the exponent is odd, the left hand side of the equation will always be positive, and so any value in the set of reals will work for x.

But the solution here, etoix.wordpress.com/category/calculus-by-spivak/page/2/ , has a different answer. How did they arrive at that? Pic related. Ultimately we both had the same final answers, all reals, but I'm curious as to how I may have arrived at the same answer differently. My method for arriving at this answer was a straightforward completion of the square. It looks as though this person used a similar method, but I don't understand how they could've done anything differently.

They essentially did the same thing as you but took the square root of both sides.

You should know your elementary number theory (congruences, bezout, chinese lemma, prime factorization, valuations, etc.) since it is the only one that will prove useful whatever math you end up doing. Moreover, it is obviously a prerequisite for any other type of number theory and it's really easy to learn.
And, even if you end up doing analytic stuff, you might need to know some of the algebraic/geometric theory in some areas (eg automorphic forms)

No one cares.

Note that the function f(x) = x^2 - 2x + 2 is continuous and f(0) = 2. If, at any point c, f(c)

Jesus Christ what are all these autistic solutions? Just do [math] x^2 - 2x + 2 = (x-1)^2 + 1 [/math] which is obviously always positive.

Is this proof right?

Theorem: The set [math]V[/math] of n-tuples of orthonormal vectors of [math]\mathbb{R}^{n+k}[/math] is a closed subspace in [math]\underbrace{S^{n+k-1}\times...\times S^{n+k-1}}_{n \;\text{times}}[/math].

Proof: The functions[eqn] \Psi_{ij}:S^{n+k-1}\times...\times S^{n+k-1}\to \mathbb{R}[/eqn] defined by [math]\Psi_{ij}(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)= x_i\cdot x_j[/math] are continuous, for [math]i,j\in\{1,...,n\}[/math] and [math]i\neq j[/math], with [math]x_i\cdot x_j[/math] the standard inner product. Hence, the set[math] \bigcup_{i}\bigcap_{j}\Psi_{ij}^{-1}(0)[/math] is closed and it is precisely the set [math]V[/math]. Q.E.D.


The main worry I have is that the set defined from [math]\Psi_{ij}[/math] isn't actually [math]V[/math].

Can you expand on this? I never took the square root. Here's my step by step process:
>[math] x^2 -2x +2 > 0 [/math]
Complete the square, of the form [math] ax^2 + bx + c [/math]
>[math] [x^2 -2x + 1] - 1 + 2 > 0 [/math]
Factor and rearrange
>[math] (x-1)(x-1) > -1 [/math]
And thus
>[math] (x-1)^2 > -1 [/math]
And it is clear that this will always be positive, for all reals.

Thanks. However I was really trying to prove it by only working within the axioms given.

I didn't want someone to waste their time explaining it, faggot. Doing otherwise would discourteous and selfish.

That is the solution I came to, as stated in my original post. I'm asking how the other person arrived at their answer.

>faggot
Why the homophobia?

They used the quadratic equation, which is the same as completing the square and then taking the root. They proved the original statement by using the fact that [math]f(x) = x^2 - 2x + 2[/math] is concave up and that [math]f(x) = 0[/math] has no real solutions therefore [math]f(x) > 0[/math]. Basically taking an analytic approach over an algebraic approach.

why the faggotry?

How would you convince someone of the joys of mathematics? This particular person comes from a standard, run of the mill USA background: little exposure to trig, no exposure to any maths other than that of a tedious computational nature (and the accompanying lack of rigor), and, given the circumstances, this person carries an appropriate disdainful attitude towards the subject.

I was thinking Stillwell's "Elements of Mathematics" would be a good fit, and maybe "What is Mathematics?" by Courant. I don't want sources that 'teach' or 'drill' necessarily - really I want the opposite. Things that just might initiate an interest. Any videos, resources, etc would all be welcome.

>pic semi-related

To clarify, the quadratic equation was used to come up with [math]f(x) = 0[/math] having no real solutions.

It all makes sense now, thank you.

just define [math]\Psi \colon V \to \text{Mat}_{n+k}(\mathbb{R})[/math] sending each [math](n+k)[/math]-tuple to the Gram matrix. then [math]V = \Psi^{-1}(I)[/math].

But is what i did right or not?

