/sci/ - Science & Math

I've got more than a fuck ton of math books, what area do I read about first

>>15260496
Tropical geometry

New thread new problem!!!!
THIS IS EASY. YOU CAN'T SOLVE IT? YOU'RE DUMMY DUM DUM. Haha jk. You're not.
I appreciate anyone who replies whether or not they got the correct answer. If you're stuck, feel free to ask for help as I've solved it and I believe I can give hints that may be helpful. Feel free to create new variations to this problem if you think of something interesting. And good luck!

>>15260581
0 when s>r

I have been going through alternative calculus books which I think will be useful to advanced students in physics and engineering. Courant's Analysis seems perfect for this, but I rarely see it recommended; which is surprising because his "What is Mathematics?" is quite popular. Perhaps because it's "old"? In any case it seems excellent and is in two volumes. Those that have read it, what did you think? [There's an even older original version in German written in the 1920s, but this was updated in 1965]

Another nice one that I came across is Higher Math for Beginners (Mostly Physicists and Engineers) by Zeldovich Y. and Yaglom I. a Soviet era book published in English by Mir. Seems quite comfy for someone that just wants to get busy using the apparatus of calculus. Easier than the above, but it isn't dumbed down for "no child left behind" Americans.
Link: https://archive.org/details/HigherMathForBeginners/mode/2up

Anyone know what's the difference between this? If written -3^2 on the calculator the answer is -9 and if written -3 * -3 the answer is 9.

The correct answer should be positive because two negatives are multiplied. But if you use the symbol ^ for some reason it remains negative.

Also, if I put sqrt(3.5) in calculator I got 1.87 and not 1.75.

>>15260682
It depends on your calculator, but the problem is with operator precedence of '-'. Your calculator is parsing the syntax as -(3^2). So to get what you want you need (-3)^2.

>>15260591
Good observation my friend. However I believe you meant to say n>r as there are n different kinds of cards, which have been all printed s times. "A company produces s series of n cards". So they made s many series of cards which have n variety. At least this is how I understand it. It also seems to agree with the result given in the book.

Like I said, great observation! You can use it later to check it your answer seems to be getting the right answer. Good luck solving the entirety of the problem fren! I believe in you!

>>15260682
>Also, if I put sqrt(3.5) in calculator I got 1.87 and not 1.75.
And is 1.75*1.75 equal to 3.5? No, it's not.

>>15253151
>>15259394
There's no need to compile counterexamples, because the subject is quite intuitive and clean. You can always think of counterexamples on your own, and you don't usually need any contrived ones.
Analysis and topology on the other hand are quite messy.

>>15260707
>Just search your mental space of projective perverse sheaves for couterexamples. Search till you gottem

>>15260711
Skill issue

>>15260642 (me)
Another I neglected to mention is the classic Differential and Integral Calculus by Nikolai Piskunov.

>>15260688
Makes sense, thank you.
>>15260696
Ah, I mistaken it as an addition, my bad.

>>15260642
Good book, exercises are decent but could use some simpler ones. It is a bit wordy and Courant has a weird obsession with plane graphs. It could be half as short without losing much.

Jech's Set Theory or Enderton's Elements of Set Theory? Or is there a better book anyone could recommend? Halmos seems to be more 'casual' than either mentioned.

>>15261530 (me)
Basic research says Jech's is more thorough but if Enderton (or any other book) is also worth reading, I'm open to suggestions.

>>15260571
Tropical geometry is racist
>The origins of tropical geometry date back about twenty years. One of the pioneers of the theory was Imre Simon, a mathematician and computer scientist from Brazil — which is by the way the only reason for the peculiar name “tropical geometry”.

Would any of you recommend good introductory books on category theory (funny arrows and funny words and even funnier concepts) and ZFC (funny thing about bags within bags)? Thanks.
Forgive the humorous descriptions.

>>15260487
>lawvere - sets for mathematics
>aluffi - algebra: chapter 0
>bradley, bryson - topology: a categorical approach
ive been told by some smart people in my college to avoid going too much into category theory before learning basic concept. but these books seem to just start with those concepts in a category theory way. ive read a bit of sets for math and kinda like it. the other, i need for my degree anyway(the lecturer recs are hard to read(artin, munkres)).
do i continue reading these, or just read non-cat books first?

im gonna get drunk and do some graphing tonight probably

Would this be a good room to study mathematics in? Looking for an English country house.

>>15261909
Uhm, its up to you. LIke this one french nigger likes to do math in a train, so there is no good or bad place to do math

>>15261921
I'm just worried that if I study in the wrong spot, I may hold back mathematics research for a month. Need to level up my Erdős number. There's lots of space here too. if you look at the outside.

>>15261530
I think people get at those books for the wrong reasons. It will depend on what you want.

>>15261630
I've written a list at one point, with links. The first two that are still online are
https://arxiv.org/pdf/1612.09375.pdf
http://pi.math.cornell.edu/~dmehrle/notes/partiii/cattheory_partiii_notes.pdf
Of course there might be other options if you download them yourself, but there you go

>>15260487
What are some advanced books that are actually affordable? Field doesn't matter, I'm just wondering if you guys encountered any on your path

This function is continuous at the origin. I'm asked to determine whether or not it's also differentiable at the origin. I think it is, but I'm not 100% confident in my work:
[math]
\lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}}-0-\begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} x-0 \\ y-0 \end{pmatrix} \right|}{\| (x,y) \| }
= \lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}} \right|}{ \sqrt{x^2+y^2} }
= \lim_{(x,y) \to (0,0)} \frac{x^2\left|y\right|}{\sqrt{|xy|(x^2+y^2)}}
= \lim_{r \to 0^+} \frac{r^2\cos^2(\theta)\left|r\sin(\theta)\right|}{\sqrt{|r^2\cos(\theta)\sin(\theta)| \cdot r^2}}
= \lim_{r \to 0^+} \left|r\right| \frac{\cos ^2\left(\theta\right)\sqrt{\left|\sin \left(\theta\right)\right|}}{\sqrt{\left|\cos \left(\theta\right)\right|}}=0
[/math]
Thoughts?

>>15261934
>>15261909
Hm...try this...

Move around slowly in the vicinity of the English
country house thinking of math. You'll eventually
zone out thinking of math and your body is
doing the walking for you. Naturally, you'll move
away from some spots and stick around in others.
The place where you stick around the most is
the spot to do your math research. And if not
in the vicinity of the country house, try it elsewhere.

>>15260695
>>15260695
Adding to it,
when r<n prob=0
when 2n<r prob=1
so it can be some linear function for n<r<2n
say prob = (r-n)/n

>>15260695
No wait, shouldn't it be 1 for r>=n
assuming the company packages the cards in fixed order?

>>15260695
I was wrong, the probability is not 1 for r>2n
but it probably is for r > (n-1)s

>>15260695
I think I have the answer now,
[math]\dfrac{n\cdot\binom sr\cdot\binom{n(s-1)}r}{\binom{ns}r}[/math]
is it correct?

>>15260695
i was wrong again lol
what about this one?
[math]\dfrac{s^n\cdot\binom{n(s-1)}r}{\binom{ns}r}[/math]

>>15260695
I think the answer is
[math]\dfrac{s^n\cdot\binom{n(s-1)}r}{\binom{ns}r}[/math]

>>15261934
I wouldn't worry about it
the harrowing anguish of realizing that what you thought was true is a mere phantom
the embarrassment
the lust for knowledge unfulfilled
how can you not dull the pain with the hubbub and bustle of city life
and what will your neighbors think
mucking around in the dark dank dungeon of...of...of...denoted demonic diabolic symbolic formality!!!
you have chores
there are responsibilities
upkeep of the grounds
general respect for society
hygiene
why
you are fixing to turn yourself into a hobbit
mathematicians are like Jack
don't fool yourself
touch the blackboard
feel the blackboard
her cold touch
so distant
so innocent
she beckons to you
"Come to me!" she says
"I must feel the tip of your stick" she moans
you press the chalk stick onto her green grassy bosom
she sighs

>>15260695
correction
[math]\dfrac{s^n\cdot\binom{n(s-1)}r}{\binom{ns}{r-n}}[/math]

>>15260695
hmm
[math]\dfrac{s^n\cdot\binom{n(s-1)}{r-n}}{\binom{ns}r}[/math]

>>15262109
>>15262146
>>15262207
>>15262218
>>15262224
(I'm under the assumption all these belong to the same person, I apologize if that's not the case.)
Okay anon, first things first I'm really happy to see you attempting to problem. You have some really correct ideas and I think you're on your way to the correct answer. Unfortunately so far, you're not quite there yet.
So, to begin
>when r<n prob=0
you're right! This is of course very easy to see and it's not really clear how it'll help you get the answer, but it may help you do a quick sanity check on your answer. Does it agree with this? When you plug in numbers such that r<n, does it equal zero?
>No wait, shouldn't it be 1 for r>=n
>assuming the company packages the cards in fixed order?
They do not. You buy r packages randomly.
>I think the answer is
unfortunately not, that's not the answer.
>correction
Bratty problem!!! Answer correction is needed!
Your answer here in >>15262224 has the correct denominator as the actual answer, however the topside is quite wrong. I'll say, the answer is not as clean and pretty as any you've written so far(for the topside[numerator]). It's actually a summation, which might give you a clue as to how to get the correct answer. GOOD LUCK! I'M HERE FOR YOU! If you ask for help, I'll do my best. I'm not saying too much here because I don't know if you want to me give help or if you want to do it yourself more.

Which country is the best in maths and why is it France?

>>15262245
yes, it was all me lol
let me try some more without help, for now

>>15261876

hello stats friends please help me out
im pulling my hair out over this when i just KNOW it's something stupid and simple that im missing
>>15260023

>>15262254
Russia, UK, France. Germany and Switzerland get an honorable mention, but their best days are behind them.

