/sci/ - Science & Math

>>16035182
latex question, why use the matrix environments when array exists?

>>16035182
what be a good curriculum for middle school-level?
haha

what's the threshold for "real" maths again? topology?

>>16035800
Abstract algebra

>>16035707
Serge Lang's Basic Mathematics. Good book by a well known author

>>16035800
Anything that uses proofs. Analysis, Abstract Algebra, Number Theory. Typically Calculus and some proofwriting experience is considered sufficient for getting into "advanced" topics. Differential Equations is required for the modeling or applied side of advanced math.

>>16035182
I wonder how much adderall it would take to get through the topics list in OP

>meme'd into applied maths grad program
>too brainlet for pure (just not schizo enough to make it)
>fucking hate coding
>doing too much physics nonsense
Is there a happy medium? I intentionally stayed away from le AI meme, but it's unironically looking appealing now not just as an escape plan

>>16035926
>Serge Lang's Basic Mathematics.
Lang is a meme.

>>16035431
https://tex.stackexchange.com/questions/324455/matrix-vs-array-environments

a) can be proven with five lemma I believe, but doesn't b) contradict it?

Hey guys, just fucked my first differential equations test. Do you guys know any places I can find first order practice problems? Don't want to fail this semester.

>>16035800
Algebraic geometry.

>>16036877
No. You just want the map induced by the inclusion to not be an isomorphism; this says nothing about H_n(A) and H_n(B) being isomorphic though.

How do I become a high school math teacher?

>>16036889
>he failed 1st order ODEs
just give up
there's no fucking hope for you
if I was a hiring manager and I found out you failed 1st order ODEs I wouldn't even hire you as a fucking janitor
WHAT THE FUCK

>>16036889
Your textbook faggot

>>16037559
This but unironically.

>>16035800
Not getting filtered buy the concepts of "countable" and "uncountable."

If you can wrap your head around basic set theory, continuity and what it actually means for a function to be continuous, it doesn't matter whether you learned that concept in a real analysis course, a topology course, or an algebra course. You've started actual mathematics at that point.

>>16037665
I'm apparently getting filtered by spelling, lmao.

>>16035800
proofs I would say, just because there is a creative element.
Maybe you're applied and you just play on your computer all day.

I'm being raped by sylow theory
finite group theory in general is really shitty
only the biggest of nerds even bother to learn it

>>16037665
> "countable" and "uncountable."
>basic set theory, continuity and what it actually means for a function to be continuous
that's literally first year stuff.

>>16035823
God I love algebra

what’s a good book for Calc 1, 2 and 3 that’s not Spivak or Apostol? Stewart and Strang are generally recommended, are there others? looking for concise yet comprehensive books. I found this one but it’s a bit short and the exercises are somewhat difficult with little to no hints.

http://intellectualmathematics.com/dl/calculus.pdf

>>16037553
But if they are isomorphic, won't the relative homology groups be the same since you get the same exact sequence?

>And by a categorical argument, we see that if follows trivially that...

>like algebra
>dislike "real" analysis
How is it possible to be this based?

>>16038003
Mathematicians use "we", not "I".

>ruins math
Why did he do it?

>>16035182
What's the issue with neural networks supposedly not having a solid mathematical foundation? I notice that for simple enough examples it's easy to follow the explicit calculations they'll carry out (e.g. neural networks learning XOR is fairly trivial), and the linear transformations going from layer to layer is also fairly straightforward.
So what stops this from being as "mathematically solid" as people want?

>mfw haven't been here since 2018 and went back through the archives looking at all the comfy and fun threads

>>16038145
It's mostly based on combinatorics and not maths.

>>16038150
there's no substance to this post except you announcing you're retarded and can't understand counting problems for some reason

>>16038145
Because computer scientists havent actually proven anything. NNs take advantage of certain properties of the Gaussian and softmax, but theres no actual proof this architecture is optimal. Also what mathematicians are looking for is universality, and it hasnt been proven that any AI architecture can exhibit universality.

Without a proof AI people essentially just have to keep training up bigger and bigger models until they see behavior which apparently looks universal (AGI), so not very mathematical.