I don't know, you tell me

Give him a #1 Olympiad problem, an hour, and promise a blowjob as a reward if he gets it right.

He's too far away for a bj in a reasonable amount of time. Plus he's not gay, and I don't think he's into incest.

Does Apostol's Analytic Number Theory cover elementary as well?

He sounds like a fag.

Would you recommend this then?

Not the guy you are talking to but Apostol is my bible. Apostol is a weird beast. In his first chapter, he presents the basics of divisibility theory (divisors, gcd, factorization, etc.) and just that. This is technically all you need for the next 3 chapters, which are 100% calculus. Then on the 5th chapter, he goes back to elementary number theory, this time for congruences. He covers everything you need to know about congruences. After this he goes back to calculus, but he actually sprinkles a ton of really heavy algebra (at least for an undergrad) aswell so be ready for that. Then finally after getting ass-raped by algebra and calculus, again in chapter 9 Papa Apostol goes back to elementary number theory. This time for quadratic reciprocity (quick hacks to solve quadratic congruences). Then he moves on to primitive roots, which is also elementary number theory. This is the last he will touch of elementary number theory, and it is, in my opinion, everything that counts as elementary number theory (except for diophantine equations, but every now and then he covers a little bit of that). Even stuff like primitive roots is slightly pushing it in the direction of algebra, but primitive roots were in my elem. number theory course so it counts.

So Apostol will teach you elementary number theory but I can tell you this much: He covers those topics as if you were a 500 IQ grad-student who needs no explanation for anything, or if you already know everything but just need a quick refresher in preparation for his problems (which count as statutory rape in my state). He mentions in his introduction that his book is indeed for grad students and very advanced undergraduate students. If you are looking to be able to solve his problems, you will end up severely depressed (unless you have 500 IQ) as you will barely be able to solve his first problem, and very likely be considering suicide by his #10.

You sold me. I'm not that user but I am downloading the book right now.

That's ok. If you are an advanced student then definitely go for it, it is a really fun ride. I should mention that while some of his problems are standard olympiad-level stuff, some others are full of research problems. In chapter 2 there is a problem that is literally a result that was published back in like the 1900's. And other problems are so weird you will probably need math software to crunch some numbers in order to find out what the deal is with the problem.

After chapter 2 I personally had to re-read the various chapters multiple times before I could actually understand Apostol enough to tackle his problems. If you are going to push through I recommend having advanced knowledge of:

>Inequalities (at the analysis level, not petty high school shit)
>Integration
>Linear Algebra
>Algebra
>Combinatorics
>Creativity

For when you give up, there is a blog online with all the solutions to all of his problems. Have fun.

not that guy but
>solutions to all the problems exist
thanks i can just read the book and pretend like i wouldve gotten the answers by myself eventually after looking at the solutions

thanks now i have 500iq

Reading a solution only gives +0.001 IQ. Coming up with a solution gives +0.1 IQ.

this is top quality sci right here

thanks for the recommendation my dude

thanks user, 'preciate it. have you read any text in algebraic number theory that you might endorse?

No, I haven't touched anything in that direction except for the quick rundown Apostol gives in his book.

>where the fuck do i start with number theory? elementary, analytic or algebraic? getting sick of the attacks on /mg/
Ireland and Rosen

Theres a lot of arithmetic, algebra and geometry books. Can you recommend any computation, combinatorics and logic books?

>/mg/ - Book Recomendations & /sqt/

More like
>/mg/ - /b/ but for math

Classical then whichever camp you like more (protip analytical is the bestest)

Veeky Forums-science.wikia.com/wiki/Mathematics#Primers_in_Combinatorics_and_Graph_Theory
Veeky Forums-science.wikia.com/wiki/Mathematics#Proofs_and_Mathematical_Reasoning
Veeky Forums-science.wikia.com/wiki/Mathematics#Introductory_Logic
Veeky Forums-science.wikia.com/wiki/Computer_Science_and_Engineering#Automata.2C_Computability_Theory.2C_and_Complexity_Theory

cute marisa

don't do that

Ok. I'm probably going with Ireland and Rosen then Apostal, based off what I've read here and elsewhere. Thanks, I'll come back for some algebraic down the road

>Ok. I'm probably going with Ireland and Rosen
Make sure you get the most recent edition, it's been heavily updated

I have Kaczynski's dissertation. I am not going into its substance as such, but I am studying its layout. Pic related is a diagonalization argument figure presented on page 27.