UK and France probably will fall off within a generation, as they're being "enriched".

>>15260581
Why is this incorrect?

[math]\frac{r \choose n}{ns \choose r}[/math]

My logic is that number of possible configurations of the chosen set is the denominator, and the number of those possibilities which satisfy the problem (i.e. contain all n coupons) is in the numerator. I don't see what's wrong with this logic, please explain anime girl poster.

As the other anon mentioned, this is obviously only valid for r >= n, and the probability is 0 otherwise.

>>15262296
n = 10, s = 2, r = 10

r= 10prob + 10

>>15262254
Grothendiek is not french. Also no lmao.

>>15261876
>>15262271
These are cool anon you just spiked me into planning a night of getting completely stoned and doing graphing
I've been wanting to make my own visualizers for my music from scratch but have been so lazy

>>15262266
I wish you the best! I hope you succeed my fren.
>>15262296
>Why is this incorrect?
Well, the denominator is actually correct, so nice job!
As for the numerator, let's see if I can manage to explain why it's wrong. I might fail, so apologies about that. If we have,say, n=r=1 and s=2, your answer seems to give (1 choose 1)/(2 choose 1)
which is 1/2 if I'm not making a careless mistake. However, this doesn't make sense as there is only one kind of card and we're taking 1 card. So we should have gotten 1 as our answer instead as out of two 2 cards available, no matter which one we pick we'll get all the different kinds of cards there are. So I hope this does show that your answer is wrong. As for WHY it's wrong, which is definitely a lot more useful, ehghghhmhgh. I'm sorry. I can't really explain. It doesn't give you all possibilities I guess, which include picking the n different types plus useless ones we won't need. Maybe that's why? Not sure honestly.
>this is obviously only valid for r >= n, and the probability is 0 otherwise.
I think (n choose k) is defined as 0 unless 0<=k<=n so I don't think you need to say this as your answer would have given 0 on the numerator if n was bigger than r. The correct answer also of course gives 0 for that case of course, you don't need to say anything else about it for when r>=n doesn't hold.

I hope I didn't say anything stupid or made any careless mistakes, that'd be embarrassing.

>>15262280
UK and France enriched, while Russians are simply disappearing.
Which country will carry on the Math legacy?

>>15262320
i didnt make the second one. some other anon did some photoshoop on it. pretty good job considering there is alot of gotcha in terms of the colors used.

ive spent years graphing and even some of my earlier work is all stuck on a hard drive not hooked up to anything so that sucks

>>15262445
SPQR

>>15262445
What language/program do you use to make these?

>>15262478
c++ on gcc on linux mostly. but i also have a version on windows visual studio i just dont like it as much, I used some functions for writing pixels to a bitmap. then use another program to kinda turn it into either gifs or jpgs or whatever. it can be slow and there is plenty of automation i could do but really the most time consuming part is writing the math so its not that big a deal to me

>google easybmp if you want to get started easily its old but basic

most of the actual math looks like
>for loops
>do some math
>write to red
>write to green
>write to blue pixel buffers
>end for loops
>write to bmp

hope that helps

>>15262271
https://mp3.hardnrg.com/morgan/Morgan-Decepticon.mp3

>>15262495
Thank you very much! Thought it might have been julia for some reason.

>>15262506

>>15262573
https://github.com/gearsprog/gears/blob/main/gears.c

https://pyramidslabel.bandcamp.com/album/spiritual-continuum

>>15260581
My guess using generating functions
[x^r] ((1+x)^s - 1)^n / [x^r] (1+ x)^(ns)
= [x^r] ((1+x)^s - 1)^n / (ns C r)
= Sum[ (n C k)*(sk C r)*(-1)^(n-k), {k,0,n}] / (ns C r)
>Proof: think

>>15262045
Anything from Dover. Sometimes you'll find math books in used bookstores, but usually a very limited selection. Your local math department might have a table or a shelf of books which are free to take.

>>15260581
[math]\sum_{k=0}^{\lfloor \frac{r-n}{s} \rfloor} (-1)^k {n \choose k} {{r-sk-k-1} \choose {n-k-1}}[/math] out of [math]sn \choose r[/math]

The numerator should be the number of integer solutions to [math]x_1+x_2...+x_n=r[/math] where [math]1 \le x_i \le s[/math]

>>15262725
Try r=n=s=2.
You get 1/6 with your method.
The actual answer is 4/6.

You're on the right track.
Maybe try weighting each solution you count by the number of ways it can happen.

>>15262725
Also, however you count solutions in {1,2,...,s}^n for the numerator should give the correct denominator when counting solutions in {0,1,...,s}^n

>using latex
>not using unicode
>2023
Check out what i can write:
∂ₜu = Δu
∫_M dω = ∫_∂M ω
φ(a⋅b) = φ(a)⋅φ(b)
Hₙ(S1) = a
Λ0(M) Λ1(M) Λ2(M) …
K = {⅟ₙ | n ∈ ℕ, n > 0} = {1, ½, ⅓, …}
∑⅟ₙ2 = ⅙π2
eᴵˣ = cos(x) + isin(x)

>>15262065
man yall rly don't like continuous math huh

Okay so basically im trying to show [eqn] e^{-\pi}+e^{-3^2\pi}+e^{-5^2\pi}+\cdot\cdot\cdot=\frac{(\sqrt[4]{2}-1)\Gamma(\frac{1}{4})}{2^{11/4}\pi^{3/4}} [/eqn] Its pretty obvious that the sum on the left is [math] \frac{1}{2}\theta_2(e^{-4\pi}) [/math] and its also obvious that is equivalent to [math] \frac{\eta^2(8i)}{\eta(4i)} [/math]

But how do I actually evaluate the eta function? I can relate them to the Euler function [math] E(q) [/math] (or [math] \phi(q) [/math] it is sometimes denoted) but then the problem remains how do I evaluate that function?

How extensive/difficult is math above the introductory level? Past discrete math, past most undergrad calculus classes, etc. Is math above that level just the tip of the iceberg, or just fucked up bloat

pls help math bros >>15263244

>>15260581
Finding the number of ways is easy: just stars and bars, but the probability is where I stumble, since it is not equally likely. Anyways, a non closed form solution for now is:
[eqn] \frac{1}{{sn \choose r}} \sum_{ \mathbf a \in A} {s \choose r - \sum_{i=1}^{n-1} a_i } \prod_{i=1}^{n-1} {s \choose a_i} [/eqn]
where [math] A = \{ \mathbf a \in (\mathbb N \cap [1,s])^n : \mathbf 1^{ \mathsf T} a = r \}[/math], i.e. n-tuples of positive integers that add up to [math] r[/math], but each integer less than [math] s[/math].
I haven't tried finding the complement using inclusion-exclusion, perhaps that would be a way.

>>15262224
I get what strategy you have used but this is incorrect because you are double counting.

>>15262664
Anon, initially I was going to say that looks different from the correct answer in some ways,however upon further inspection they seem to be the equal unless I have made a careless mistake. I declare your answer correct. I'll check it again later today when I'm at the library but I'm 95% you're correct.

So great job my friend! You did it! So quick and short too. Generating functions are magic. Thank you a lot for your time and effort solving this problem. I hope you found it fun. Later today(maybe tomorrow) I'll post the solution from the book which does it differently from you,so be on the lookout for that as that might be interesting for you. Have a pleasant day!
>>15262725
I don't think this is correct anon. If we had n=r=1 and s=2, your solution seems to give (-1)^0*(1 C 0)*(0 C 0) out of (2 C 1) which should equal 1 out of 2. Apologies if I'm making a mistake here but that doesn't seem right to me. Again I'm really sorry if I made a mistake here but yeah, doesn't seem right to me.
>>15262820
Who are you? You're not me.
Regardless, you seem to know the solution since you're replying to another anon about their solution being incorrect, which I agree. So thanks for that I suppose. I assume you found the solution but aren't sharing to not spoil others? That's cool! I assume that's the case. Please clarify.
Are you this person >>15262664?

>>15263264
Sorry anon, your message came when I was already writing >>15263281 this so I didn't reply to you there.

Now for your solution....
What the frick is that thing anon?!? There is a sum inside a sum and there's like a product going on there too. What the hell is happening!
I can't even say if it's correct or not because I don't really understand what it means. I'll say that the real answer is hell of a lot less complicated. You're correct on the denominator side of things, it really is (sn C r) however the rest I can't make sense of. Again, maybe it's correct and equal to the much simpler answer somehow but I personally don't see it.
>Anyways, a non closed form solution for now is
The answer from the book is also nonclosed so that's okay.
>inclusion-exclusion
Please try that instead. That should get you the answer really quickly. That's how I did it, that's how the book does it. It gets you the much simpler much prettier albeit non closed answer.

Now that Wildberger is retired, who's math's lolcow?

>>15263459
I'd like to volunteer John Gabriel and his "New Calculus"
https://www.youtube.com/watch?v=fCA5kGOgBgY&t=8s
https://thenewcalculus.weebly.com/

>>15263476
And to quote from his "textbook":
>After Euclid and before me,not a single mathematics academic, ever understood what is a number. The fact that academics called the requirements axioms or postulates is proof of this, because in order to derive numbers, these requirements must have been established. Mainstream academics failed dismally to understand what Euclid was attempting to write down – the perfect derivation of numbers.

