/sci/ - Science & Math

Yes

PSA: /mg/ has moved to >>>/r9k/48322652

bump

>>10026916
obviously only brainletlets would take the most simple of math classes, and would obviously complain

the more time that passes the more I realize Calc II was the hardest class I ever took

>>10027857
Nice bait, even reddit spaced. 8/10

>>10026916
That's what you get for not teaching them math in the first grade.

Post news on the pic related

>>10026712
No

https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view

>>10028118
>5 pages
Atiyah has really gone senile huh

>>10028148
Well, he did say it was a simple proof.

>>10028148
kek, that's shorter than some definitions in IUT III

How long does it usually take for referees to confirm whether the proof is valid or not?

>>10028297
There are cases where it took decades.

>>10028092
Positive

>>10028118
LOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOL

>>10028372
I can't even say he's wrong, but I feel like he's overlooking something that will just fuck him in the ass

The squeeze theorem works for infinite limits too, right?
My course material specifies that lim f = lim h = L has to be a real number but it doesn't make sense to me.

>>10028522
Not sure I understand you. If you have two things going off to infinity, how possibly could anything be "squeezed" between them. And if that were the case, then it would just be infinity itself, no?

>>10028531
What I mean is, if f(x) <= g(x) <= h(x) and both f and g's limits are infinite as x approaches a bound of the interval they're defined on, wouldn't g's limit be infinite as well (thus making the theorem applicable)?
I don't understand how g could have a real limit in this case.

>>10028537
if you can prove g is bounded below by f and above by h on the interval, sure.

>>10028547
Yeah I was assuming f <= g <= h. Thanks.

how long does it usually take you guys to self study a new book?

Zero is not a number.

>>10026916
Does this count for european universities too or just burgermania?

https://old.reddit.com/r/math/comments/9igre1/ask_reddit_what_is_something_you_wish_you_knew/e6jn8e0/

Burger here, is this guy over on lebbit wrong or should I just kill myself now? I feel seriously depressed after reading that, I thought I was in an OK place with some research experience and a good honors thesis I plan on splitting a publication off, but apparently you're fucked if you haven't been doing cohomology since high school.

Maybe a wage-slave software engineer is all I'm destined to be. Fucking end me.

>>10028598
Post the comment on pastebin or something, not everyone has a reddit account

>>10028611
You don't need an account to read reddit (I don't have one myself), but here's the parent comment for reference

>>10028620
He's full of shit and trying to sound smart.
Just apply for the PhD programs relevant to your field of interest, you don't have to do all that shit to get in. The hard part will be postdoc anyway.

>>10028568
What zero is or isn't is none of your business

>>10028628
>>10028598
Generally speaking, academia is not really a good field to pursue a career. Academia is supposed to be for the very top of the university students. So if you aren't confident that you are among the best 0,1-0,5% of your classes, don't pursue it. Luckily for you, there are a lot of interesting fields for math graduates that pay well, are interesting, and are also contributing to society in meaningful ways. So in all honesty, if you don't think you are a really special talent and are not willing to victimize your life for academia, don't do it.

>>10028639
>there are a lot of interesting fields for math graduates that pay well, are interesting, and are also contributing to society in meaningful ways
Really?
Because outside of statistics/probability applied to finance, actuarial science and ML, I don't see it. And those three (aside from the latter, arguably) don't even contribute to society much if at all.
It seems that every well-paid applied mathematician I've seen in the private sector is doing R programming and statistical modelling for a bank or tech company.

>>10028639
My problem isn't a lack of confidence in my ability it's that I switched into a math major my second year and haven't done most of the shit that guy is saying is essential. I guess I could delay graduating but I can only afford the uni I'm at now because they gave me a fuck load of financial aid but that only lasts for four years.

>>10028648
Well, what do you think you would be doing in academia most of the time? You teach students and correct their stuff, do basic work for papers, etc. Compared to that, striking a job at a tech or finance company where you are going to be a leading/creative part of a project can be really attractive.

>>10028632
You're a zero, I'm a hero.

>>10028660
>a job at a tech or finance company where you are going to be a leading/creative part of a project
You just need a PhD in (any area of) mathematics and this becomes a tangible opportunity?
I thought those positions were only filled with candidates from top 15 universities.

>>10028598
>Burger here, is this guy over on lebbit wrong or should I just kill myself now? I feel seriously depressed after reading that, I thought I was in an OK place with some research experience and a good honors thesis I plan on splitting a publication off, but apparently you're fucked if you haven't been doing cohomology since high school.
He's mostly full of shit. Look, it's not like the top schools in the nation aren't difficult to get into, but not having grad courses won't kill you. Having a strong showing in your undergraduate coursework, strong letters of rec, and solid research experience plus a thesis will do wonders for you. At the very least you'll have a good shot at top 100 unis. Now I wouldn't say don't take grad courses, usually graduate topology and algebra are pretty easy going, just munkres/hatcher/lee and D&F, nothing terribly rough and there are plenty of online resources to help. Just focus on doing good work and don't get hung up on other people's accomplishments.

>>10027903
this is a good picture. i will do this with anyone i get close to, thanks my nigger.

>>10028598
He's just sucking his own cock. The massive overemphasis on coursework makes me think he probably didn't have much else going for him.
If you have a decent GPA, some legitimate research experience and good letters you're going to get in somewhere decent. Probably quite a few decent places if you cast a wide net.

Even among people who get into top 5 schools like Princeton, there's no causation at all from coursework, just correlation because they're all very smart and like math a lot.
The things that get people into places like Princeton or Harvard are exceptionally good research and letters from exceptionally respected mathematicians. Not a fucking functional analysis class.

>>10028620
>scroll down further
>find this post
this is the ding-dong you're considering taking advice from.
Moral of the story; stay off reddit. Even better, don't take life-altering advice from the Internet at all.

what are some interesting topics in algebraic geometry and algebraic number theory that are worthwhile exploring for a phd?

>>10029118
>let me scoop you
How about no?

>>10029118
Interuniversal Teichmuller Theory

>>10027903
god i wish that were me

>>10026711
hey guys, how can i prove this? i know there is a way to solve every limit with this form, but i don't remember it.

>>10029214
try proving that the limit of the logarithm is zero instead

>>10029261
thanks

>>10029118
Perfectoid spaces

>>10029118
local-global principles (Brauer-Manin obstructions)
mirror symmetry
perverse sheaves and representation theory
tropical geometry and F1 geometry
p-adic Hodge theory
automorphic forms and Galois representations

You know it's national help a brainlet day? Why don't you pitch in!
>>10029357
>>10029363

>>10029364
yes it works

>forgot most things ive learned in high school
>thinking about stuying math next year
Is this a good idea? I have some time to catch up on everything. How do I know if Im talented enough for a math degree?

>>10030191
By trying

>>10030191
I'm in a similar situation, anon, let's just do it. We only live once.

>>10030244
True, well even if it turns out that I have no future in pure math it can't hurt to refresh and broaden my knowledge in math.

>>10030512
Good luck, hope you'll achieve your goals.

>>10030653
We're gonna make it

Friendly reminder that >>10030668 is wishful thinking.

>>10030191
>>10030512
Have you done any serious proof-based work? If you haven't you don't know what you're getting yourself into desu.