I bought Lang's Basic Mathematics to start my journey. Wish me luck

>>16038554
Ur getting filtered if you don't know algebra

>>16035182
That infographic has to be a joke right no way it is humanely possible to do that

>>16038582
Well someone has to build all the shit

>>16035182
Hello /sci/, bachelors student here. I was looking into multivariable integration since I have it next semester and all resources I found are very strange. Baby Rudin's ch 10 11 do it very weirdly and very short/terse manner, and I did not like Spivak Calculus on Manifolds either.
I tried reading Munkres Analysis on Manifolds and even he has that portion not as well as he dealt with the multivariable differentiation

Any good suggestions for this topic? Anything is appreciated. I enjoyed Munkres Analysis on Manifolds treatment of differentiation a lot.

>>16038133
He is jewish that's why.

I have to show that [math]\lim(2n-n^{3})=- \infty[/math] by definition, is this correct?
[math] 2n-n^{3}<-E \iff n^{3}-2n>E \iff n(n^{2}-2)>E\text{ , let n>=2, then }\leftarrow n>E \text{ now let }N=\max \{2, \lceil E \rceil+1 \}[/math] .

>>16038559
no i won't

>>16038133
serre was unironically better than grothendieck, he's just less famous because he was less of a weirdo freak

>>16038885
none of you are well read enough to seriously have a conversation like that

Combinatorics: probably of dubs or higher? (Anywhere in number)

>>16035800
Anything not taught in engineering. So proofs.

>>16038635
what the fuck is this mess
>>16038885
Serre is a fucking psycho

>>16038585
Help me sci.

>>16039971
>Serre is a fucking psycho
how
he seemed very normal in the abel prize interview with him

>>16038585
It's done in Pugh (chapter 5) nicely, I think.

>>16039971
What is wrong with it?

>>16040333
Wtf Pugh has it even shorter and more condensed, you sure it is good?

>>16040443
>its le bad!

>>16038585
Each time your integrate you kill a variable so as long as you're integrable in each variable it should be fine.

I'm about to leave my PhD program with a masters. No regrets. Math at this level feels like such a waste of time. So much of it can only be understood by such a small group of people, and almost all of it will never matter in 'real life' or work.

I've frankly become quite jaded to the subject and lost much of my interest.

I've met a lot of nice people and a lot of straight up weirdos that cannot function outside of a math academic setting. Most of my peers have zero skills or knowledge outside of doing math.

For those who really love this stuff: I'm happy for you and I have all the respect in the world for those who continue on the path and actually make significant contributions to the field. But having done it now for 3 years, it feels like a pointless pursuit.

>>16038002
Yeah, I'd say after your first year of undergrad, you are doing real math. It's not research level math, but it's real math.

How to get good at maths that requires dealing with numbers and counting?

>Last thread has high quality yet it was archived at 107 posts.
>The rest of the board is the usual garbage.
Did the pseuds get bored of /mg/?

>>16041512
Wtf is this shit

>>16041696
Hey man I am an undergrad, any advise for me on pursuing math phd?

>>16037556
Acquire the requisite qualifications and apply to a job

>>16036078
If you hate coding you'll hate AI research trust me.

>>16042193
Who is gonna check this daily just for two lazy shitposts

Is there a successor to mathjobrumors or mathchan?

Does anyone have good resources in the construction of Lebesgue-Stiltjes measures? I'm having trouble understanding how you go about constructing a LS measure on the reals from a generic right continuous set function.

>>16036238
>Lang is a meme
No, AIDS is a meme.
>>16038145
Computer scientists fetishize “rigorous” methods like automated verification through logical calculi, proofsof termination through fixed point theorems etc. That kind of rigor is by definition oit of reach for even medium size practical computer programs, much less a fucking neural net you trained on 100 gigs of data you bought. Even the question of Well actually, there is a big theoretical AI community, it’s been talking about markov chains and context free grammars and modal logics for 60 years, it hasn’t accomplished jack shit

>>16038145
NN's are an extension of an MLP, and MLP's are "universal function approximators" (meaning that given a diverging size training set, any MLP can learn any "sufficiently smooth" function mapping Rn -> Rm where n is the dimension of the input layer, and m is the dimension of the output layer).