>>15263281
>Are you this person
Yes.
>you seem to know the solution since you're replying to another anon about their solution being incorrect
I started with the same lattice point approach and made a similar error.
>initially I was going to say that looks different from the correct answer in some ways,however upon further inspection they seem to be the equal unless I have made a careless mistake
Here's an explanation:
Let each of the s*n packages be represented by a (1+x) term. 1 represents not collected, x represents collected.
Selecting the coefficient of x^r in the numerator and denominator restricts to the case that exactly r are collected.
The denominator is straightforward.
In the numerator you can represent each of the n types by the term (1+x)^s -1.
The (1+x)^s represents all of the ways you can collect or not collect the s duplicate coupons. The -1 excludes the case where none of them are collected (which is what the problem demands).
Expanding the outer exponent of ((1+x)^s - 1)^n with the binomial expansion then extracting the x^r coefficient of each (1+x)^(s*k) gives the final form.
>Generating functions are magic
Yes they are. Every way of collecting any number of packages was emulated all at once through the algebraic expansion of polynomials!
>Have a pleasant day!
No u

Ok soooo the continuity equation
[math] \partial_t \rho + \partial_x (\rho v) = 0 [/math] is positivity preserving if [math] v(t, x) \in C^1(\mathbb{R} \times \mathbb{R}; \mathbb{R}) [/math]... right? If the flow field has sufficiently high regularity then mass can only get moved at a certain maximum speed, so you get a lower bound exponential decay for the minimum of the density. Why have I never heard of this before? Am I missing something?

>>15260581
You're a cute boy. I would like to kiss you.

Simulink/matlab bros, how do I get the yellow set of data samples to be a curve like the blue one? For the yellow I am using a constant values of 12 or 0 for the scope block..

>>15263576
This is amazing, lol. Where can I read up more on generating functions?
>>15260695
>>15263359
If possible, can you explain me how to solve this with traditional methods?
For example what i did in >>15262224 was for each type of coupon, I chose one of them hence [math]s^n[/math],
and then chose rest [math]r-n[/math] coupons randomly from rest slots [math]n(s-1)[/math] hence [math]n(s-1)\choose r-n[/math],
but now seeing the correct answer, I realize that I was using permutation and not combinations,
so what is the correct method to do this?

>>15263576
Beautiful anon. Respect! You've done a very elegant job fren! Thank you a lot for your explanation too. I appreciate it.
>No u
U HAVE AN EVEN MORE PLEASANT DAY THEN, DUM DUM!
>>15263704
Don't make this weird, weirdo. Do I know you? Anyway, solve my problem at the very least if you're going to say weird things to me. Weirdo.
>>15263790
>If possible, can you explain me how to solve this with traditional methods?
So anon, you gotta know there is this thing called PIE. Principle of Inclusion Exclusion. I'd love to write more here unfortunately I have to go right now! I have a 3 hour exam now so I gotta go. I'll give you this solution from the book the problem is from. I hope it's understandable but it might not be. To understand it you need to know PIE. You can find it in basically any combinatorics book. Mossinghoff and Harris' Combinatorics and Graph theory book has a chapter for it I believe, that'll probably be a nice place to look. I'm a beginner myself btw just to be clear.

>>15263835
>I have a 3 hour exam now
Best of luck for the exam

>>15263835
>>15263855
Thanks Anon, I understand now. Altho, I couldn't solve it myself, it was quite fun.

>more than likely have a B final grade in Linear Algebra
No no no no no not like this... not like this......

>>15264045
grades are overrated dont worry too much
>t. literally failed a few classes and still got a PhD and good job

finally learning how to count...

>>15260487
>Absolutely destroyed the exam for the hardest course ive ever taken
Feels good lads, seems that dream of a good phd position will come true

Why wasn't Aldous-Fill published?
https://www.stat.berkeley.edu/~aldous/RWG/book.html

>>15264462
in a post-LTCM world, publishing this definitely seems like the right thing to do

>>15263835
thank you anime girl poster, good luck in your exam and I appreciate the quality posts

>>15262442
Poland just wait until I graduate

>>15264402
To add to this, it was an exam on control of pde using an abstract DE approach.
Has anyone else here done something similar? I'd love to discuss it.

Im also down to discuss anything related to nonlinear functional analysis and functional optimization as im writing my master thesis on the topic.

If you want to know more about any of this, please do ask!

Bros... Tonight I again dreamed about my favorite set of measurable subsets of the reals!
The sigma algebra [math] {\mathcal B}({\mathbb R})\subset {\mathcal P}({\mathbb R}) [/math] generated by all intervals finite intervals [math](x,y\,][/math].

ITT:
What are your favorite functions selecting members from each non-empty set in [math] {\mathcal B}({\mathbb R}) [/math]?
Can't think of any right now, but will post once I got one 4u

I've got some fun questions!

1. Are there any fields of math similar to algebra, or potential expansions/alternatives to it?
2. Are there any alternatives to backpropagation? What was used before backpropagation? >>15264616
3. How do you guys 'create' math? Do you have a specific process for going from what you want to do to doing it?
4. Do you know of any 'failed' or fields that didn't 'catch on' which were created and forgotten about? If not, do you know where we can find out?

all math can literally be done by a retard and I'm tired of pretending it can't be

Define a *bolt* to be a set B with binary operation * : B x B -> B and distinguished element e <- B such that
1. there is an element e <- B such that
>eb = be = b for all b <- B, and
2. for all b,c,d <- B
>b(cd) <- (bc)B.
The element e <- B is the *identity*, in symbols e = 1.
Let B be a bolt. Define a (right) B-*target* to be a set T with binary operation T x B -> T such that
1. for all t <- T
>t1 = t, and
2. for all t <- T and all b,c <- B
>t(bc) <- (tb)B.

>>15264632
>4. Do you know of any 'failed' or fields that didn't 'catch on' which were created and forgotten about? If not, do you know where we can find out?
Many things mathematicians work on are nices, many work in the fringes, many topics are not well known about by most people. Historically speaking, universal algebra or super niche variants of Hopf- or quantum group structures come to mind. Niche logics. Niche geometries. I'd post links to things but you seem to be at the start of the journey. People also just spent their time differently. Like computing logs and shit by hand. Generally, synthetic geometry was taught a whole lot, even in school, still hundred years ago, but it's more and more being faded out. 223 years ago, math papers were also more verbose or literal than today.

>>15264649
I am the retard all math can be done by.
I am the retard who can do all math.
>a retard can, like, totally do all math, fr fr, and I'm tired of pretending it can't be JUST DO IT
https://www.youtube.com/watch?v=ZXsQAXx_ao0
JUST
STOP USING THE PASSIVE VOICE
YES YOU CAN
STOP PRETENDING THE PASSIVE VOICE CAN'T STOPPED BEING USED BY YOU
DO IT

>>15264653
Dude you're here for a while now - can you just start using \in instead of <-?
Like even if you don't TeX it, it would make it more readable.

Not even discussing your vocabulary introducing post.

This edition of Home Math Companion brought to you by 3M and the Archer-Daniels-Midland company
https://soundcloud.com/user-114329832/sets/howls-moving-castle-ost

>>15264658
>universal algebra or super niche variants of Hopf- or quantum group structures come to mind.
I think stuff like that is... Overcomplicated nonsense, personally. It looks esoteric but it's... Not practical, looking at who invented it, my brain just completely skips it as even an option. I used to think differential equations were hard, but they're actually super easy, it was that nobody explained them in a simple way to me.

>>15264663
___?
___|
___v
?->x<-?
___^
___|
___?

>>15264682
___!
___||
___v
!=>t<=!
___^
___||
___!

>>15264678
Universal algebra is algebra over a logic which is more quantifier free in a way, which makes some things nicer to work with. But a lot of this style of research was then displaced by category theory.
Hopf-algebras are a sort of natural abstraction popping up, for example, in differential operator theory, and that's not too far from e.g. differential equations. It also has combinatorical flavour and I do not call that niche in itself.
But people tried to extend it. A lot of this is motivated by the aim of breaking up Hilbert space theory (for QM) in a way that still permits a nice theory. "Desperate" attempts to get new theories, so it's in a way also motivated - just didn't work much.

>>15264632
>4. Do you know of any 'failed' or fields that didn't 'catch on' which were created and forgotten about? If not, do you know where we can find out?
No such thing, things maybe you consider failures were all likely the impetus of a larger more successful theory or were incorporated into one. Some things remain more niche than others, but thats not due to a failure of the theory merely a failure of advertising.
But everything in math is built on itself, and if a single brick serves its purpose in a wall you cant call that brick a failure for not being the shiniest or highest of all bricks.
For instance Klein once lamented how he witnessed the theory of abelian functions fall from a very active field to virtually unknown.
But that fall from stardom was not due to the failure of the theory; rather, abelian functions became less studied due to the enormous success of the theory. The study of abelian functions formed the backbone of complex algebraic geometry, and although those roots would become obscured somewhat by the revolution commutative algebra brought to the field, one can still see the connections with a good understanding of theories both past and present. Our modern understanding of algebraic groups and abelian varieties would never have happened without the masters of old studying abelian functions.
If a niche topic hasnt "caught on" in mathematics, then either it will in the future or it already has, and had caught on so successfully that it was rapidly incorporated in larger theories to the point you consider subsequent developments of that niche topic to actually be a development within the larger field.
All that said, sometimes one person will fully develop a theory of some niche topic. Then in your mind as subsequent developments in that area run dry even though very few people worked on the theory, you may consider that it never caught on. But in reality the theory served its purpose fully and can be applied to other theories of other areas

>>15264114
I just got my grade back and it was an A. Could possibly have a C in physics II, which has been the hardest course I've taken as its made no sense to me, and that was with going in with all As in calculus I - III..... that C will also be my lowest grade, but I'd rather have a D in a non math course than a B in a math course.

Thank you /mg/ for being good peers and keeping me feeling inadequate.

>be me, reading an introductory discrete mathematics book
>never done discrete math stuff before
>brain feels like killing itself
am I done for

>>15262216
Please don't fuck the blackboard, she's going to exploit you for child support money.