>>10030813
No lol. I just don't know what I'll study next year since I haven't developed a passion yet. I'll see if I like university math once I refreshed my highschool knowledge and start doing uni stuff

>>10030866
If you university has a "intro to proofs" or even just a basic discrete math class you should take that ASAP.

>>10030875
I'm not enrolled in any university yet so I don't think I can check out any courses.

>>10030894
University course catalogs are usually public, as long as you have an idea where you'll be going you can check

>>10030702
Suck a donkey's dick. Die choking on it. We're all going to make it.

>>10030875
I never understood the advice that people should jump into proofs to decide if they like math or not.
Deciding whether you like math based on your intro to proofs course is like deciding whether you like chemistry based on your lab safety orientation.

Sorry to shit up the thread lads, but can someone recommend a decent Calc 3 textbook that is structured like Stewart, but offers more rigor? I've read the wiki and looked through a few. I really like the unified approach from Hubbard, but it is too far off from what I will be tested on.

>>10030894
Are you going to place into upper level courses? If not you're going to have to take a boat load of prerequisite any ways. I think by the time you finish the calculus sequence you're bound to know if its for you or not.

>>10026711
Una vela

>>10031042
I'm not sure what you mean by "rigor". Classes based on Stewart almost exclusively test your ability to solve problems, so any rigorous proofs at all are going to be totally unrelated to what you'll be tested on.

If what you're asking for is a supplementary textbook because Stewart sucks you could take a look at Div Grad Curl, which was more or less written to be supplementary reading for vector calc courses.

>>10031040
Yeah, and decided to major in math based on the calculus track is like deciding if you like chemistry because you liked making baking soda volcanos in elementary school. It's all relative.

>>10031061
It's not "all relative at all". The point that you should try to get into an actual math course and see if you like it is sound, but do aim for an actual math course. Any type of discrete math or graph theory or whatever is a fine choice, so is something like number theory.
Personally I think if somebody's university has a good math department the best thing they could do is get into the math major section for linear algebra.
This will let them decide if they like reasoning about abstract structures, and if they don't, they may as well leave math before they waste any more time on it.

>>10031048
I dont know what youre talking about. Im located in europe, we have a set schedule in the first semesters iirc.

Let me see your cutest group and tell me what she smells like.

>>10031157
SL(n) best group

>>10031052
Apostol is the crux of calculus with good rigour. Next step is analysis

>>10031157
Tits group

like an old book

>>10026916
"Many students come into College Algebra unprepared. With no ability to work with fractions or distinguish between multiplication and addition in algebraic notation, there's little chance they can pass"

>college
>doesn't know difference between + and *

what the actual fuck?

>>10028491
He didn't use a single property of the Riemann zeta function (besides it being analytic). If this argument applied, it would show any non-zero analytic function would have no zeros outside the critical line.

>>10030894
check out Book of Proof, helped me when I started

What is the best way to learn tensor notation w/ Einstein summation convention?

I’ve tried doing simple problems and proofs to see if I can just work in the notation in my thought process, but it’s ultimately confusing as fuck when you’re working with two terms that are really meant to denote 27 other terms.

I just have no intuition in working with them.

>>10031722
I can't help you man, I gave up on self-teaching general relativity for the same reason. Fucking physishits and their obsession with using the most autistic vague notation available to them.

How do I get good at dealing with Dedekind cuts? I'm a freshman and we're learning about them in my analysis course. It's a very interesting concept, and I feel that I have a good understanding of them, but showing that two real numbers are equivalent is quite a difficult thing to do.

For pic related, how should I go about showing that [math]\alpha_i + \alpha_j = \alpha_k[/math] where k = i + j? Showing that [math]\alpha_i + \alpha_j \subseteq \alpha_k[/math] is easy, but showing that [math]\alpha_k \subseteq \alpha_i + \alpha_j[/math] is more difficult to do. I tried showing that any element in [math]\alpha_k[/math] an be written as a sum of an element in [math]\alpha_i[/math] and an element in [math]\alpha_j[/math], but I didn't get anywhere. Wat do?

>>10031824
>For pic related
Forget this. Was gonna post my work (but obviously didn't)

>>10031409
I think he's saying they don't know that a(x + 1) is a * (x + 1), not that they don't know the difference between + and *

Can someone tell me how you find the 4th root of a number like 625?
I run in to a lot of trouble understanding how to do this easily without a calculator or memorizing the awnsers

>>10031824
>How do I get good at dealing with Dedekind cuts?
Huh?

>>10031826
>still doesn't post his work

lmao.

No one help this brainlet.

most people who write technical documents have a colleague that can proof read for the obvious stuff
>must be nice
Anyone ever consulted the errata of a text book? Here's Sean's:
https://www.preposterousuniverse.com/spacetimeandgeometry/
>must have been stoned

>>10031824
Do some actual math already, literally nobody gives a shit about Dedekind cuts.

>>10032108
You should know thats 5 almost instantly, just by looking at it

for a more general method, try thinking "in base what is this 10000

>>10031722
perhaps you should learn what tensors are beforehand, at least geometrically, before you start thinking about the rules with them.

It's not that hard either, one up one down means sum over them, and the metric can be used to contract indices

>>10030702
i love this 2hu anon

>doing math exercises
>it's difficult
>get stuck, have to look up solution
or
>doing math exercises
>it's difficult but I'm getting through it
>finish, feel proud
>check solution to make sure
>everything I wrote is wrong
Why am I such a fucking retard? I want to git gud

Just graduated with a degree in math, I covered:

- Analysis I&II
- ODEs and PDEs
- Convex Optimization including Calculus of Variations
- Markov Chains and Stochastic Processes
- Probability and Statistics
- Linear Algebra I&II
- Financial Mathematics up to Ito Calculus
- Machine Learning and Data Mining
- Numerical Linear Algebra and other Statistical Computation

I feel pretty good about my knowledge base, but I'm wondering if there is anything particularly useful that I've missed? Anyone have any insight? Note, I didn't include classes I took involving applications of mathematics like econometrics, game theory, etc. Just looking specifically for math courses I might want to take next.

>>10032637
Where the fuck is Abstract Algebra I&II, Point-set Topology, and Complex Analysis? Your degree isn't worth the paper its printed on.

>>10032649

>> Missing 3 topics, all of which are very basic pure math extensions of existing courses
>> In what is clearly an applied math degree
>> useless degree

I'm asking for further math topics I can actually apply to the private sector, not pure math autism, thank you :)

>>10031409
Don't be racist, you Nazi.

http://newsupdatessa.site/education-is-not-for-africans-it-came-with-the-whites-reveals-minister-of-basic-education/

>>10032654
>complex analysis has no applications

What's your name so I can tell HR to throw out your resume on sight.

>>10028372
Actual autism

So what the heck is a Todd function?

>>10029118
Idk if it's actually a popular research direction, but I think the primes <---> knots thing is cool.

So if K is a number field, with residue field F at a prime p.

Then we view the maps SpecF ---> SpecO_K in analogy with Knot embeddings in 3-manifolds.

The reason we think about SpecO_K as a 3-manifold, is because from the perspective of etale cohomology it behaves like a 3-manifold behaves in singular cohomology.