>>16038240
This just isn't true. As much as I hate to give credit to computer scientists (because in truth most of the mathematical proofs were done by EE's and Applied Mathematics people) but the universal approximation theorem has shown that NN's can approximate any Lipschitz continuous mapping from Rn -> Rm to an arbitrary degree of precision (provided infinite computing precision and diverging training sample size, which of course, can't be realized by real computers with floating point precision and memory limitations).

There are still a lot of questions about the boundaries of these learning models (of which there are interesting research papers being published on "over-parametrization" as well as statistical properties of NN's) but we have some pretty solid theorems about their theoretical limits for continuous function approximation.

Are you as smart as Romanian 12th graders?

>>16042309
math board on econjobrumors

brainlet here
how do i find this graphs function?

>>16043276
Take
[eqn]H' = \{4^k | k \in \mathbb{Z}\}[/eqn]
and the homomorphism [math]\mu: H' \to \mathbb{Z} \\
4^k \mapsto k
[/math]
It can't be extended to a homomorphism [math]\bar \mu: \mathbb{Q}^* \to \mathbb{Z} [/math] since you would need
[eqn]1 = \bar \mu(4) = \bar \mu(2) + \bar \mu(2)[/eqn]
but there is no number in [math]\mathbb{Z}[/math] that gives 1 when added to itself.

>>16043964
If you studied any set theory you would understand that the graph IS the function.

>>16044050
well then latex didn't study any set theory either because it needs the numbers

>>16043964
Every smooth graph is equal to at leaat one polynomial so just estimate it

>>16043964
assuming it's a rational function, use asymptotes to find the denominator and zeroes to find the numerator, and play around in Desmos

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

>>16044560

>>16041696
I went through a pretty similar thing mastering out of a PhD recently. I actually liked the teaching parts of the job a lot but the "research" was unbearable.
For me it's not even so much that it will never matter in real life, it's the feeling that it doesn't even matter in math. Everything people do is so contrived and the reason math has splintered into a billion topics that 5 people have heard of is because most of it has zero interest other than getting you papers. You have to force yourself to invest such huge amounts of mental energy into things which are monstrously complicated, very poorly documented and explained unless you are in a room with the author, and monstrously boring.
My mental state was completely fried by the end, I'm fairly sure I would have an hero'd if I had tried to stick it out another year

>>16042236
Honestly I'm not sure what advice to give you.

For me personally, the things I have found to be important are:

>having a hobby outside of math that I enjoy doing and that I do regularly
>having a group of friends I can be relaxed around within the department and work together with. For me, math is a team sport.
>do things with that group of friends that have nothing to do with math. Ex. in my program we get together for weekly basketball games.
>finding out who the nice professors are. This field attracts a lot of social rejects and people who think way too highly of themselves. find the good people and stick with them.
>realize that math is hard, no matter what the tards on sci say, its hard and its ok to struggle.
>Network with other departments. You never know who might be able to help you with a job one day.
>Don't over commit to a single field immediately. So many people entered my program thinking they wanted to just do algebra for the rest of their lives - most changed their minds.
>Don't take my advice as the word of allah. Your experience could be different than mine and keeping an open mind is always a good thing.

Good luck with your studies though bro. Undergrad grades are important. If you can do undergrad research, do it. I tried for years to do research at my undergrad and was always told there were no opportunities - it was really disheartening. My current school has a lot of great opportunities for undergrads and its cool to work with them and meet them. I've met some incredibly talented people who I plan on keeping in touch with; who knows what they might be doing in the future its always good to have connections.

>>16044624
As this anon said: teaching can be a lot of fun.. but it can also suck if you are stuck with students who obviously do not care at all. I don't blame them for not liking the field (I hate it at times too).

/g/ knows how monads work. Do you?

>>16035800
I'd say complex analysis because you're no longer restricted to the real line or some cartesian product of it

>>16038127
Some narcissistic ones do.