>>15264792
>n introductory discrete mathematics book
You mean an intro to proofs book? Thats all the initial part of a discrete math book is, essentially. Most math is just feeling like the dumbest person who ever lived, hopeless, filtered at this exact stage of math study and topic, until it just clicks and you feel like a 300IQ God and the smartest creature to ever bless the universe, nothing beyond your grasp.

>>15260642
Yeah I still don't see a better calculus book than Stewart. I know people here hate it, but its the ideal textbook for what it means to do. Wait until you're in an analysis course to go over analysis.

>>15262216
https://voca.ro/1eFvngHSIc8x

>>15265277
https://voca.ro/1mXI1QGmk3Qa
https://fundamatics.net/article/little-miss-polynomial/

>>15263855
Thank you lots fren. It went okay.
>>15263880
I beg to differ! I think you could have solved it yourself, but in this instance you may have lacked the few tricks that make this problem quite easy.

Now that the exam is over, I'll actually explain how I went about solving it

First I labeled every card depending on it's variety and which series it belonged to. So cards printed in series one were (1,1),(1,2),...,(1,n). And for series two it was (2,1),(2,2),...,(2,n) and so on until (s,1),...,(s,n).
So we have sn cards from which we can pick from and the total number of ways to do that is (sn C r) which will be our denominator.

To find how many ways we can collect r cards so that there is no type missing, I subtracted from the total ways of picking cards, the number of ways to pick our cards such that 1 type is missing, and to find this number we must first choose which type of card will be missing from our collected ones, we have (n C 1) ways to do that, next we'll collect r cards from the remaining ones normally. Since we removed one type, we are left with s(n-1) cards. So we can pick r cards in (s(n-1) C r) ways. Notice however that doing so removes the number of ways to collect r cards such that type 1 AND 2 are missing twice. Once when we removed the number of ways to collect r cards such that type 1 is missing, and once again for 2. This is the case for all pairs of numbers so we add them back (n C 2)(s(n-2) C r). But now we have added the number of ways such that 1,2 and 3 are missing too many times so we remove them and so on and so forth. This gives us the number of ways such that no type is missing.
[eqn]\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{s(n-k)}{r}[/eqn]
Apologies if I made any mistakes here. The book's solution is of course so much shorter and nicer. This is just to show I went about it.
>>15264370
Enumerative combinatorics?
>>15264495
Thank you a lot fren!
>I appreciate the quality posts
You make me blush. I don't deserve your praise.

>>15264626
for me? It would have to be f((x,y]) = x^y*y^x

>About to finish my master's thesis on PDEs
>Realize I know jack shit about PDEs
How the fuck did I make it this far

>>15265432
>masters
it's over

>>15265432
Oh I get it friend.
I've done courses on:
>Pde numerics, FDM, FEM, FVM, etc
>Modern linear pdes (stationary)
>Modern nonlinear pdes (stationary)
>Pde control with semigroups and so on
I still think I dont know squat about pdes, that whole part engineers and physicists do on solving certain pdes analytically with like separation of variables and the fourier transform and so on I know it pretty badly, I understand the methods and how to use them theoretically but I would be slow and clumsy in solving thise kinds of problems.
I also know Jack shit about characteristics.

What I know is basically abstract operator theory and how to show existence and uniqueness for certain kind of pdes.

I'm a mentally ill fucker who is tired of waging away at life. All I want to do is contribute something to humanity for me to justify my existence. What are some fields of math should I study that have a substantial need for more research/papers/studies/whatever the fuck

So what's the answer you fuckers

Is it worth it to ask a local community college if I can teach some of their math courses remotely
Still working through my bachelors so no degree, but proficient in most levels of math

>>15265564
D, obviously.

>>15265564
>>15265693
I guess I should explain why. Maybe you really need it.
Let q(x) = p(x) + 2, so by construction 3 is a root of q(x), and thus has (x - 3) as a factor.

Re-arranging, we have p(x) = q(x) - 2. What happens when we divide p(x) by (x-3)? Look at the right hand side. Since (x-3) is a factor of q(x) it divides cleanly with no remainder, whereas (-2) is obviously not divisible by (x-3), and that's your remainder.

Lets do it /mg/ how do we get our nobel prize

>>15266021
We invent a time machine and give Nobel a good beating with a tire iron until he agrees to make a Nobel prize in Mathematics to begin with.

You should understand gradients

https://people.math.harvard.edu/~knill/teaching/summer2011/handouts/34-gradient.pdf

>>15266030
Sad

>>15266021
>how do we get the jews to award themselves more prizes?

>>15260151
I don’t have straight answers, but some comments and thoughts:

I don’t know why you say
>But what we really want is to show that there is no injection out of the power set.
since the argument is really only about the surjection non-existence. Anything about cardinality and thier order relation comes later. That said, your question about injections on their own are interesting.

For some notation, f:X->Y, x1 in X is why X is inhabited, and R:=range(f), with R subset Y.

1. What certainly doesn’t work is just to throw the elements of R back to their preimage, and otherwise map to x0.
This is because R will not be detachable from Y.
Consider e.g. Y={0,1}^N which contains the zero sequence y0 that fulfills the (for Brower undecidable) predicate isZero(b):=forall(n in N). b(n)=0. Take X equal to Y\{y0} (which is inhabited, e.g. x1 the constant 1 sequence is in it). If we take f to be the identity on X, then given y in Y we don’t know if y=y0 and so we don’t know if we should map it back to y iteself or x0.
So I doubt it’s provable (although, again, I now focused on a particular inverse).

To 2 and 3, it’s clear that the existence of such an injection is impossible once we adopt Bernstein Schröder, since there’s also a trivial injection x\mapsto {x}, while there can’t be a surjection (and thus also no bijection) A to P(A).

I also know that Bauer showed that an injection from N^N to N realizable in some weird topos model or whatnot, but of course N^N can be made „smaller“ than P({0, 1}) is.

>>15265564
well A,B,C are obviously wrong. are you asking WHY D is true? are you asking if the question is wrong? consider the polynomial p(x)+2, which has x=3 as a root

>>15266157
hitler was a psychopath

https://www.youtube.com/watch?v=fcrexKS8kxA

=>

>>15265564
if p(x) is divided by x-3 then there exists q and r such that

p(x) = q(x)(x-3) + r(x)

since p(3) = -2 we have
-2 = q(3)(3-3) + r(x)

therefore r(x) = -2. it's fairly easy if you actually know the machinery behind this

Does anyone have the Kindle version of picrel?

>>15260581
That problem illustrates perfectly why I despise math quizzes. There is not enough information to solve. The only way to solve is by making implicit assumptions, which the authors claim are obvious, but which are not.

TLDR GT FKD

There are all those different pictures of different parts of the exterior next to the mandelbrot set at different levels of detail and they all have different colors. Does every image exist near the boundary of the mandelbrot set?
It seems like a dumb question. The mandelbrot set, or the boundary we're familiar with, is a chunk of the complex plane where every point is associated with a number (the "escape time") based on how it behaves under an iterated map. But if we modulo the escape time by the number of pixel colors (2^30) we get colorful pictures (this is how it's normally done).
A picture of the mandelbrot set is an m*n rectangular grid of lattice points and a coloring function modulo some p. What's the smallest rectangle, for some p, that can't be found?

Math level ?

given two sets X={1,2,...,n} and Y={1,2,...,m}, there are |Y^X|=m^n sequences of length n with values in Y.
Consider two values yA and yB (w.l.o.g. take 1 and 2).
Let F \subset Y^X be the space of sequences such that 1 and 2 never appear right after one another.
What's the cardinality of F?

Is there a more general theory of the sizes of the sequence spaces, given similar rules about what values are and are not allowed to appear after one another?

>>15262853
>∫_M
>∑⅟ₙ2
>eᴵˣ
>⅟ₙ

>>15266793
>1 and 2 never appear right after one another.
Isn't that another way of saying "1 and 2 can't both be in the sequence"?

When we need to prove that a function is continuous, we use the squeeze theorem and the absolute value of that function - f(0,0).
I get why we're using the squeeze theorem, but why the absolute value?

>>15266963
No, I'm speaking of direct sucession.
If n=10 and m=5, then
>3454321234
is not legal, but
>3454322134
is

>>15267072
Have you tried recursion?
Make three functions:
>f(L) = number of strings of length L that do not contain 12/21, and end in 1
>g(L) = number of strings of length L that do not contain 12/21, and end in 2
>h(L) = number of strings of length L that do not contain 12/21, and end in neither
Then figure out their recursive formulas by sticking new numbers at the end.
f = g in this case due to symmetry.

>>15265531
I got it by picking the correct courses in undergrad, so it's basically free.

>>15265545
>I also know Jack shit about characteristics.
Same, holy fuck I'm a fraud. Funniest thing is that I was a TA for our PDE course, thank god I got by.

>>15266793
For your case (1 and 2) let a(x) be the generating function for the number of all admissible strings (of any length > 0) that end in 1 and b(x) be the g.f. for strings that end in anything other than 1.

a = (a+b)*x + x
a*x ~ there is 1 way to go from a to a by adding another symbol to the end of the string.
b*x ~ there is 1 way to go from b to a by adding another symbol to the end of the string.
+x ~ account for the string "1" of length 1

b = ((m-2)*a + (m-1)*b)*x + (m-1)*x
(m-2)*a*x ~ there are (m-2) ways to go from a to b by adding another symbol to the end of the string.
(m-1)*b*x ~ there are (m-1) ways to go from b to b by adding another symbol to the end of the string.
+ (m-1)x ~ account for the strings "2", "3", ..., "m" of length 1

Solve for a and b then calculate [x^n] (a+b)
a = x/(1-m*x+x^2)
b = ((m-1)x - x^2)/(1-m*x+x^2)
a+b = (m*x - x^2)/(1-m*x+x^2)

You can just use partial fraction decomposition and get a closed form solution.
Wolfram tells me the answer is ChebyshevU[n,m/2].