>>10032587
>find math interesting
>am bad at it

>>10032587
>>10032880
Well, did you try learning from the solutions? A lot of times when I encounter a hard problem I either try solving an easier version of it, try to find the relevant theorems given the assumptions at hand, try breaking it into parts, or use previously successful techniques to attack the problem. When you looked at the solutions, did you actually digest and understand why that solution work and how you may have gotten to that solution? Frequently one can just mess around with a problem, moving around parts or making guesses and actually hit upon the right path. Don't underestimate the power of just trying out shit and seeing what happens. Even if you can't solve hard problems initially the important thing is to learn so that you improve. Next time you get really stuck on a problem and you have to look up the solution, try to recreate the logic of the solution step by step to understand the thought process of how to solve the problem. Think about how and why someone used that technique, or split the problem into those parts, or etc.
>>10032863
Not him but this sounds cool, are there any introductory sources for this sort of thing?

>>10032889
>Not him but this sounds cool, are there any introductory sources for this sort of thing?

http://hyoka.ofc.kyushu-u.ac.jp/search/details/K002688/english.html

He has written various notes and a book on the topic.

>>10032889
I understand solutions and can break them down perfectly. I fully comprehend HOW the person managed to get that result, but I can never replicate that way of thinking or "initiative" myself, if that makes sense. It's like I have absolutely no imagination when it comes to mathematics.

>>10032915
>but I can never replicate that way of thinking or "initiative" myself, if that makes sense. It's like I have absolutely no imagination when it comes to mathematics.
I used to have similar problems when it came to combinatorics. Is this specific to one field or to all of mathematics? It's not uncommon for people to have weak suits when it comes to certain fields so maybe you just need to find the one you're good at. Also when you say you know how but not why what do you exactly mean? Do you mean you understand each individual step but not why someone would use the technique or method in that particular way? If you have an example I could try working through it myself if that helps. That's something else that may work you, just getting help from a classmate or your prof.
>>10032893
Thanks

>>10032201
It's for a course, dipshit

>>10032982
It's not specific to one field, I get that problem all the time. The most recent example was a problem with a sequence where I had to prove it was monotonically increasing, then prove that the sequence was always superior to its rank n. I more or less knew what I had to do, the first step required me to compare the sequence at rank n with rank (n+1) which I did by dividing them, and the second was a proof by induction.
Even though I was aware of which methods to use, I still ended up stuck for both steps, either because I'm bad at calculating and fail to notice obvious patterns (thus ending up switching around factors and terms aimlessly and going nowhere), or because I never know how to actually apply the method correctly (my proof by induction was a clusterfuck).
I admittedly could know my theorems and basic formulas a little better, but even when relying on cheat sheets for all that stuff I have a hard time determining WHAT exactly I have to do given the information I have in order to arrive at the required result.
>Do you mean you understand each individual step but not why someone would use the technique or method in that particular way
I understand both of these things, but I lack the "initiative" to use those techniques myself, especially when the problem requires a little bit of creativity.
And when I have to do something that requires a very straightforward method (like calculating a derivative to study monotony), I manage to fuck up on the calculations.

Sorry for the longpost.

>>10033026
You might want to go over your fundamentals again then. Instead of doing harder problems try cleanly solving a lot of easier ones until you stop making those minor mistakes. Also carefully go over what you just wrote down before going to the next step to avoid making minor errors that mess things up later on. I can't say this will solve all your problems but I did find that in my experience teaching calculus that most people had problems not because the material was hard but because they lacked experience and key skills were simply unpolished rather than them being incapable.

>>10033057
Yeah, I'm going to go over the basics once again. I've learned to go slower and try to make sure I'm not writing down garbage before moving on to further steps. What I really need to get better at is problem solving: taking an expression, equation or whatever and using the right methods and operations to arrive at a clean result instead of complicating things and leading to gibberish like I always do.
I remember back in high school I'd write down entire pages of calculations and theorem applications only to arrive at a dead end, and then I got presented with a clean and elegant solution that seemed to obvious in retrospect.

>>10033072
>I remember back in high school I'd write down entire pages of calculations and theorem applications only to arrive at a dead end, and then I got presented with a clean and elegant solution that seemed to obvious in retrospect.
I know this feeling painfully well. I can recommend some books that helped me with problem solving if you like.

>>10033117
Sure, I'd appreciate that. How long did it take for you to get better?

How much time per day do you devote to independent study and how do you usually manage your studying sessions?
Natural math prodigies need not respond.

>>10026711
hey I have a stupid question but theres no sqt up. so here it goes. (pls help bros).
I dont know how to answer this question that appears on a test model for my upcoming calc 3 test.

if [math] \phi [/math] is a differentiable function, prove that for the function [math] f [/math], given by:
[eqn] f(x,y) = x^3 \phi(x^2 - y) [/eqn]
it holds that:
[eqn] x\frac{\partial f(x,y)}{\partial x} +2x^2 \frac{\partial f(x,y)}{\partial} = 3f(x,y) [/eqn]

truly apologize for my retarded question. wolfram wont help.

>>10033247
fuck, missed a y

[eqn] x\frac{\partial f(x,y)}{\partial x} + 2x^2 \frac{\partial f(x,y)}{\partial y} = 3f(x,y) [/eqn}

So, is Mochizooka basically done?

>>10033120
I used polya's how to solve it plus his other two volume set, engel's problem solving strategies, and larsen's problem solving through problems. I wasn't terribly good at math back in high school since I most half assed homework assignments. It was only until I took calculus bc my senior year of high school that I realized I really needed to hunker down and get better. I almost failed the class but ended up getting a 5 on the bc exam. It took another year but I finally got into the groove of things and was able to start taking upper level math courses and even grad courses during my undergrad days. So about two years give or take, but that was coming from a place of being pretty bad at mathematics overall. It was time consuming at times though. When I wanted to learn basic real analysis I actually went through all of baby rudin by attempting to prove every theorem before looking at the proof and solving every exercise. Suffice to say it took a while but after that analysis has been a breeze, at least so far. Still in the process of learning, right from Stein's harmonic analysis books.

>>10033247
Just compute the derivatives
[math]f(x,y) = x^3 \phi(x^2 - y)[/math]
[math]x\frac{\partial f(x,y)}{\partial x} +2x^2 \frac{\partial f(x,y)}{\partial y} = 3f(x,y)[/math]
[math]2x^2 \frac{\partial f(x,y)}{\partial y} = 2x^5 \frac{\partial \phi(x^2 - y)}{\partial y}=-2x^5\phi'(x^2 - y)[/math]
[math]x \frac{\partial f(x,y)}{\partial x} = x \frac{\partial x^3\phi(x^2 - y)}{\partial x}=2x^5\phi'(x^2 - y)+3x^3\phi(x^2 - y)[/math]

>>10033262
I see. Thanks for your advice, it might take some time to get good but it's still encouraging.

>>10033274
Huh it was actually really simple.
Thought i had to use that theorem for differentiating compound functions with rsspect an external variable and got stuck.
Also, my professor recommended the stewart as the book to guide us thru the course but I fucking despise that book.
Are there any better options out there for multivariable calculus??