>>16042234
f(x,y,z) = f(x) prove me wrong

>>16037037
I would say so

>>16041696
Any advice for switching from an undergrads in math to a PhD in physics(QFT)?

>>16044688
it’s just tag you put on your variable, similar the “const” etc. you see in most languages. because everything in haskell is supposed to be a term like f(g(x,y)) , they decided to express the tag as just another symbol in the term, like f or g. the “monad” axioms turned out to express how they wanted to be able to move that symbol around within the term

this is all ancient history of course, haskel trannys have long since 41%. a new generation of troons have come along and invented the wheel yet again, this time under the name “rust safety”

>>16043964
It's some sec(x) or csc(x) bullshit.

>>16035800
Everything up to but not including limits.

Is there a 'dodo bird hypothesis' of math research? If something can be proven in math that has precisely zero connection to real world use, can it be said to be 'equivalent' to the uselessness of other useless results? Why not just write some autoprover that goes down a tree of true statements from axioms and publish it?

>>16046452
>Why not just write some autoprover that goes down a tree of true statements from axioms and publish it?
why don't you then?

>>16046454
maybe I just might.

>be software dev
>country requires a BSc in math to teach high school math
>enroll inmath degree part time for fun
>classmates have 0 interest in anything and do very little work, they just want to teach
>they're very negative and always complain about shit being too hard
>after 1 year two privileged brainlet fuckers start to bully me
>literally cant show interest or ability in any way without being treated like filth by them
may their time in Jahannam be painful

>>16046926
Here's a list of what causes them to lash out on me:
>i present a problem in class
>i ask a question in class
>they asked me what my favorite class was, and they did not like my answer
>someone mentions a topic and I say I struggled with learning it
>I express any sort of interest in math
>I show an interest in learning about practical uses of a math topic in e.g. CS or stats

>>16046285
You don't know how monads work.

>>16042241
>>16042411
Why does AI research not seem to be much about theories of how human brains work?

>>16046926
> >literally cant show interest or ability in any way without being treated like filth by them
are you 12? tell them to fuck off.

What are the most beautiful applied math fields?

>>16046999
not only is it how they work, it also summarizes how and why they were included in the first place

>>16046213
Sorry I dont have much advice to give for that. I took a QFT course and was lost on the physics concepts but was able to follow the math for the most part, but I also had none of the physics prereqs for the course i.e. grad level QM.

If youre still in undergrad it'd prob be a good idea to take some upper level physics courses or try to take grad level physics course and see if your math department will give you credits for it.

>>16035182
Btw the daughter of the man who created the picrelated curriculum is to graduate 3 years late, she took 7 years to finish a 4 year BS degree in mathematics.

hello, I am working through Gelfand’s Algebra. Is there any other books like it at the same level that I can read covering Algebra? Thanks

>>16047446
>had to stay for 2 generations of football teams

>>16047510
Maybe she would even stay for longer. She's currently in the third year.

Looking to pivot out of healthcare into a field like electrical engineering but my math skills are lacking. Was never bad at it in school, just didn’t see the practical application so I cared little for it. Pretty much starting from the ground up and a lot of the infographic suggestions I see seem to be more geared towards high level stuff. Been going through Khan academy to refresh on the basics but beyond that it seems to be not recommended. Any suggestions for good starting points or trajectories? I have got to do something different with my life, anons.

>>16043964
[math]f\left(x\right)=\frac{-16875x^{2}}{\left(x-10\right)^{3}\left(x+10\right)^{3}}[/math]
Close enough, I'd say

>>16035182
is it possible to find the curve which gives the line integral with lowest value (i.e. shortest path) between two points in a non conservative vector field without numerical approximations/graph algorithms?
picrel is a random pic i found when i typed vector field

>>16047525
>I have got to do something different with my life, anons.
Lol I’m in a math program, with the same feeling—like I should try for medicine, at least I would be helping people for once. Anyway, khan academy is a great start. Other things that come to mind
> https://www.edx.org/school/mitx
These are online intro university courses from MIT. They are free (but they nag you to upgrade to a “premium” version). There is a schedule, with online lectures, homework due dates, and a functioning online grading system. I like this because its lectures are very good and it imitates the structure of a real course, which hooks into your brain a little better than
>Opencourseware
MIT’s older online courses. These are good, just less polished and less advanced in terms of user interface, and without the enforced structure
>leetcode
Computer programming and math and closely related skills

Anyway even if you do stick with healthcare, it is always a good thing to learn something on the side. Studying out if interest is one of the easiest hobbies out there other than beer and video games, and obviously a lot better for your life

>>16041696
>>16044624
>>16044671
That's what happens when you fall for the pure math meme without being equipped with enough schizophrenia and autism.