The body of the text is 75 pages with constant lemma/theorem proving and occasional comments. It's straight, terse business throughout, basically.

25 lemmas and 10 theorems are proven, so the business averages a result proven every two pages or so.

Why did he show the diagonalization argument? Are these his own personal proofs or is he just repeating other people's shit?

Am I doomed because I cant get Lang's Algebra?
Im still babby tier starting 2nd semester now and I gotta do a seminar about the first like 30 pages.
I understood it pretty well up to factor groups but then he started listing theorems with literally nothing more than bare bone proof sketches.
He doesnt write functions in his graphs, he doesnt write down Isomorphisms, how the fuck am I supposed to understand this.
Now he introduced subset towers and Im pretty much lost.
I cant develop any intuition for how factor groups and normal subgroups behave. I just dont know what's "allowed" and not with the notation.

Lang is a meme. Try a different algebra book.

In the context of the dissertaion, Kaczynski uses a form of diagonalization argument in the service of a particular lemma, which has to do with complex analysis, topology, and geometry. That much is perfectly clear. Admittedly, I haven't checked details because those are over my head, but I have enough education to get the general thrust. Your statement, "THE diagonalization argument" suggests to me that you think that that K is just aping Cantor en.wikipedia.org/wiki/Cantor's_diagonal_argument , when I instead simply wanted to post a fun pic from the dissertation for some color.

The thing itself also contains several results which are really his, as stated, and also contains several things which are slight modifications or else fully-cited repetitions of previous results. "Repeating other people's shit", when done properly and with the right context, is also known as "academic research", but you don't know that.

His main work was in complex analysis. You can trivially think of a bijection from R^2 to C (a+bi is equivalent to (a,b)). I suppose the diagonalization argument was to then form a bijection from R to R^2 (you can akshually algebraically prove a bijection from R to any R^n, but this diagonalization argument gives us a neat visualization for the specific case of R -> R^2).

what is that supposed to mean

>what is that supposed to mean
It means you should read something else.

That feeling comes and goes, just keep going.

It was wrong, I forgot that I needed a union over all the possible [math]x[/math] in my space and that wouldnt have been necessarily closed. Thanks bud

If you've never done any algebra it would be stupid to learn from Lang. Try Dummit Foote

ok thanks for the advise.
I just got the book because it was listed as reference material.

From Spivak's Callculus 1.1.5:
>[math] (x^2 + X^{n-1}y +...+x^2 y^{n-2} + y^{n-1}x) - (yx^{n-1} - y^2x^{n-2} -...- xy^{n-1} - y^n) [/math]

Which readily simplifies to:
>[math] x^n + x^2y^{n-2} - x^{n-2}y^2 + y^n [/math]

But the answer is supposed to be [math] x^n + y^n [/math], and I'm failing to see how to simplify further. Is it something to do with the expansion in the ellipses? I'm having the same problem this guy is: math.stackexchange.com/questions/580397/spivak-calculus-chapter-1-problem-1v


It seems the top answer by 'oks' simply continued the expansion for one more step, is this the solution?

The original problem is detailed in the link provided, please check that so I don't have to type tex for 15 minutes. It should be very easy to tell, I'm just too skeptical to assume that and move on without confirmation.

My original attempt was done identically to oks', except I didn't add [math] x^{n-3}y^2 [/math] and such because as you can see in the OP it wasn't there. Is the only honest solution to go one more step down the expansion like oks did?

>Which readily simplifies to:
>>xn+x2yn−2−xn−2y2+yn
That's not true.

>In the OP it wasn't there.
But it is.

It was in the expansion, is that what you're saying? And I was supposed to poke around in there?

I mean there's obviously a symmetry there between the terms and cancellations that I understand would reduce to x^n - y^n, but how do 'prove' that that? What if the problem wasn't set up so that the next term didn't fix everything? It seems the top answer on the link just added one more explicit term and then everything canceled smoothly, but perhaps in a future situation it may not conveniently be the next term in line that evens everything out.