In general, you can express your rule as an automata (finite state machine) then have a generating function for each node then calculate the relations (similar to above) based on the transitions.

>>15267068
Not sure if I misunderstand you anon, but it sounds like you need to recall your epsilon-delta definitions and that absolute value is distance.

>>15260151
1. It's not valid constructively because (as you suspected correctly) it implies EM
2. That case is indeed provable constructively (and so could be combined with Cantor's theorem to show that there is no injection PA -> A)
3. Yes this too works out constructively

Does anyone have suggestions for youtube channels that explains college/university level math clearly and concisely?
my professor is a fucking retard and if i continue learning from him i am going to fail. i want to supplement the lectures with actual information.

>>15266358
r(3) = -2

im trying to read about elliptic curves. can someone explain why the chord-tangent method is referred to as "doubling" a point? i get that it gives you a new point, but what motivation would you have for considering it "doubled"? i think i read somewhere that it had something to do with doubling the value of some integral, but there wasnt a ton of explanation in that regard. also, if anyone knows what "algebra" theyre referring to in the highlighted part of pic related, that would also be very helpful.

>>15268208
He's obviously doesn't understand the "machinery," just chink style mechanized plug n' chug.

>>15267926
Yeah, you understood me well, thanks Anon

>>15266411
Hmmm, skill issue. Git gud, anon.
No need to make excuses because you can't solve something. The author in question is by the way Martin Aigner. The assumptions needed are indeed extremely basic and you can see from the replies there are people who managed to solved it and some who got close to the real answer(not you though, lol).

>>15267946
Got details on how 3 is proven?

>>15267504
Cool I'll try to follow that explicitly later

>>15268086
MIT channel

>>15266631
Interesting question but maybe just focus on the 2*2 grid example for all p. I don't see why all 2*2 examples must exist for arbitrarily large finite p.

>>15268668
If [math]f : PA \to A[/math] is injective, then taking [math]D=\{f(X)\mid X\subseteq A,f(X)\notin X\}[/math] gives us [math]f(D)\in D\leftrightarrow f(D)\notin D[/math], which is the familiar contradiction.

>still getting filtered by Yoneda lemma
Why is category theory so hard, bros?

>The Gaussian function cannot be represented by elementary functions.
>The Gaussian function is smooth.
>The Stone-Weierstrass theorem says that every smooth function can be represented by elementary functions.
What am I missing?

>>15268848
Stone-Weierstrass only applies on a compact interval

>>15268848
>>The Gaussian function cannot be represented by elementary functions.
It is literally the concatenation of elementary functions.
>>The Stone-Weierstrass theorem says that every smooth function can be represented by elementary functions.
Can be represented *as a limit* of elementary functions *on a compact set*.

>>15268863
>It is literally the concatenation of elementary functions.
I assume he means the integral thereof and is just dumb.

>>15268230
it actually comes from the group operation on the picard group. just so happens there's an isomorphism from picard group to hyper elliptic curves in the genus 1 case, but not so for higher genus. genus 1 hyperelliptic = elliptic curve.

just so happens that riemann roch theorem is ez to explain as purely geometric but the chord and tangent is not the reason for it.

check out washington chapter on divisors, section on riemann-roch for more info

>>15266833
Ok, 4chan filtered some unicodes, but everywhere else i tested things like the exponents worked.

>>15268961
Fixed the integral one:
>∫ₘdω = ∫ₔₘω

Let U be the subcollection of binary sequences s in N->{0,1} that are not computable. U is uncountable.
Is there a predicate P(s) such that {s in U | P(s)} is nonempty but finite or countable?
Everything I think of either doesn't weed out enough, or weeds out too much (everything fulfilling the restriction would also be computable)

>>15268671
It will get more complicated for more complex rules.
It should be possible to get a formula for any regular expression.

You can just represent your finite state machine as an adjacency matrix M = a(i, j) = number of symbols that transition to node i from node j.
If you let v(x) be the vector of generating functions f_i for the nodes, the equation defining v(x) is:
v(x) = x*Mv(x) + v(0)
v(0) should just have a 1 in the component representing the initial empty string node.
Solving gives v(x) = (1-xM)^(-1) v(0) = (1 + xM + (xM)^2 + ... )v(0)
The coefficient of x^n describing the length n strings is just (M^n)v(0)

>>15269002
Fix an uncomputable sequence x.
Let P(s) be "s differs from x in finitely-many places".
This gives a countable set.

>>15268810
nice

>>15269171
Curious.
Did you make hat up yourself or does this find any use.
I suppose what happens here is that you formulated a set which is but computable w.r.t. x.

>>15269112
I lost track of which number equals the final answer, counting the number of strict now involving the subsequence 12
A sum over all v(n)? Or is it already one of the components. Because the generating function a of "ending or not ending in 1" alone doesn't pin down the final number, does it?

Also, do you got a reference? Is this in generatingfunctionology?

>>15269207
My pic explains the process. I omitted the generating functions since you seem unfamiliar with them.
The matrix explanation comes from https://en.wikipedia.org/wiki/Adjacency_matrix#Matrix_powers
You can also read some of the citations in https://en.wikipedia.org/wiki/Regular_language#The_number_of_words_in_a_regular_language

I recommend getting Flajolet & Sedgewick.
It is free.
https://ac.cs.princeton.edu/home/AC.pdf
Page 57 of the book (71 of the pdf) basically gives the answer I gave.

>>15269207
My pic explains the process. I omitted the generating functions since you seem unfamiliar with them.
The matrix explanation comes from https://en.wikipedia.org/wiki/Adjacency_matrix#Matrix_powers
You can also read some of the citations in https://en.wikipedia.org/wiki/Regular_language#The_number_of_words_in_a_regular_language

I recommend getting Flajolet & Sedgweick.
It is free.
https://ac.cs.princeton.edu/home/AC.pdf
Page 57 of the book (71 of the pdf) basically gives the answer I gave.

Big Nose™ Home Math Companion
https://www.youtube.com/watch?v=xM2D9egIpyg
>it isn't the way you calculate
>it isn't the style of text in your proof
>you see, math is a feeling
>that nobody really understands
>unless you're deep into the vibe
>deep into the vibe of math

>>15269360
right right

>>15269188
>Did you make hat up yourself or does this find any use.
I did. Because it's easy to come up with.
It's the same idea as the classes of R/Q. (R the reals, Q the rationals)
R/Q is uncountable, but elements of each class differ from each other by a rational number, i.e. the set of differences, Q, is countable.
In sequences over {0,1}, the set of of "finite" sequences (all entries are zero except finitely-many) is countable. These take the role of Q.

>>15269407
Okay, nice you came up with it yourself.
The R/Q situation might have some similarities, but it's still quite different. The "finite" sequences in the last line are all computable and so not in U anyway, and the R/Q construction also doesn't talk about singling out any particular x. I wouldn't have made that jump.

>>15269431
>The "finite" sequences in the last line are all computable and so not in U anyway
Maybe you misunderstood me, but of course these aren't in U; adding them to x gives you elements in U.
The set that you get from applying the predicate is the set of x+f (entries mod 2, of course), where f is "finite", and each of these is uncomputable.

>>15262665
Thanks for the response, sadly I haven't had any luck with bookstores yet and my math dept doesn't have what you described (luckily they lend some books for up to 1 year's time). As for Dover, how do they compete with the canon AMS/Springer books? I'm not quite there yet, but I doubt they have anything on the level of Hartshorne, for example?

ACK I wanna study math in my free time but I have hyper ADD and don't know where to start... recently made it through half of Calc 3 but got bored at multiple integrals.... WHAT DO

Metaphysical Starship™ by Home Math Companion with New Capability Evolution Assistant
>Only $19.95!
>Operators are standing by!
>Order now and we'll throw in a free t-shirt and swag bag!
>Home Math Companion
>*Where fine math products are sold!*

>>15269377
I think I have the matrix transposed improperly but the vectors are still good.

Anyone know of a non-meme book on quantum computing? Sounds like an interesting subject, but most sources I find are either dumbed down or papers that don't introduce the field.

>>15269721
Did you try Nielsen and Chuang? It's the standard textbook.
For me, I studied from an edX course offered by MIT, taught by none other than Peter Shor himself (who came up with Shor's algorithm).
This is the course: https://courses.edx.org/courses/course-v1:MITx+8.370.1x+1T2018/syllabus/
I think the course might not show up in the search results anymore, idk.
Nielsen and Chuang is what they were using for the course.

There are two more parts to the course:
https://courses.edx.org/courses/course-v1:MITx+8.370.2x+1T2018/syllabus/
https://courses.edx.org/courses/course-v1:MITx+8.370.3x+1T2018/syllabus/

Opal Dragon™
https://www.youtube.com/watch?v=OBC5c6jnmtk

>>15269747
Great! Thank you for the sources, Anon.
Also, lucky you being taught by Shor. Is he a good teacher?

>>15265248
>for what it means to do.
Teach Ameriniggers?

>>15269953
I took the course almost exactly 4 years ago, so I don't remember much, but I think he was good. Maybe also a little awkward.
I used what I learned from this course to win in an online international quantum computing competition lol.

Quantum mechanics is actually implying math it self doesn't exist, and is in fact just a construct used to describe macro-level interactions.

Quantum mechanics implies that rational numbers don't exist, just probabilities. As a side-effect of this, quantum mechanics can also explain qualia. How? Quantum mechanics indicates that a quantum measurement of a system is dependent on the reference frame of an observer. A measurement in quantum mechanics is generally represented as a probability distribution. Classically one would believe that this probability distribution approaches one answer that is the correct answer, but what it's really indicating is that there are multiple correct answer depending on ones frame of reference.