>>16047563
Appreciate it, anon. I was half expecting to get shit on for asking here so thanks for the genuine info. But yeah, good chance that I do end up sticking in healthcare because it’s definitely not a bad place to be, but either way I’m going back to school and need to brush up on my math regardless. Also I’m finding it to be weirdly enjoyable.

>>16047319
I dont know but A Very Short Introduction to Applied Math by Goriely was short but good read

>>16042140
post lewd jahy pics and ill give you the answer

>>16048439
Just take the arithmetic mean of A, B and C, bro.

>>16048608
How do you prove it results in equal areas

>>16048439
solve |det(AB,AP)| = |det(PC,CA)| = |det(BC,BP)|.

>>16047966
I'm not much into degeneracy, anon.

Does the Finnish topology sister still post here?

How to get good at curves, affine varieties, number theory, alg. geometry and other maths needed for cryptography? I missed most of highscool math but studied some more "abstract maths" on my own years ago. When I read something basic about algebraic curves, I understand all the words and definitions like a robot would, but cannot gather them together to make the same intuitive jumps that are assumed all over the text.
I wish to gain the intuitions I missed out in school.
Pls send help.

>>16038060
God I FUCKINg HATE this negrotic 'gigachad' phenomenon. The retard looks like an inbred mongoloid with too much plastic surgery... And the smell and hair is just.... Fuck u Anon, this is what's holding evolution back.

if a human is able to prove the mathematical statement it is not real math

>>16049286
It's a photoshop you retard.

>>16041696
NOOOOOOOOO YOU ARE SUPPOSED TO FEEL SUPERIOR TO THE INFERIOR VERMIN OF THE WORLD THAT CAN'T SYMBOL SHUNT LIKE YOU CAN THAT SHOULD BE ENOUGH OF A MOTIVATING FACTOR NOOOOOO

>>16049286
If anything, Dr. Manhattan is where it's at. Note the subtle jaw, smooth face, large cranium and B R I G H T eyes, a stark contrast from the Hapsburgian, unkempt, slit-eyed, slant-headed abomination that is le 'gigachad'.

>>16049352
I feel something like Dr. Manhattan myself

>>16048676
Shoelace

>>16049286
>God I FUCKINg HATE this negrotic 'gigachad' phenomenon. The retard looks like an inbred mongoloid with too much plastic surgery... And the smell and hair is just.... Fuck u Anon, this is what's holding evolution back.

>>16048439
Going off of >>16048760 I found the x and y
coordinates. It's was some spaghetti I did.

>>16049352
thanks to stable diffusion we can slowly but subtly replace all gay reddit memes with better ones

What undergrad mathematics topics do you guys think are an absolute "must know" before moving on to more advanced/grad level math?

>>16044624
I too have a PhD fellow /sci/ anon.
Aren't we all so very smart?
I loooove debates about different versions of string theory.

>>16049843
Elementary number theory, real and complex analysis, abstract algebra up to basic Galois theory, point-set topology, commutative algebra, basic differential geometry and basic algebraic topology

>>16049843
>Analysis including measure theory, complex analysis, ODEs and functional analysis
>Abstract Algebra up to Galois theory
>Discrete Math (Combinatorics, Graph Theory, Search Algorithms)
>Basic Numerical Analysis (Stability and conditioning of problems, polynom and spline interpolation, root-finding algorithms, matrix factorization algorithms (LUP, QR, Cholesky), numerical quadrature, Runge–Kutta, FDM, FEM)
>Linear Programming
>Probability Theory up to the theory of martingales