>>10033320
google.com

>>10033320
Hubbard and Hubbard Calculus, Linear Algebra, and Differential Forms
https://ia601608.us.archive.org/14/items/HubbardJ.H.HubbardB.B.VectorCalculusLinearAlgebraAndDifferentialFormsAUnifiedApp/Hubbard%2C%20J.%20H.%3B%20Hubbard%2C%20B.%20B.%20Vector%20Calculus%20Linear%20Algebra%20And%20Differential%20Forms%20-%20A%20Unified%20Approach%20%28Prentice%20Hall%20698S%29.pdf
Maybe try khan academy or one of the other youtube channels that do something like that.

>>10032637
What is Linear Algebra II? Adventures in tensors? Meanwhile
>I didn't classes I took involving applications of mathematics
>Lists ML+Data, Numerical Lin.Alg, Markov Chains+Stochastic Processes, Convex optimization, Finance math (business calc?)
Complex Anal when? Or is that Analysis II? Even those on an applied math track have to take Complex Anal, Abstract Algebra, Lin.Alg (the real one. Not the one engineers take) and proofing at my uni

>>10033365
Thanks bro, appreciate it.

Serious question. Why can't I learn from books like I can from class? Do I just have shitty books? If I didn't go to my calculus classes, I would've been fucking clueless but everyone else tells me they just read the book

>>10032637
>No geometry or combinatorics

You didn't do any real math

>>10031722
I'd say just find a reasonably readable intro, and reread it (and work excercises) until you've got it. Zee's "Einstein Gravity in a Nutshell" had a fairly good intro imo.

I'm really not a fan of it. Sometimes it can be confusing to tell whether or not a summation symbol is being omitted, I'd prefer if it just didn't omit the symbol in the first place.

>>10033562
No, he just didn't do any useless math.

If [math]d=\gcd(m,n)[/math], how do I show that [math]\gcd(2^{m}-1,2^{n}-1) = 2^{d} -1[/math]?

>>10033612
>>>/toy/

>>10033612
>No, he just didn't do any useless math.
I'm not a "he".

>>10033644
>If [math]d=\gcd(m,n)[/math], how do I show that [math]\gcd(2^{m}-1,2^{n}-1) = 2^{d} -1[/math]?
What have you tried?

>>10033644
Alright, I think I got it. Took longer than I'd like and the solution isn't terribly elegant but whatever.
[math]x^a - y^a = (x-y)(x^{a-1} + x^{a-2} y + ... + x y^{a-2} + y^{a-1})[/math]
Set [math]x=2^d[/math] and [math]y=1[/math]
then
[math]2^{ad} - 1 = (2^d-1)(2^{ad-d} + 2^{ad-2d} + ... + 2^d + 1)[/math]
If we now consider the expansions for
[math]2^{m}-1[/math] and [math]2^{n}-1[/math] it's pretty easy to see that [math]2^d-1[/math] divides both of them, it remains to be shown that this is the greatest common divisor. So how do we do this? Assume wlog that [math]ad=n>m=bd[/math]
Let's assume that [math]2^d-1[/math] and when can subsequently divide [math](2^{ad-d} + 2^{ad-2d} + ... + 2^d + 1)[/math] and
[math](2^{bd-d} + 2^{bd-2d} + ... + 2^d + 1)[/math] by some odd number [math]k[/math]. This number can't be even because neither sum is divisible by [math]2[/math], just saying that unless it wasn't clear.
Since [math]a>b[/math] let's just write this as [math]a=b+j[/math] for positive [math]j[/math].
Then in the expansion [math](2^{ad-d} + 2^{ad-2d} + ... + 2^d + 1)[/math] the sum of the terms [math]2^{ad-jd}[/math] and below are equivalent to the sum
[math](2^{bd-d} + 2^{bd-2d} + ... + 2^d + 1)[/math]. This should be easy to see but just in case observe
[math](2^{ad-jd} + 2^{ad-(j+1)d} + ... + 2^d + 1)=(2^{bd-d} + 2^{bd-2d} + ... + 2^d + 1)[/math]
So if [math]k[/math] divides both [math](2^{bd-d} + 2^{bd-2d} + ... + 2^d + 1)[/math] and
[math](2^{ad-d} + 2^{ad-2d} + ... + 2^d + 1)[/math] then this means that it also divides [math](2^{ad-jd} + 2^{ad-(j+1)d} + ... + 2^d + 1)[/math] and [math](2^{ad-d} + 2^{ad-2d} + ... + 2^{ad-(j-1)d})[/math]
But this is clearly false since [math]k[/math] is odd and [math](2^{ad-d} + 2^{ad-2d} + ... + 2^{ad-(j-1)d})[/math] is even. There's almost certainly a better solution, probably using Fermat's little theorem, but my number theory isn't strong. I think this actually generalizes fairly easily.

>>10034032
>Let's assume that [math]2^d-1[/math] and when
Let's assume that [math]2^d-1[/math] isn't the gcd and one can subsequently

>>10034032
Ah, found another error, set [math]a=b+j+1[/math] and everything should be fine though.

>>10034032
>>10034043
[math]a=b+j-1[/math] fuck me what is happening today, I need to sleep.

>>10034032
>>10034038
>>10034043
>>10034049
I just came back and reread my post because I felt something was off about my argument and realized it's wrong. The last line doesn't work since, duh, 6 is even and can be divided by 3. What I should've said is that since it's composed of power of 2 it shouldn't be divisible by an odd integer which may be true. If you look at
[math](2^{ad-d} + 2^{ad-2d} + ... + 2^{ad-(j-1)d})=2^{ad-(j-1)d}(2^{jd-2d} + 2^{jd-3d} + ... + 1)=2^{bd}(2^{(a-b-1)d} + 2^{(a-b-2)d} + ... + 1)[/math]
We can now try something kind of weird. Since [math]2^{bd}[/math] is a power of two we know it can't be divided by [math]k[/math] so the other part must be divisible by [math]k[/math].
As before we could note if [math]b-1>a-b-1[/math] and redo the arguments before to then claim that [math]k[/math] must divide
[math](2^{bd-d}+2^{bd-2d}+...+2^{d}+1)-(2^{(a-b-1)d} + 2^{(a-b-2)d} + ... + 1)[/math]
Same with if [math]a-b-1>b-1[/math] just reversing the order. It seems like you could keep iterating this procedure until you force the difference to be a power of two and thus not divisible by [math]k[/math] giving rise the contradiction for odd [math]k>1[/math]. I think this works, maybe not.

>>10026733
>/r9k/ neets
>Knowing any math at all
Why would you even think a /mg/ thread would work on a board full of lonely high school dropout brainlet losers? It least when /diy/ tried it they could handle some babby math (albeit only at an engineerlet level).

>>10028620
>>10028598
I did all of that in my undergrad double degree program and I still feel like I'm not good enough for grad school. Seeing that post and then seeing that there are people worse off than me makes me think that maybe I should reach out to some researchers/supervisors who are working on stuff I'm interested in.

>>10034296
We need to proselytize mathematics to them to save them from their terrible lives

>>10034296
Blaise Pascal was an incel

>>10028319
You should'ofe say Maybe

>>10026711
What should I do over the summer /mg/?
>paid work in computer science, probably working on something like optimization, high performance computing, parallelism or garbage collection
>unpaid maths work. Ask a professor for a project in computational algebraic number theory
The maths is obviously more interesting. I probably wouldn't actually get to do anything useful though. Whereas the computer science (code) would end up being used probably.