The location of an atom can never really be defined because "location" is just a concept with no real meaning. When a macro level object moves around, the probability function representing the location of the entangled atoms approaches finity, but will never reach it. Things only appear to have finite structure, but quantum mechanics is indicating that an objects mass/energy is neither finite or infinite, nor can it be represented as a rational number.

Even calculus tried to tell us that true curves don't exist and that "curvature" is just something we perceive.

>>15270158
Brainlet post. Math exists independent of the real world, and its uses for modeling the real world don't bind it.

>>15270158
>rational numbers don't exist, just probabilities
I don't follow at all

>>15268408
at no point did you make an argument about how the coupons are distributed in the packages.

All the responses are INCORRECT. Die alone.

>>15270219
There's one coupon in each package. It says that in the problem.

>>15270158

I'm fucking tired of being directionless, which areas of mathematics can I contribute to in research?

>>15270219
Right there it says that the company puts one card in each of sn packages. There are n different types of cards, the company produces this series s times. We have sn cards and there is one card per package. From the image itself "...and puts one coupon in each of sn packages.". I guess 3 lines of text was more than you're used to so you got confused. Honestly the way you're talking is so sad and pathetic I feel bad calling you out on your excuses.

Some of the responses are correct, some are not(but all were good attempts). But none of them are your responses because you can't even read a simple problem and at least attempt a solution. You'd rather make excuses about how you hate "math quizzes". Just sad for you.
>>15270226
He can't read, fren.

>>15270587
>directionless
I'd recommend linear algebra then

Would Amann-Escher be worth going through fully (all 3 volumes) as a physics student? Is it useful or should I focus on other areas.

I doing analysis atm and we seem to be skipping around, doing things much less in detail than Amann-Escher. Thoughts?

>>15268779
thanks

>>15270587
theyre testing the effects of polynomials in the bloodstream. you might lose your hair but you get fifty bucks

>>15270009
>I used what I learned from this course to win in an online international quantum computing competition lol.
Nice!

> Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".

This bothers me a lot. Source

https://en.wikipedia.org/wiki/Model_theory

Does it mean math is not rigorous?

>>15270158
>qualia
Stopped reading here. Wrong thread philosophy history faggot. You are not studying nor do you understand math or logic. You will never be a philosopher, you are not philosophizing, but simply studying the history of philosophy and effectively foreign literature with various errors and misconceptions due to translations.

>>15270580
what this means is that we can render any system of equations with an unknown calc map variable, i.e. a compiled program that must pass a test suite such as
>e(a[i]) = c[i] for all i <- { 1,...,n }
where the variable e is unknown as a system of equations in the monoid A x E
>(1,e) (a[i],1) = (c[i],e^a[i]) for all i <- { 1,...,n }
in short: the solution to systems of monoid equations is powerful enough to cover all algorithmic computation
here the calc is taken as a model of discrete algorithmic step-by-step computation where the domain is a free monoid or a submonoid of a free monoid

>>15260581
Nice book drawing fren.

Consider the set formed of the canonic basis I_n, its opposite -I_n and of every vector whose coordinates are either 1 or -1.

I want to find a vector v that is as far as possible from its closest neighbor in S (in terms of angles).
How should i proceed ?
Example in dimension 2:
http://sketchtoy.com/71046449
here it is easy to find such a vector (i only drew one but there are many others). In dimension >= 3 however i'm not sure.

Textbooks vs. lecture notes? I feel like texts are overrated and far too wordy, whereas lecture notes are succinct but lack exercises, which is the main benefit of textbooks. How wrong am I?

>>15270875
No, swallow the formalist view and put proof over semantic arguments. (Semantic as in proven by model theoretical means.)
It's fairly difficult to disentangle chains of justification if you involve model theory results. Usually a model is a vast set and they are not afraid of making their case via Zorn's lemma. It's not super foundational in a sense.

>>15271813
It's said you're fairly wrong. I don't think textbooks are too long, but what the real case against lecture notes is, is that books go through editorial and have hopefully less errors or vague formulations. Lecture notes are just some guy patching together something for, well, a 16 hours of lecture.

>>15271862
That makes sense, thx

I just had my first real analysis class today and the professor said he was going to be using Rudin as the main book for the class, but like... Isn't Rudin a book for graduate students? Is the guy out of his mind? I'm kinda scared now...

This might be a stupid question, but:

If [math] 0\rightarrow X\xrightarrow{f} Y\xrightarrow{g} Z\rightarrow 0 [/math] is a short exact sequence in some Abelian category [math] \mathcal{C} [/math] , then in general are [math] \{ F \in \mathrm{Hom}_{\mathcal{C}}(Y,X) : F \circ f = 0 \in \mathrm{End}_{\mathcal{C}}(X) \} [/math] and [math] \{ G \in \mathrm{Hom}_{\mathcal{C}}(Z,Y) : g \circ G = 0 \in \mathrm{End}_{\mathcal{C}}(Z) \} [/math] ?

>>15272024
Are they isomorphic, sorry forgot to write "isomorphic" at the end

>>15271908
Rudin is like… for undergrads
If a professor was using rudin in a grad class I’d be like… OMG WHAT!

>>15271908
Grad students at low end schools.

If [math] 0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 [/math] is a short exact sequence of (left) modules over some ring [math] R [/math] , then in general are [math] \{ F \in \mathrm{Hom}_{\mathcal{C}}(Y,X) : F \circ f =0_X \} [/math] and [math] \{ G \in \mathrm{Hom}_{\mathcal{C}}(Z,Y) : g \circ G =0_Z \} [/math] isomorphic as Abelian groups?
What about as [math] R [/math]-modules, when [math] R [/math] is commutative?

>>15272100
I wish I could learn math just to troll people like you

>>15271908
It's written for undergrads but your professor is a retard

>>15272109
>I wish I could
Who says you couldn't?

>>15272114
geeeeze dude i dunno fucking o rings and sets and cardinality and shit, I just don't know.

>>15272119
>15272119
meant to post this

If [math] 0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 [/math] is a short exact sequence of (left) modules over some ring [math] R [/math] , then in general are [math] \{ F \in \mathrm{Hom}_{R}(Y,X) : F \circ f =0_X \} [/math] and [math] \{ G \in \mathrm{Hom}_{R}(Z,Y) : g \circ G =0_Z \} [/math] isomorphic as Abelian groups?
What about as [math] R [/math]-modules, when [math] R [/math] is commutative?

>>15272130
okay you're getting me good

>>15272147
What?
Also pls help or go away

>>15271491
Let R be an abelian unital ring. We can form the monoid ring R[A x E], formal linear combinations of "calculating elements with output" a.k.a. elements of A x E with coefficients in R.
Example. Take A = natural numbers and R = integers.
This gives a formal notation to "traditional Hollywood" or "discrete probability class" style notes tracking the evolution of states of some process such as
>take an n sided die
>write down all of the numbers in { 1,...,n }
>repeat: throw the die, cross out the number showing on the top face
>stop when all numbers have been crossed out
How do we do this?
Take A to be the free monoid generated by d[i] for i <- { 1,...,n } and R to be the integers.
We are interested in the "bingo" calc e : A -> A. It will emit d[i] the first time it is seen, then never emit it again. Once all d[i] have been seen, e is the trivial map on A. So, e simply gives the unique occurrences of its argument in the order given. We can now take a very simple probability calculation effort that counts observations and calculate by hand
>(1,e)( Sum[i=1]^n (d[i],1) )^k.
Tabulating observations after k throws yields n^k total observations.

>>15272197
Why are you taking basic notions from abstract algebra and combining them with random words like "Hollywood"?

I would have studied mathematics if I hadn't chose chemistry,what could have been

>>15272197
So if we want to apply a Python class such as
>>15244337
then we first have to parameterize E because it's too big. We want to select e^A where e is the bingo calc map above.
Now let p : P({ d[i] : i <- { 1,...,n } }) -> E be p(j) = e^{ Prod[i=1,d[i]<-j]^n d[i] } = e^{ Prod j }. We have p[P({ d[1],...,d[n] })] = e^A.
In Python, we can simply use a subset of { 1,...,n } to represent an element in the domain of p, our parameterization map.
So, the empty set will stand for e and { 1,...,n } will stand for the trivial map.
A tuple of integers in { 1,...,n } can stand for an element of A, with product concatenation of tuples. Let R be the integers.
Assignment: adapt the Python code listing above for elements of R[A x e^A] using the parameterization map above.

If I gave on average, 3 hours a day to this list, how long would it take me to complete? I start my undergrad in Math next fall. I'd like to be over-prepared. Is doing this list a feasible route in that time-frame?

>>15272297
>13 books to get to the level of a 1st semester undegraduate
>>15272297
https://www.youtube.com/watch?v=XbOx8TyvUmI

>>15272330
does it make sense to read so many books at once? I feel like I would forget the stuff from the first books by reading the latter books

>>15260487
I always loved math and studying it but i hate how genetics-based it actually is, it takes me 2x more time than for the average person to understand a concept even though i study regularly. I don't think i'll ever be able to learn a grasp of what a math major learns during his degree even if i study it for years

>>15272473
It's partly about exposure. Isn't math cumulative anyway?

>>15272528
no, it's accumulative

Home Math Companion™ theorem accumulator/humidifier now on sale!
Check twice daily for new theorems.
WARNING: mutant creatures have been known to appear suddenly in theorem trap. Exercise caution when opening device.

>>15272246

https://www.youtube.com/watch?v=XaYSz1IWOxY

What is the correlation between No Fap and Mathematics?