>>16049845
The post you're passive-aggressively malding about is talking about quitting his PhD
Seems like you might be even lower than 89

wilesposting

What are some good books on Proofs beyond How to Prove it?

experimental mathematics

>>16049843

>>16044624
>>16041696
I was close with one of my professors in undergrad and he had a similar outlook. He was an old man who had been involved with this all his life. He wasn't bitter, regretful, or depressed about it. But he was extremely lucid and self-aware that the work of his entire career was basically meaningless. He was in an extremely tiny subfield of a subfield of a philosophy that was already being left behind. He rarely even found people to talk to at conventions and already knew everyone vaguely involved in what he was doing. He keeps publishing and being a mathematician and professor because that's his trade at this point. I found he was noticeably disinterested in dealing with the mathematics undergrads and in fact preferred meeting with the rare humanities students. He was the only professor I had that was like this, maybe that's what drew me to him.
Meeting him, and seeing that the only graduate school bound pure math students were so utterly disconnected from reality on every level and said things like "I don't care if what I do has no purpose or if nobody ever looks at it I just like moving these symbols around." definitely made me reconsider continuing.

I'm not saying everything should just be applied math or something, but I just didn't see the appeal in that kind of thing for me and being surrounded by severely autistic people for the rest of my life.

>>16050446
A=B
:^)

How would you explain operator algebras to a layman? I often ask them if they know Linear Algebra and build from that but if they don't I just give up

>>16050609
I forgot to add that I want to know because I get asked often what I do by non-math people.

>>16050609
Remember that even the normies who've done "linear algebra" think matrices are just arrays of numbers

>>16050446
There are too many books covering the same topics as Velleman's (there was time when the clascic was Daniel Solow's), so maybe this one is at least an outlier. Journey Into Mathematics: An Introduction to Proofs by Joseph J. Rotman. On the other hand, if you reallyjliked Velleman's, he wrote a book about rigorous calculus/introductory analysis.

Is there any way of uniquely representing the real numbers? Every representation I know of has some redundancy, usually at rational points.

>>16050452
married bachelor

Retard trying to get into the Army here.
This is a question on a practice exam.
How do I actually find the ratio without brute force division in a calculator (we're not given one in the exame anyways)

The ratio of 3(1/4) to 5(1/4) is equivalent to the ratio of
>a) 3 to 5
>b) 4 to 7
>c) 8 to 13
>d) 13 to 21

>>16050815
Subtract 1/4 from both sides to get "3 to 5"

>>16050815
the fact that both sides are being divided by 4 suggests that you should multiply both sides by 4, for starters

>>16050818
>>16050821
But that's not the correct answer...

>>16050824
multiply both sides by 4, for starters

>>16050824
Must be a software bug

>>16050825
nigga, multiplying by four does nothing
it gives you back 3 : 5, which isn't equivalent to 5.25 / 3.25

the actual answer is 13 : 21 because is equal to 5.25 / 3.25

I'm just wondering how tf I'm supposed to arrive to that

>>16050831
The answer is not 13 : 21

>>16050831
assuming you are attempting to denote things with mixed numbers, which is what you seem to be trying to do judging by your insistence on 5.25 and 3.25
[math]5.25 \times 4 = 21[/math]
[math]3.25 \times 4 = 13[/math]
assuming you genuinely mean [math]\frac{3}{4}[/math] and [math]\frac{5}{4}[/math], which is what you have written, you end up with 3:5 as expected

>>16050833
It is
I figured it out
3.25 = (4 - 3/4) = (16-3)/4 = 13/4
5.25 = (6 - 3/4) = (24 - 3)/4 = 21/4

13 : 21

And also the book says it's D, and division says the same

>>16050834
Yeah thanks anon, but I'm not allowed to use a calculator on the exame so I was looking more for the process and I just split the problem into more manageable parts
Divide and conquer is a good problem solving method

>>16050693
Sure, just use the decimal representation except add a requirement that it not end in 9 repeating.

>>16050446
Mathematical Proofs by Chartrand, Polimeni and Zhang

>>16047644
What sort of healthcare are you in

>>16050971
>Zhang et al.