>>10034510
>unpaid anything
anon what are you doing? Anyway you may be able to get paid maths work, I managed it at least and I'm a brainlet

>>10034551
In undergrad?

>>10034556
yuuup

Is there a writeup anywhere that covers, in broad strokes, the proof strategy for the classification of finite simple groups?

>>10034625
Yes

>>10034639
Could you link it anon?

>>10026711

https://sadguruband.bandcamp.com/releases

really made me think https://thehighergeometer.files.wordpress.com/2018/09/mochizuki_final1.pdf

>>10034312
No way, could you imagine having to collaborate with a retard like that? There's a reason most autistic people never reach grad level math.

>>10034651
I'm glad he typed up those notes. He also has some posts on Woit's blog.

http://www.math.columbia.edu/~woit/wordpress/?p=10560

https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/

This might be relevant to those interested in Atiyah's Riemann Hypothesis argument beyond rebbit and motl shitposts.

hey /mg/.
im an EE undergrad currently taking an AC and our teacher presented us with the following notion:

let [math] f: \mathbb{R} \rightarrow \mathbb{R} [/math] such that [math] f [/math] is a periodic function.
given that [math] f [/math] is periodic, the average value of [math] f [/math] is given by:

[eqn] F = \frac{1}{T} \int_{0}^{T} f^2(t)dt [/eqn]

where: [math] F [/math] is the average value of [math] f [/math] and [math] T [/math] is the period of [math] f [/math].

does this make any sense mathematically?
this is what is used to compute rms values for things like voltage. (the 120v measurement youre familiar as the voltage in any homes power socket is computed exactly like this).

does the notion of average have any meaning in math?

>>10034707
Aperiodical put up a summary of the situation at a more layman level.
https://aperiodical.com/2018/09/atiyah-riemann-hypothesis-proof-final-thoughts/

There's a math overflow thread discussing the Todd function as well.
https://mathoverflow.net/questions/311280/what-is-the-definition-of-the-function-t-used-in-atiyahs-attempted-proof-of-the

I know how to prove that, given a group of order mp^n, with p not dividing m, there is a subgroup of order p^n. That's how the teacher taught us the first Sylow theorem (using induction on Cauchy's I think), and that's how Artin's and DF's present it.

However, Herstein's call this as a corollary (and also the Wikipedia article on Sylow's theorems) to the theorem stated as

"A finite group G whose order |G| is divisible by a prime power p^k has a subgroup of order p^k"

Herstein's proof uses weird combinatorics tricks, but Wikipedia uses the very same ideas that are used in the Artin's and DF's proof of their version of the FST.

Basically my question is, are these people being purposely retarded for not presenting the whole result as it is? The version we were presented to in class is so limited compared to this other result. What the fuck?

>>10026711
Didn't want to make a thread and break the rules

I was wondering is it possible for me to get into a Masters of Science program and study applied math.

I have an undergraduate degree in computer science however my grades were mediocre. (~2.8 GPA) and I didn't cover much math in my course either (discrete math, some linear algebra sans proofs, some calculus 2/3. Is there a chance for me to get accepted into a program to study? I have started studying math in my spare time and am working toward it.

>>10034959
It's definitely a long a shot.

A really good GRE score and maybe applying for more computational programs at mediocre schools could do it.

>>10034959
It depends on the
>country
>university
>supervisor
>research project
>type of masters (course based or thesis based)
If you have a strong interest in a compelling research project and there is a supervisor who wants to work with you then your odds aren't that bad. Conversely, a math undergrad with a perfect GPA may have a hard time getting into a masters if they don't have a passion or interest in a specific research project and don't really have a supervisor in mind.

>>10034977
>>10034984
Thank you guys. I probably won't go into everything an undergrad in applied math might go into. I am stilling working on calculus and then plan to move onto differential equations and going deeper in linear algebra.

>type of masters (course based or thesis based)
As of now its looking like its going to be course based.

>>10034993
>As of now its looking like its going to be course based.
I get the sense that these are easier to get into and aimed at people who just want a masters for a job but not for academia.

>>10034994
That's my impression as well. I want it primarily for my personal edification, and secondly for the employment prospects. I might just do a Masters in CS.

>>10034852
There should be a square root somewhere in the equation.

The concept of an average does have a place in math, but there are many possible definitions.

The mean would be the usual average for discrete cases, it's simply the sum of all of your values divided by the number of values, it's what you would typically think of when thinking of an average.

There are other notions of average though. For an abstract probability distribution, the average is the expected value i.e.

[math] E[x] = \int x P(x) dx [\math] where [math] P(x) [\math] is a probability distribution of a single variable x, and the integral is over all possible values of x (if x is discrete then this will just be a sum). The calculated mean should go to the expected value as you increase your number of observations.

The RMS average used by your prof is useful when the usual mean doesn't capture a lot about the distribution, such as when it is skewed or similar.

>>10035008
Shit, I can't latex on 4chan

>>10034852
That's not the average. If you take the square root of that, then you get the RMS (root-mean-square) value, which is the DC voltage/current which delivers the same power to a resistive load (P = V^2/R = I^2/R).

E.g. "240 V" mains is (roughly) a sine wave with a peak voltage of 340 V, as that delivers the same power as 240V DC.

What is a good book to learn group theory in depth?

>>10035091
Groups and representations

>>10031446
How can you say he didn't use any property of it? We don't even know what the fuck k the Tod function is.

If I am given an epsilon delta limit statement like this:

[math]0 < |x-a| < \delta = min (1, c\epsilon ) \Rightarrow |g(x) -L| < \epsilon[/math]

how do I prove that the limit of the function to the power n is just

[math]\lim_{x\rightarrow a} g(x)^n = L^n[/math]

Been racking my brain over this forever

>>10035204
With the assumption that g is continuous at a, you are allowed to move the limit inside the function, that is, compute the limit and then take the power. Think about why

>>10035145
>We don't even know what the fuck k the Tod function is.
His paper on the fine structure constant talks about it more
https://mathoverflow.net/questions/311280/what-is-the-definition-of-the-function-t-used-in-atiyahs-attempted-proof-of-the
From the definition of the Todd function, as others have pointed out, you can still run the arguments with other analytic functions and get a similar, yet false result.

>>10035223
He never explicitly defines the function, he only listed a bunch of its properties which others have used to try and reverse engineer it.

>>10034510
if you want to do a phd, 100% second option. If you're unsure about doing a phd, but perhaps yes, then 80% the second option. If you don't want to do a phd, first option 100%

>>10035014
the math tags should have / not \. If you're using 4chan X, you can just use alt+m or alt+e

>>10035334

>>10026711
What nice properties (e.g. Hausdorffness, connectedness), if any, are preserved under initial/final topologies?

>>10035238
So can we expect a follow up paper?

Is being cyclic, abelian or simple a property preserved under a group mono/epimorphism?

>>10035569
>Is being cyclic, abelian or simple a property preserved under a group mono/epimorphism?
What have you tried?

>>10035569
>>>/s4s/

>>10035569
under any (valid) homomorphism, a cyclic subgroup gets mapped to a cyclic subgroup and an abelian group gets mapped to an abelian subgroup. From this it is obvious to check whether injectivity or surjectivity affect the target.