>>15262065
My thoughts are, why do you need any further derivations?
Isn't the ratio of powers enough to state the limit?
T. Math noob
If someone can teach me, I'll be thankful

>>15271515
Thanks fren but that's not my drawing, I just edited the book in her hands(unless that is what you were referring to?) . And hey, if you want something like this, let me know. I could do one for you with whatever book you wanted. Of course in exchange I ask you attempt my problem afterwards! It's okay if you don't end up solving it.
Apologies if I sound incoherent here, I woke up 10 minutes ago or so.

>>15261909
Looks good may I move in
>>15262045
Don't be stingy
>>15262254
France punches above its weight to a ridiculous degree but the US is still the best

>>15272130
Sorry I keep messing up the notation. Correct version:


If [math] 0 \rightarrow X \xrightarrow{f} Y \xrightarrow{g} Z \rightarrow 0 [/math] is a short exact sequence of (left) modules over some ring [math] R [/math] , then in general are [math] \{ F \in \mathrm{Hom}_{R}(Y,X) : F \circ f =0_X \} [/math] and [math] \{ G \in \mathrm{Hom}_{R}(Z,Y) : g \circ G =0_Z \} [/math] isomorphic as Abelian groups?
What about as [math] R [/math]-modules, when [math] R [/math] is commutative?

Homotopy Groups

how long will it take a retarded undergrad like me to understand any paper in this: https://arxiv.org/list/math/recent

>>15273089
5 years

>>15273131
What books would I have to work through to cover all or most of this?

>>15271908
>Isn't Rudin a book for graduate students?
real analysis is an undergrad topic, I mean the field has been dead for centuries, it's a foundation that you study quickly and early

>>15271576
still stuck

>>15262065
The absolute value there says me it is not diferentiable. I am too lazy to do this but try to find the partial derivative when [math]y = -1[/math] and find the limits of the definition of derivative.

>>15272119
>>15272122
who says you can't just read this?

>>15273415
It starts with Analysis by Schwartz or Zorich so just start with that I guess

>>15273415

>>15272326
Anon, if you don't solve this, then I'm going to make a cute little pdf with code listings and formulas and references to undergrad texts in math AND COMPUTER SCIENCE and I'm going to SCREAM! And I'm going to post that little pdf in /mg/ and you're going to HEAR ME SCREAM when you read it!
https://www.youtube.com/watch?v=wNXMfI1D8Xw

>>15274368
Notation: Za means integers
Hint: in
>>15244337
the 'nd' variable collects like terms using the hash table in Python's "dict" data structure for high performance
If we have any monoid A and we're using a hashable data type to represent elements of a subset of A (such as the integers (elements of the set of integers) that can be represented using Python's big integer) then we can use the dict object to collate like terms
Now switch back to the context of the problem where we are trying to represent elements of Za[A x e^A] and adapt the Python class to calculate products, scalar multiples, and sums of elements.
This means using the "frozenset" data type (a hashable object) when it comes time to mint a new element of A x e^A because frozenset is hashable and the Python "set" data type isn't.
Remember: a tuple of Python data types is hashable as long as each item is hashable. So, we can exploit a "Python-math pun" and use a Python pair (L,R) to represent an element of a set A x B that is, as a mathematical matter, technically a set of pairs.

I'm retarded, what does it mean when you divide a probability by a probability

>>15274425
here's picrel

>>15274451
Think of P(A) as the new sample space. Use a figure where you have the events A and B within the sample space.

>>15274362
thanks

>>15272122
Every book there is a waste of time.

Can you do integral transforms without postulating that the class of natural numbers is a set?

Had a thought regarding something I lack the understanding to even begin learning about.

Given some infinite degrees of freedom, the thought came up on asking whether one could apply the concept of inversion to a degree of freedom itself, a dimension. That is, not points on a dimension, but the degree of freedom itself given an infinitely-dimensioned topography. Topographical solutions don't seem to fit in the concept of infitely many degrees of freedom. All I could reason was the inversion of any given dimension of a given infinitely-dimensioned topography ends up being the same dimension with respect to infinite dimensions.

I'm clearly lacking the requisite technical knowledge. I'm probably asking something stupid as a result.

>>15275205
>concept of inversion
What do you mean by that? What is it you want to invert?
And why do you speak of topography, that term is not much used in math anyway

>>15275225
>What do you mean by that? What is it you want to invert?
The dimension itself. I can conceive of doing so representing finite-spaced finite degrees of freedom and a given topographical representation, as there'd be a relative axis of symmetry and the corresponding topography of that space would necessarily change. So a kind of "meta-space" within which finite spaces could be said to have such features.

I don't know how to explain it. I just had the thought and couldn't figure out how to even ask the question.

>>15263576
>((1+x)^s - 1)^n
An expression with an expansion corresponding to all possible choices of boxes, further subtracting and re-adding choices which don't contain some subset of the n ticket types, in an inclusion/exclusion way.

>Expanding the outer exponent of ((1+x)^s - 1)^n with the binomial expansion
I think you're saying that you are now combining all the inclusion/exclusion terms together based on how many levels of inclusion/exclusion a term was listed under.

>the x^r coefficient
After expanding the entire expression, these are the choices resulting in r total tickets.

>then extracting the x^r coefficient of each (1+x)^(s*k) gives the final form
I assume "extracting" means that you did a further binomial expansion of each of those inclusion/exclusion terms, took the x^r level coefficients, and rearranged them somehow. I don't understand how you got your solution
>Sum[ (n C k)*(sk C r)*(-1)^(n-k), {k,0,n}]
from that, though. The set of coefficients is going to vary in size so that seems hard. Is there a general way to solve that problem or is it case specific algebraic witchcraft?

It seems that the k in your solution can be viewed as the level of inclusion/exclusion that you're on. Then the solution would correspond to listing all box choices which avoid k ticket types, and multiplying that by the number of ways to choose k ticket types to ignore. Then doing inclusion/exclusion as the number of ignored ticket types k goes up to n.

Are you saying that the algebraic witchcraft manipulated the binomial expansion coefficients you had into polynomials corresponding to those choice functions? Sorry, I'm not trying to be combative. I just don't know if I'm missing something.

How should i proceed to solve pic related ?

>I can only study pure mathematics for a few hours a day before my brain tries to kill itself and loses interest
is it over

>>15275308
>I assume "extracting" means that you did a further binomial expansion of each of those inclusion/exclusion terms, took the x^r level coefficients, and rearranged them somehow. I don't understand how you got your solution
((1+x)^s - 1)^n = Sum[(n C k)*(1+x)^(sk)*(-1)^(n-k), {k,0,n}]
You can interpret this expansion as an inclusion/exclusion principle.
the x^r coefficient of this is:
Sum[(n C k)*(sk C r)*(-1)^(n-k), {k,0,n}].
No fancy rearranging.

>>15274630
Enderton and Apostol are great books, imbecile.

>>15272122
which of these books will make me a prodigy?

>>15275806
take a break, go for a walk in the sun, eat a meal and make yourself something to drink. Sit down to work only when you are refreshed and relaxed again

Why do people here hate "Intro to Proofs" books? I've been reading Velleman and it's really good.

Any good books on differential geometry (differentiable manifolds, tangent and cotangent spaces, differential forms, Sard's Theorem and Riemann surfaces) besides Tu and Lee?

>>15276735
Do Carmo.

>>15276775
Differential Geometry of Curves & Surfaces
or
Differential Forms and Applications?

>>15260487
Why is the Golgi’s staining method mechanism still unknown?

Is this correct? It doesn't look quite right to me, but maybe im just cloudy with my definitions...

>>15276856
Riemannian Geometry.

Markov's principle

[math] \forall n. \big(P(n)\lor \neg P(n)\big)\implies\big( \neg \neg \exists n. P(n)\implies \exists m. P(m)\big) [/math]

Yay or nay?

>>15276889
Fuck off.

>>15276864
As usual, no, although at least it's fairly close this time.

>>15276864
I think that we should have a version of GPT that spouts categorical nonsense to prove whatever it's asked. Something like "this is a trivial consequence of the adjoint functor theorem applied to the coslice category along [math]\kappa[/math]-small limits where the pullback square commutes on the nose". You can't even say it doesnt make sense because there is always a way to make sense of such catego-astrological bullshit.

We have to do it. Now.

r8

>>15273131
I don't get it, is this a troll or a geniune guide on how to stay ahead every year of education and get a phd quickly

>taking abstract algebra
>its an 8 week long course
>professor is known to be a hardass and tough grader
>midterm and final worth 70% of total grade
>I'm super rusty on proofs and only got a B on intro to proofs a year ago

hey fags, mathcel here. what courses/resources would you recommend if I really want to get balls deep in the mathematics behind neural networks? the most advanced courses I've taken are 3d calc, matrix algebra, some probability courses, and an intro to pure math/proofs course

>>15276997
Starting early, if I put in around 30 hours or more before class starts on monday, I won't be blindsided and humiliated as hard...

>>15277010
see >>15273131

>>15277010
AI needs a lot of measure theory, so read through Fremlin's books on that

>>15277010
You have the fundamentals to understand how methods like older methods like gradient decent and perceptron.

For emerging techniques like Transformers, areas that might be useful are Natural Language Processing, Numerical Analysis, and Information Theory. Or read their paper.

>>15276893
Is this a yay or a nay?

>>15277043
>chapter 0: prove fundamental theorem of arithmetic, prove every common multiple of a and b is a multiple of lcm(a,b)

>>15276997
Show me your Intro to Proofs test.

>>15277502
Why would I have that test on hand? I'm paranoid enough to assume that you're the professor teaching my course, he seems like someone that would browse here. Its also why you believe it likely that I keep a test from a year ago.
I'll post one exam if you tell me your intentions.

>>15277696
>'m paranoid enough to assume that you're the professor teaching my course, he seems like someone that would browse here
so what are you worried about?