No summation notation allowed because that hasn't been introduced yet. Excuse my blabbing, just trying to make sure there aren't mistakes in my thoughts because I feel there are.

What is mathematical maturity, and how does one gain it?

Reminder that string theory is not physics.

Basically, do that opposite of this guy

>studying
Been out of school for about 4 years.
Trying to grind out math from basic algebra all way back to calc.
So far I'm up to college algebra. Shit is going to take me some time.

it's a measure of the ease of which you can work through, digest and retain new and/or difficult text.

you have to work through, digest and retain a lot of new and rigorous text

Nice job user, It's worth it, keep it up.

also, see:
>terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

basically it's a measure of the length and intensity that your nose has been at the mathematical grindstone

Do we have a definition of n-category that allows us to compute coherence laws for any n?

Thanks, so I just gain mathematical maturity by just doing it.

basically, but you can't be doing it mindlessly and expecting solid results. while drilling is great, you should also emphasize looking for patterns and how the pieces connect and you should always be working toward gaining deeper insight:

>calnewport.com/blog/2008/11/14/how-to-ace-calculus-the-art-of-doing-well-in-technical-courses/

and I'd definitely try to read text that demand a more mathematically mature reader. Compare Spivak to Stewart in their expositions on Calculus and you'll see what I mean.

It's still an open problem for n = 1.

...why

It's hard. Cubical TT is getting there but still unable to fully generalize to HoTT - which is only for n = 1 from my understanding. Unless I'm not getting what you mean by "compute coherence laws".

I also don't really know what I'm talking about, I'm just a filthy type theorist and category theory is mostly incomprehensible for me. So take what I say with a lot of salt.

ok...I thought you were trolling lol. There are no coherence laws for 1-categories. I'm talking about things like the pentagon identity. It's extremely hard to work out what they are supposed to be in higher dimensions, but I thought HoTT or something might help with that. Unless I'm mistaken we only know up to n = 3 or 4.

ncatlab.org/nlab/show/coherence law

>Geothendieck
So sick of this meme. Etale orbifolds can be topologized without reference to topos.

My bad, I was thinking of [math](\infty, n)[/math]-categories. I'm definitely not clear on exactly the differences between all the different higher categories. HoTT handles higher dimensions as just higher homotopies, which are all equivalences. Or in cubical TT, hypercubes. I've seen the pentagon law done for n = 2, and I imagine one could theoretically pick any n and do it for that. But I'm not sure how you would go about doing it in one shot for any n, if it's possible in HoTT at all.

>What have you been studying, /mg/?

Lately, synthetic geometry and the physics of spacetime

>Prove that the sequence 10^-n converges to 0
>Prove that sqrt(1+(sqrt(1+... = golden ratio
>Prove that the bounded sequence [math]x_n[/math] such that [math](2 - x_n)(x_n+1) = 1[/math] converges to 1
>Prove that the sequence a_n = (1/n^2)([x] + [2x] + ... [nx]) converges to x/2

What
the
fuck

I'm going to start a math major in like a week and these are some of the problems the last weeks of intro to calculus. I'm really good at self-studying, I've studied a lot of undergraduate stuff these months and I've managed to master a lot of exercises from books (Spivak, Herstein, Enderton)

But I feel like a fucking brainlet whenever I check sequence related stuff. I can do convergence tests and proofs related to infinite series but I can't prove a fucking sequence convergence. Heck, I'm even able to prove some functions' limits through epsilon-delta

So, any help? What's a good book that'll help me prove sequence theorems/exercises? Rudin is too topologic, Spivak is too simple, Stewart doesn't even feature proofs.

[spoiler] also... am I panicking too soon for not being able to do these problems? [/spoiler]

the third one should have a [math]x_{n+1}[/math] as the second factor

>am I panicking too soon for not being able to do these problems?

yes. that's what your math major is for.

>Doing my assignment in the library today
>Guy next to me stares at my problem sheet for a solid minute then looks at me like some kind of freak

Does a math career guarantee no social life?

I've read (cover to cover) texts in precalc, calc, discrete maths/combinatorics, linalg and analysis yet I still get stumped ju stupid shit like quadratic equations, factoring, trig, algebra and a long list of similar issues. How do I repair these deficiencies?

stop humble bragging faggot

>Guy
Did you fuck?

do the exercises