Simpleness is much more subtle, given that the target group can be much "larger". However, a surjective morphism from a simple group is clearly an isomorphism or trivial anyways.

>>10034556
I made ~$20K from paid RA positions throughout my undergrad. Oftentimes (especially at smaller universities) departments/professors have leftover money they need to use and are more than happy to massively overpay their favorite students for doing not really that much.

Adult student here and lifelong(until recently) hater of math. Alot of math issues people have are the fucking shit teachers that teach math. Its usually a guy that went to get a degree in math, figured out that was worth jack all, and decided to teach it. These type of teachers understood and were good at math to begin with. So it's the old i'm going to show you once and if you don't get it too fucking bad, here are your 30 or so homework problems i'll see you tomorrow for the same bullshit.

Thankfully i was able to get a math teacher that really loved math and loved to teach. He would take time to answer even my dumbest questions just because i was curious. And he took things slow and explained everything out. Now i really like math. I'm doing things i never thought i would be able to do.

>>10035675
>(valid) homomorphism
what did he/she mean by this?

>>10035716
you know what i mean

>>10035727

>>10035727
>you know what i mean
define "(valid) homomorphism"

>tfw don't know any topology beyond incredibly basic shit like open / closed / compact
What do? Munkres?

>>10035918
Any map f:G -> H which preserves the group operations on G and H.
f(ab)=f(a)f(b)

>>10036266
I believe the point the anon you're replying to is making is that is just "a homomorphism". There's no such thing as n "valid homomorphism", it either preserves the group operation or it doesn't.

>>10036281
Yeah, I'm a bit baked, I caught that after I posted. Oh wellllllll.

The math building at my school is unironically the comfiest building on campus. It feels more like a small art museum with offices in it. I remember getting rock hard as a freshman walking through those halls. And the lounge is great.

>>10036647
Our math building is small and old and somewhat dilapidated but still comfy af. It also feels like it has actual character and history unlike say, the brand new electrical and robotics engineering building which feels like something out of a meme silicon valley campus.

>>10036655
Our's is very small too but relatively new, it's not as new and flashy as the new engineering buildings but not as old and uncharismatic as the physics building. And when I say uncharismatic I really mean it, the physics building is a depressing joke. But hey there are probably some parts of it that I am unaware of.

Is my work correct?

>>10036647
My math building was a 50's brutalist cube where each hallway looked the exact same.

>>10026711
How do I show that any infinite subset of the natural numbers is isomorphic to the natural numbers (under >, ordering)?

seems like it's just obvious based on the definition....

>>10036853
N is well ordered, so each non empty subset has a least element. Use that.

>>10036853
f: N -> A
map 1 to the least element x_1 of A
map 2 to the least element x_2 of A- {x_1}
map 3 to the least element x_3 of A-{x_1,x_2}
...
obviously bijective

>>10036897
>>10036878
Thanks, guys. This makes sense. This is my first time doing this kind of math; it's very strange.

It's also not clear to me how this well-ordering property applies to non-empty finite subsets?

>>10036956
What do you mean? The well-ordering property doesn't distinguish between finite and infinite subsets.

>>10036769
UBA?

>>10035366
Does anyone know or is this too general of a question?

>>10036765
No.
You should understand that the binomial distribution is calculating the probability that 5 messages are high priority AND that the other 15 messages are not high priority. You're going to calculate the probability that the other messages are something other than high priority again later, so you shouldn't be multiplying by (0.9)^15 now as well.

>>10036647
>tfw the math department at my university occupied floors 9 and 10 of a building with 1 elevator

>>10026916
>sub-pre-sub-remedial course
my sides

>>10035366
>>10036997
if X a topological space, Y a set, and f:X->Y such that f makes Y a topological space with f continuous, then clearly f is continuous by definition. Hence connectedness, compactness and other things that are preserved by continuity are preserved. Hausdorffness is not such a property (in particular, the quotient topology is defined as such and is well known to not be Hausdorff in many cases).

If in the opposite situation, ie: f:X->Y to make X a topological space, then surely you can do a similar analysis, but it's been 2+ years since i did any general topology, so i can't remember what properties are preserved under preimage, and cba to derive it

>>10037024
>tfw the university system will continue to degrade into an extension of high school and will get more and more corporatized
>tfw the old university culture and experience will never be recovered
>tfw you'll never be able to walk into your university, proud to be in a place of scholarship and knowledge and knowing that you're surrounded by like-minded, intelligent people
It's not fair.

Are there any lecture course on trigonometry and precalculus and algebra?

>>10037042
my heart

Why is all the stuff in the math tags not showing up properly to me? it just looks like normal text.
Am i retarded?

>>10037072
adblock off

>>10037075
nah thanks i don't want to get raped by popups

>>10037092
there's no pop-ups, there's only non-intrusive ads at the top/bottom of the page (which you can target with the zapper mode in ublock)

How would you guys recommend I study for Calc 4? What tips could you assist me with?

Can someone please explain to a brainlet how it's possible that it takes months or years to verify a math demonstration?

Isn't math the most rigorous and formally consistent area of knowledge? You don't need to verify any data, any experimental methodology, any equipment.

Why can't people tell for sure whether the old man's RH thing is right or wrong in definitive terms by now?

>>10037042
Please anon, my heart can't take this

>>10037545
the truth
will set you free

>>10037042
Most public Unis are already like this. But some majors are purer than others.

>>10032637
>no abstract algebra
>heavy focus on analysis
Fucking based

>>10033546
I know what you mean, it's much easier for me to learn from lecture than from a book. I think it comes down to a combination of me learning better verbally, and the fact that class time is set aside and as such is more focussed.

>>10035707
You're hopeless. You have no chances in mathematics. Pick up Rudin Real and Complex Analysis, read it cover to cover, work 95% of the exercises (trivial) and come back when you are worth our time.

>>10036238
That's about all topology is
But yes, Munkres. It's very, very good.

>>10037019
Berkeleyanon?
Evans is objectively the worst building on this campus even though I basically live in it.

>>10037535
>Why can't people tell for sure whether the old man's RH thing is right or wrong in definitive terms by now?
If you go look up any mathematics paper, you'd probably find that you don't understand what any of the words mean, or how the words interact with each other. Basically it's like a different language.
This is how people even in research areas very close to Memechizuki feel about his work, because he's invented so many words and objects that did not exist before.

Imagine if you were asked to proofread 1000 pages of Sanskrit, but you aren't allowed to have a dictionary and the only person you can ask for help is the man who wrote it.

>>10037687
>Evans is objectively the worst building on this campus
The math building is invariably among the shittiest buildings on every campus I've been to.
>tfw watching administration spend millions of dollars building a giant, viciously air-conditioned glass box complete with detached swivel office chairs, indoor gardens and waterfalls in the open floor-plan hallways, and a built-in Starbucks for the engineering students while the rooms for my TA sections have holes in the walls

>>10030702
We're all gonna make it except this nerd.

>>10036769
VT?

>>10036985
Didn't know there was a fellow arg anon here.

>>10037019
>>10037687
>>10037802
Fucking this.