>>16036078
Applied math major is a fake major for schizo yet midwit drones who have constantly illusions of grandeur and think they can tackle a wide subject such as maths. So they made applied maths as a major in order not to hurt their feelings and dump them next in finance and/or computer science, or any field related where they can wageslave for the entirety of their pathetic confused life.
>t. Applied math grad

>>16049286
Greasy fat hands typed this

>>16035823
That it is called "abstract algebra" is a sign that it's not real maths. Mathematicians just call it "algebra". They only say "abstract algebra" when they're teaching American undergrads or writing a textbook for them or something like that.

>>16050815
that's hard for me

>>16050815
What I did was 3 + 1/4 is 13 eighths, and 5 + 1/4 is 8 more eighths, seeing the answers, but that was after like 3 minutes of it being in the back of my head and my IQ is above average and you're supposed to do it in like 30 seconds to a minute

However I think this is more of a cognitive test than studying math

>>16051295
>eighths
1/4ths i mean

>>16038145
>>16042445
>NN's are an extension of an MLP, and MLP's are "universal function approximators" (meaning that given a diverging size training set, any MLP can learn any "sufficiently smooth" function mapping Rn -> Rm where n is the dimension of the input layer, and m is the dimension of the output layer).
i'm aware of this, I don't see how it answers the question though - what is it mathematicians are unsatisfied with about this?
I hear mathematicians talking about how "the maths is a mess" and it makes me think that the complaints are just the sounds of irrelevant boomers since it's not even clear what they have a problem with and neural networks are used for practical tasks anyway. Rigour is the least of anyone's concerns.

>>16051384
The problem is that you can't use it to prove anything useful about what they'll do.

>>16051500
what do you mean by useful?

>>16051384
>what is it mathematicians are unsatisfied with about this?
If your model can represent anything, it's basically useless. That's why all these commercial models are so big: The need all that data to come close to doing something useful. If the technique was really intelligent, you wouldn't need to show it a billion apple photos before it could recognize an apple.

Also, mathematicians worked out the basics of statistical learning theory back in the 1970s.

>>16051583
>it's basically useless.
why?

>>16051616
Because of overfitting. "If someone can explain anything you ask, their explanations are going to be bullshit."

>>16049110
Pls help anons.

Is the uncertainty 1-p(((~a1 v b1)&(~a2 v b2)) | a1 v a2) no greater than the sum of uncertainties 1-p(b1|a1) + 1-p(b2|a2)?

please check out pic related.

Lets say for a NEET who hasn't done any of these questions even in highschool. How long until I can ace these type of questions if i self study it?

>>16052182

>>16051672
>it's useless/bullshit because it's universal
Not the response I expected from a mathematician tbqh
Fourier series are universal in a certain sense (though "complete" is probably the better word) and you don't hear mathematicians calling them useless

>>16052362
Lang is a meme.

>>16052387
How so?

>>16050815
It's my first time visiting sci so I have no clue if this post or the replies are sarcastic. Just turn it into an improper fraction and you'll see the answer.
>3(1/4) = ((3*4)+1)/4 = (12+1)/4 = 13/4
>5(1/4) = ((5*4)+1)/4 = (20+1)/4 = 21/4
You see. Now multiply each by 4 and you get 13 and 21.
Since 13 is a prime number you can't simplify, so the ratio is 13:21
That's assuming it's three and a quarter vs 5 and a quarter, and you're not trolling with semantics by meaning something else.
BTW it's fine being bad at maths, most people can learn it to at least a high school level in a relatively short amount of time.

>>16051062
cry now laugh later. let's see how your superior intellect and purity will serve you when you are teaching highschoolers for a living while applied mathematicians are earning 200k in finance and data science with just a master's degree.

Why do grad schools have exams for kiddie undergrad stuff like Analysis, Algebra, Topology, etc., that also 2 years after you join? Doesn't it make sense to have them within the first six months and test more advanced things like Operator algebra, algebraic geometry, etc.?

>>16051882
Browse through this collection of book reviews and find something that suits you.
https://maa.org/tags/cryptography

>>16052363
That's because in math we can actually sum infinitely many terms or approximate the error from taking a finite amount of them.