>>15277768
That I'll get bullied. I am a very, very shy person. Even if I am dumb, I just want to enjoy the math I can and maybe earn a math degree you know. Its already bad enough he is making us do video introductions, I might have a heart attack.

HOW THE FUCK DOES THE MONTESINOS TRICK WORK AHHHHHH

Can ChatGPT do math proofs yet? How long before it reemplaces mathematicians?

>>15277895
Can't even do basic stats. Its just a glorified chatbot. If it could do even the most basic proofs (rather than regurgitating anything it DOES get right) it could just compute fast enough to write and error check its own code and create a better AI.

>>15277895
If you think it can do proofs then you really don't understand what the algorithm is doing.

Is there a name for impossibility proofs like Russel's paradox where you define some counterexample with recursive structure that breaks things? The only other example I know is the halting theorem in computer science.

>>15276993
It's a meme from some russian professor but on /mg/ its "how I would redo it if I could go back in time" cope.

>>15272054
>>15272054
That's the undergrad Rudin book and is used for courses at some schools. It's terse and challenging but at the end of the day its just real analysis. You DONT have to only read that books. Free to check out Abbott or Lang for alternative explanations of the material.
Rudin also has "Real and Complex Analysis" and "Functional Analysis" which are the ones used in graduate school.

>>15277980
They're all diagonal arguments, and Godel's theorems are as well.

>>15277696
I just wanted to use it to test my own knowledge on the subject, I'm studying proofs as well right now.

>>15278009
Thanks
I think I can see the similarity with the card R > card N argument

I'm looking for a statistics term.

Let's say we have a state machine with states A and B. If in state A it has a 60% chance to go to state B and 40% chance to stay on itself. If in state B it has an 80% chance to go to state A and a 20% chance to stay on itself.

There's a formula for finding what percentage of the time the state machine will be in A or B, but I can't remember what these kind of problems are called. I've already tried searching for stuff like stochastic flows. Could have sworn this had something to do with stochastics.

im trying to geometrically derive the inverse sin function for ellipses. i managed to get an integral, but it doesnt really look like the elliptic integrals ive seen. does anyone know if theres a change of variables i have to do next? or maybe i made a mistake somewhere?

>>15276735
Godinho & Natario is Do Carmo's but better. There's also two geometry books by two different guys named Lee, so if you're thinking about John Lee's book, check out Jeffrey Lee's.

>>15277010
It's just linear algebra.

>>15277991
I see, makes sense thanks

>>15260581
GPT-4 answer:
To calculate the probability of a customer getting a complete set of n different coupons after collecting r packages, we can use the concept of the inclusion-exclusion principle. Let's denote the number of ways to collect r packages without getting the complete set of n different coupons as A.

Then, the probability of getting a complete set of n different coupons is 1 - (A / total ways).

Total ways of collecting r packages are simply n^r, as there are n choices for each of the r packages.

Now, we'll calculate A using the inclusion-exclusion principle. Let A_i be the number of ways to collect r packages such that coupon i is missing (1 ≤ i ≤ n). Then, A = Σ A_i - Σ (A_i ∩ A_j) + Σ (A_i ∩ A_j ∩ A_k) - ... , where the first summation is over all i, the second summation is over all distinct pairs (i, j), and so on.

Notice that A_i = (n-1)^r, as there are (n-1) choices for each of the r packages, and there are n such terms in the first summation. Similarly, for the second summation, there are (n-2)^r ways to choose r packages such that both coupons i and j are missing, and there are C(n, 2) such terms. Continuing this pattern, we have:

A = n(n-1)^r - C(n, 2)(n-2)^r + C(n, 3)(n-3)^r - ... + (-1)^n C(n, n)0^r

Now, we can calculate the probability of getting a complete set of n different coupons:

P = 1 - A / n^r

P = 1 - [n(n-1)^r - C(n, 2)(n-2)^r + C(n, 3)(n-3)^r - ... + (-1)^n C(n, n)0^r] / n^r

This formula gives you the probability of obtaining a complete set of n different coupons after collecting r packages.

>>15278604
Can you ask GPT-4 to create a definitive /mg/ book list?

>>15278619
Here is an /mg/-approved math book list, organized by subject. This list is not exhaustive but includes some popular and well-regarded texts that are often recommended in math discussions on 4chan.

Algebra
Basic Algebra by Nathan Jacobson
Abstract Algebra by Dummit and Foote
A Course in Algebra by E. B. Vinberg

Number Theory
A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen
Algebraic Number Theory by Jürgen Neukirch
Analytic Number Theory by Donald J. Newman

Calculus
Calculus by Michael Spivak
A Course of Pure Mathematics by G. H. Hardy
Calculus: An Intuitive and Physical Approach by Morris Kline

Real Analysis
Principles of Mathematical Analysis by Walter Rudin
Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert

Complex Analysis
Complex Analysis by Serge Lang
Complex Analysis by Theodore W. Gamelin
Functions of One Complex Variable by John B. Conway

Linear Algebra
Linear Algebra Done Right by Sheldon Axler
Finite-Dimensional Vector Spaces by Paul R. Halmos
Linear Algebra by Georgi E. Shilov

Topology
Topology by James R. Munkres
Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa
General Topology by Stephen Willard

Differential Equations
Ordinary Differential Equations by Vladimir I. Arnold
Partial Differential Equations by Lawrence C. Evans
Differential Equations and Their Applications by Martin Braun

Probability and Statistics
A First Course in Probability by Sheldon Ross
Probability: Theory and Examples by Rick Durrett
All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman

Geometry
Geometry: Euclid and Beyond by Robin Hartshorne
Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo
A Comprehensive Introduction to Differential Geometry by Michael Spivak (multi-volume)

Should I start studying maths in Mathematical analysis?

>>15260487
Quick question for the PDE/Functional analysis chads in here:

Suppose I have a sequence [math](u_n) \subset X[/math] such that [math]u_n \rightharpoonup x[/math] in [math]X[/math] with [math]X[/math] a reflexive, real Banach space.

Furthermore suppose that there is some other Banach space [math]Y[/math] also real but not reflexive, such that [math]Y \xhookrightarrow^c X[/math] (compact embedding).

In this situation, we have that [math]u_n \rightarrow u[/math] in [math]Y[/math]. That is, weak convergence becomes strong convergence under the compact embedding.

Why?

>>15260487
https://mathchan.org/math/thread/346
>So, I have a cousin who wants to win medals in math competitions, especially IMO. Post resources, guides and tips for olympiads.
his prep level: he's 12yo(7th standard), has completed mathematics books upto the 10th standard level. What should be his target next? And how do I help him clear doubts? We don't have decent teachers where we live, and the internet doesn't help much

>>15278604
Wow, impressive. I chatted with new ChatGPT today thanks to my friend but I kept it bried and asked non-math questions.
>To calculate the probability of a customer getting a complete set of n different coupons after collecting r packages, we can use the concept of the inclusion-exclusion principle. Let's denote the number of ways to collect r packages without getting the complete set of n different coupons as A.

Then, the probability of getting a complete set of n different coupons is 1 - (A / total ways).

Wow, so far so good.
>Total ways of collecting r packages are simply n^r, as there are n choices for each of the r packages.
well, I disagree here. We can't simply pick the same card again and again, can we? If we had n=2 s=2 r=4, this would say that we have 2^4=16 ways of picking our cards while in reality we clearly have only one way of picking them.
>>15278632
Not bad! I'm just curious, is this really tailored towards /mg/? It mentions 4chan, did it really look at the most common book names that appear here? Really interesting.

I'm excited for my research group's seminar today frens

>>15278690
No one replied but I found a reason for this myself, its rather simple:

The inclusion map [math]i:X \rightarrow Y[/math] is linear and compact, by definition of the compact embedding. Thus, its strongly continuous, that is, for a sequence [math]u_n \rightharpoonup u[/math] in [math]X[/math] we have [math]I(u_n) \rightarrow i(u)[/math] in [math]Y[/math].

Beautiful!

Hey/sci/. I posted about a year ago on this, but just wanted to update you: I solved the Collatz Conjecture. Submitted my manuscript today to JAMS. Will let you know how it goes but I am as certain my proof is correct as epistimology lets me be.

game-theoretic statistical differential geometry

>>15278883
post summary of proof

>>15278685
Probably.

>>15278903
Not until I'm deeper into review, sorry.

>>15278911
Here's my review.

>>15279038
Fair enough. Jet fuel cant initiate a total collapse of WTC7 but go ahead and screenshot my først post. Collatz is solved.

>>15277316
>covered first 3 chapters of Gallian along with youtube lectures all night last night and since this morning.

This course is going to get crushed as long as I don't forget anything during exams and start crying.

Any protips for acing abstract algebra? Going to make a collection of practice exams from other universities some time tonight.

>>15278731
You can do it like me and abuse poe's (the AI made by Quora) limited access to GPT-4 by making a new account with a temp mail for each request. Note that the response length is limited, so it's still far from ideal.
As for the /mg/ question, I made sure to clarify that it is the math general on 4chan's science and math board, but nothing beyond that. I haven't tried the same prompt multiple times or to ask the same thing without the 4chan stuff to check, since, you know, I have to create a new account for every prompt.

I've seen some people here talk about constructive logic/mathematics a lot. Where can I learn about this? Are there mathematical logic books that cover or even focus on constructive logic? I checked some of the "big names" but they don't seem to mention it. Any other books I should be aware of that do mathematics in a constructive setting?

>>15279362
I see. Well it'd certainly be interesting what would happen if one didn't mention 4chan. But I'm not curious enough to find out personally. Also how is GPT-4 and chatgpt different. Chatgpt is the one I talked to I guess, it's what my friend sent me as a link.

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