>>10037687
>>10037802
>>10037802
>indoor gardens and waterfalls
I go to another "elite school" with similar problems. It pisses me off how funds are mismanaged like this instead of going to something that could actually be useful to the shitty quality of life of students. What the actual fuck.

>>10037969
Go to another elite university. I’m glad they actually try to manage funds better, but it is annoying how the main lecture room has chairs that squeak like a motherfucker.

How should one structure a research proposal in a PhD application? What I have sketched on paper is roughly:
>what and with whom
>exposition of the questions asked
>how to approach the problems
>significance of the results
>how i will spend my time and how i intend to fund it all

>>10038527
lol i just put "interested in algebraic geometry" and I got into a top 10, fully funded

>>10038614
Sure you did. How about you fuck off?

>>10026711
Is pic related worth reading?

>>10038618
>exposition of the questions asked
right, because they don't know them, and because an outsider like you would know what's at the forefront
>how to approach the problems
lol
>significance of the results
again, what the hell do you know?
>how i will spend my time and how i intend to fund it all
working on the degree full time obviously, and getting paid by the university to do so

>>10038647
>working on the degree full time obviously, and getting paid by the university to do so
Or using a grant or a student loan since I may need external funding.

>>10038662
yeah, but there's little opportunity for pure math, so the uni itself is funding me

Can I use Dodgson condensation to find the determinant of any square matrix (assuming I do some row operations to remove zeros)?

>>10038630
Yeah if you're new to mathematics, but it's a big book with lots of topics in shallow depth. I would just go through the first few chapters on proofs and logic and then read another book of interest.

>>10038666
Not necessarily. Many places require external funding.

>>10038695
>Many places require external funding.
What kind of dumbfuck does an unfunded PhD?

Hey guys could someone help me with some probability homework pls
>>10038907

>>10038910
/sci/ is not for homework fagget

How do you keep up to date with all the mathematics you have learned?

>>10039029
memory

>>10038865
Money is not an issue for me.

>>10027903
this is awesome

>>10039019
>fagget
Why the homophobia?

>>10038630
>Is pic related worth reading?
Why don't you read it and find out?

>>10039029
I don't, but I have accepted that.

>>10028628
He right about top 10 programs though. Anyone planning to do NT/AG/AT/RT will have taken graduate courses in at least 3/4 the above during undergrad. The result of this is that at places like MIT/Harvard an AG course covering Hartshorne will have a breakdown of something like,
1st/2nd years: 1-2,
3rd years: 5-8,
4th years: 5-10,
graduate students: 7-10.

>>10039347
>graduate students: 7-10.
So by your own breakdown even at Harvard there are more graduate students in the class than anything else,so it's clearly not exactly a huge consideration to them if you've taken a bajillion grad classes before you get there or not.
Harvard (like literally any other graduate school) selects graduate students for their research potential, and whether or not you have a semester of algebraic geometry is not particularly relevant to this (unless your research statement says you want to be an algebraic geometer, of course).

>>10029118
Kodaira-Spencer gravity.
>>10031157
Mapping class group.
>>10031722
What's confusing you exactly?
>>10036647
Pretty sure I'm doing a PhD at the same place you're at anon. The math building is fucking gorgeous
>>10038527
I'd just have one single question that I focus on, then
>exposition
>strategies
>significance
>applications and implications
That's what I did for NSERC.

>>10033227
I want to study around 4 hours a day, but usually it's more like 2-3 hours. I partition my study time into 50 minute parts.

>>10039540
Non-math here, but theoretical physics. Most grad schools will value research over classes taken. If you’ve taken a grad class they would’ve preferred if you took their class. But i have to say having 1-2 publications in undergrad is a guaranteed ticket to a grad program, plus the number of grad students with at least one undergrad publication is quite common in top 10 grad schools.

Any mathematician bros get high?

What it is like doing advanced maths while high? I never tried it

>>10033227
about five hours in the early AM. when i wake up is easiest because it's easier to study than to get up and do things.

>>10039620
It's a meme. Get high if you want but it's not going to somehow help with your math. Maybe if you take a low dose of psychedelics you'd get some inspiration, but you're not gonna be doing any actual math.

>>10039599
>>10039631
What's the best studying method? I don't know if I should just do a lot of exercises, or learn theorems/properties by heart or whatever. Up to now I got by easily just by listening in class but that's not gonna cut it anymore.

>>10039639
Read the chapter carefully, especially definitions, until you have a fair understanding of the material. Then write down what you have retained without consulting the textbook. Check the textbook if what you have written down is correct and then proceed to the exercises.

>material on problem set that wasn't covered nor mentioned in class

>>10038684
Answer my question

>>10039662
Imagine having to teach yourself something. Insane shit m8.

Can someone rate >>10033682 as a source for getting started with mathematics? I've taken Calc 1 and 2

>>10039694
Probably get bored and give up before you get to anything halfway interesting.

>>10039702
Any texts that you think would more effectively get me started with proof literacy?

>>10039712
Just read Spivak. The whole point of it is to make you mathematically literate while also teaching you calculus.

>>10035707
Yeah, unfortunately your experience is way too common (without the second part).

>>10036647
what school?

>>10039620
It's like staring at a page full of symbols and not knowing what the fuck is going on.

Any book/notes on smooth manifolds that develops everything using the definition of tangent space as equivalence classes of curves? I know its harder and the notation is tedious, but derivations are too dry for my taste.

>>10039843
do Carmo is one

>>10039853
Which one?

>>10038684
>>10039674
It seems to work for any matrix provided that when you reduce down the matrix you're dealing with has a non-zero determinant, so that you don't divide by zero. So yeah, it works
https://www.maa.org/sites/default/files/pdf/Mathhorizons/pdfs/nov_2006_pp.12-15.pdf
Here's an article that clarifies the point.
>>10033227
Somewhere between 3-8 hours, really just depends on the day. I keep track and aim to at least spend 30 hours per week studying. I mostly just stay in my office and find make a check list of things I want to accomplish for the day and get through as many as I can. If I feel too bored with one topic I instead switch to another one that I find more palatable, the whole "if you're going to procrastinate, at least do something else useful/productive while you procrastinate" type deal.

>>10039679
People who can't self learn don't genuinely love what they're majoring in, either that or they have a really weird learning style but it's most likely the first one.

>>10039862
I think Riemannian Geometry but you might as well just check on genlib

Prove that Z (+) Z isn't isomorphic to Z (under ordering). Where Z (+) Z is defined as regular Z with an additional copy of Z over it such that all of the elements of the copy are greater than all of the elements of the original. And the ordering is the same under each Z. How do?

>>10040051
>How do?
What have you tried?

>>10040051
Assume the existence of an isomorphism. Let a be from Z and b be from the copy of Z. What fact can you draw from their images?

>>10040636
Showing that f(n) = (a, b)
then for some f(n + 1) = (a + 1, b + 1)
but then for some f(m) there doesn't exist a pair (a + 1, b) or (a, b + 1)
??

Is this correct?

>>10040849
What does your notation (a, b) mean?

>>10039542
I'll try writing it like that. Thanks.

This might be a /sqt/ question but I'll ask here first
Say I was evaluating a definite integral, a (some number) at the top, 0 at the bottom.
To solve it I would integrate, then have the top one subtract the bottom one
Sometimes though I get a situation where the second one contains 0/0 or something divided by 0. Can the definite integral still be solved anyway? What I've been doing is just ignoring the second bit (just solving the first bit and using that as the answer), but I'm not sure if this gives me the correct answer.
If I can't just ignore it, how can I solve it?

>>10040891
For the part you're subtracting consider the limit as x -> 0 instead of just evaluating at zero. If that limit exists (is from the right sufficient? Help me analysis bros) then you're good to go.

>>10040857
>>10040857
(a,b) is a pair where a is in the first Z and b is in the second Z.

i'm in calc 2 and we learned an introduction to differential equations. He said that you separating dy/dx is mathematically "illegal" but we do it anyway. Why is that and how is it fixed?

>>10040971
its because dy/dx isnt actually a quotient but a symbol representing the derivative of a function. im not sure what you mean by how is it fixed but people have generally figured that it can be useful to just treat it as a quotient and it all works out

>>10040971
You learned u sub right? Anyways, because diff equations, are that, equations. No matter how convoluted and retarded your method for finding a solution is, you literally just have to plug it in the equation to seee if it works or not and no fucking autist can deny that it's as rigorous as it gets.

>>10040940
In that case I don't understand what you've written at all. What does f(n) = (a, b) mean?

>>10041149
The nth element of Z is mapped to a pair of the ath element and bth element of Z (+) Z. In other words: n -> f(n) = (a,b) and similarly: f(n+1) = (a+1, b+1)

Does this make sense??

>>10041273
Ok, sorry I misunderstood what your Z (+) Z was originally. It's actually ordered pairs of integers then? In that case I don't understand the condition places on the order, ie
>all of the elements of the copy are greater than all of the elements of the original
Does this just mean a kind of dictionary ordering? Eg,
https://en.wikipedia.org/wiki/Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets

>>10041321
Oh no you’re right, it’s a regular ordering, not pairs. All elements in Z_1 are less than Z_2 and they are ordered normally within their respective sets, it is like two merged sets of Z, no ordered pairs.

Isn’t it obvious that they are not isomorphic to Z then??

>>10041339
Yeah you just need to show that an isomorphism necessarily breaks the order.

>>10041371
What about something like this:

Suppose there’s an f: Z -> Z(+)Z, then if f(n) is in Z_2, then f(n) can be n in Z_1 and f(n) in Z_2?

Showing every n can be mapped to 2 elements?

>>10040971
the basis as to why it works is simply the fundamental theorem of calculus hidden in it, and a sort of "generalized function" à la dirac delta.

Basically, at your level, having a [math]\mathrm d x[/math] term on its own in an equation doesn't make sense; it's not defined. However, it is defined once you put an integral symbol in front of it: [math]\int\mathrm dx[/math].

When you have something like [math]\frac{\mathrm{d}y}{\mathrm{d}x}=f(x)[/math] and then write [math]\mathrm{d}y=f\mathrm{d}x[/math], you're implying that they have the same integral [math]\int \mathrm{d}y=\int f\mathrm{d}x[/math]. But the real question is "how does one get from one to the other?" Simple:

[math]\frac{\mathrm{d}y}{\mathrm{d}x}=f(x)\implies \int \frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x=\int f\mathrm{d}x[/math]. But the FToC implies that [math]\int \frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x=y=\int \mathrm d y[/math]. I genuinely don't understand why they never teach this in class. It is so simple and would solve so many problems..

I need help algebros.. I just met for the first time with my PhD adviser and he wants me to study knots and their polynomials, with their connection to algebraic geometry... He says it's a cosy nook, with almost no papers on and all the chances of getting papers out and results, perhaps even get my name on something. But I don't know lads, I'm not that interested in knots, I care more about arithmetic geometry and the more heavy duty stuff, but I don't wanna look like an idiot so early

wat do

>>10041031
what i mean by "fixed" is how it is made more rigorous or mathematically "legal"

>>10041429
idk anon that sounds pretty comfy personally

>>10041612
yeah, and i know I'm reaching, but i want to work in a field with a chance (even if 0.1%) of getting a BIG result, not some small field proving tons of results no one cares about that are mildly interesting. Plus, I wanted to do more arithmetic stuff..

>>10041664
I mean I guess there are few questions to ask.
>Is this what you'd be doing your PhD in
>Is your advisor an arithmetic geometer
>Do you / he know someone who would advise and arithmetic geometry PhD
etc.

>>10041788
>1
perhaps, i dont know, depends on how deep the rabbit hole takes me?
>2,3
yes, that's why i applied for him, but he just whips this out and he says it's interesting and there's many results to be found. He also said results in arithmetic geometry are very hard to come by and that a lot of work usually goes into not much.

>>10041664
There's a reason there aren't many papers on (k)not theory.

>>10041854
what reason? what i mentioned?

>>10041854
Explain yourself like I'm a knotlet

>>10041429
>>10041664
I mean, Vaughn Jones got his fields medal for work in knot theory and some of Witten's most cited work also falls under that category. Knot theory is in the vogue when it comes to mathematical physics. Why not do this, there are some connections between knots and primes, right?
https://mathoverflow.net/questions/50879/what-is-the-knot-associated-to-a-prime
And your advisor wants to cover the connection between knots and algebraic geometry. So this my seem super naive but consider the following, if you know there are some connections between knots and algebraic geometry and there's an analogy between knots and primes, does that mean there exists a similar analogy between algebraic geometry and arithmetic geometry? I guess what I'm say is you could use the connection between knots and primes to port results from one field to another, thus working on what your advisor wants and what you want. But my thought process is likely super flawed.

>>10041664
You're going to flop out if this is your motivation. Even if you manage to get into a field with a possibility of getting a BIG result (which is difficult because most research with the possibility of making waves across the math community goes on at super-top schools) you're just going to get discouraged and leave when it's been 3 years and you haven't revolutionized math yet because your desire for prestige instead of just having fun isn't being satisfied.

>>10041866
>>10041868
Recreational math that no one cares about

>>10042065
>Recreational math that no one cares about
fite me

>>10042065
This is a real undergrad take if I've ever seen one

>>10033026
Keep trying, the "initiative" will come if you like maths and if you don't give up at each step. Try to learn why a solution works.
I had some similar problems in the past. Mathematics is hard, especially when you need to use some obscure tricks at the beginning (for example adding and subtracting one in a proof).

>>10042037
im at a top 10 school for math

>>10042034
perhaps, it does sound interesting, but it's not really what i wanna do. I wanna do stuff related to elliptic curves.

>>10039620
Weed makes me psychotic and retarded for the next couple of days. Last time I smoked weed I thought I was being sewn into the couch, I could feel sound (which was extremely painful, it felt like razor blades) and I thought aliens were sending my nightmares and childhood fears into my head (I fell asleep had a nightmare then woke up in it.) Never again. I literally get nam flashbacks everytime I smell weed now.

new thread

>>10042555
>>10042555
>>10042555
>>10042555
>>10042555

>>10042515
>I wanna do stuff related to elliptic curves.
What kind of stuff? Anything in particular?

>>10042568
I have no clue what's in the forefront desu, i should probably ask him. I just know enough to state a simple version of Birch-Swinnerton, but I wanna go further in something like that direction