>>16052850
The graduate courses/exams for analysis, abstract algebra, etc. are generally significantly more rigorous and challenging than the undergrad equivalents.

It's the difference between Rudin's Real and Complex Analysis vs. Abbot's Understanding Analysis (as an example).

>>16052850
The graduate algebra qual is not anything like your undergrad group theory course. Functional analysis and algebraic geometry aren't even required courses in many grad programs, and many undergraduate degrees don't even require topology.

ok should i study topological manifolds or smooth manifolds first?

is the set of all infinite cardinalities countable?

>>16053399
i reckon that's not even a set

>>16049110
get a firm grasp of commutative algebra first of all. i recommend atiyah-macdonald book together with borcherds' commutative algebra lectures on youtube

>>16047319
programming language theory, perhaps

>>16037962
finite groups are peng mate

Why does what Terence Tao says seem more interesting than people on /r/math? It seems like the people on /r/math aren't even trying to say anything interesting.

does anyone else here have extreme imposter syndrome combined with horrible anxiety issues?

i can't stop thinking in the back of my mind that due to my late start in college+unpleasantness which led to my prior education being a horrific experience it's over and that I can never catch up. I look around me and see people who have literally been groomed almost from birth for success with aids ranging from tutoring to private schools handed to them on a silver platter. meanwhile, here I am, from a public school that didn't even have a proper computer lab, older than the norm to boot.

>>16053476
He's a professional math communicator. Everyone here runs with sonic gumption.

>>16054010
it's not just the "communicators" who are interesting to hear, though it may also not be everyone who is good at math (or a field)

>>16053939
You're never going to catch up to people who were sent to specialized gifted programs when they were 5
Lower your standards, those people are the 1%ers of mathematics

>>16054204
>You're never going to catch up to people who were sent to specialized gifted programs when they were 5
Do you think progress or a "success" in math is that easily predicted by statistics, like that? A person may want to believe they can be past a correlation like that, but maybe it's true it doesn't tend to be that way.

>>16053132>>16053129
Seems to be pretty straightforward to me.

>>16053435
I said "applied" math.

>>16053939
Yes about the impostor syndrome. It's never really gone away, but I kept learning and am more capable now than I was some years ago, so I suppose it doesn't really matter. You'll get there.

>>16041696
AS a normie reading this thread I honestly don't understand how you autists manage to do it for as long as you do. But yeah, academia is generally a shit career unless you truly deeply intrinsically care about the subject. I can only imagine something as abstract as math is even more so.

>>16035182
Post a better quality version of this curriculum fag. The images are barely visible and I can't figure out last two books.

>>16035800
Introduction to analysis

>>16038582
It's "possible" but neither probably nor advisable

>>16048439
>hurr durr what is the barycentre

>>16049855
I would add a bit about topology or trig series or some linear analysis (but you got that)
Abstract algebra should focus on Group theory (elementary decompositions and things like artinian noetherian modules etc, Sylow theorems) for Rings the module analysis theorems
Definitely some Complex analysis

>>16054303
That's trash from a third rate university. Literally my undergraduate algebra was more involved.

>he doesn't know about \colon
ohnonono

>>16035182

Are there infinitely many pairs of distinct positive integers, (a,b) with a≠b ,
such that a^b = b^a ?

>>16052850
Because they're "qualifiers" they're not really there to bust your balls taking exams, they just want to hold everyone at a basic standard for what's expected knowledge. Doing a Ph.D isn't about solving IMO exam questions it's about conducting research/training to be a research mathematician.

Is there a mathematical way to calculate the total number of different filled sudoku grids without having to use brute force by a computer as help?

I'm going to uni this year
so now that CS is dead, is it a good idea to go into math?

>>16054794
no.
in fact, (2,4) and (4,2) are the only such pairs.

>>16055053
math is the hardest stem major and almost all of the jobs available to math graduates are available to far easier majors.
if you like it, go for it.
if you want to jerk off your ego, maybe
if you only care about money then no

>>16055130
>if you want to jerk off your ego
>if you only care about money
i need both, CS could satisfy both but might not be very useful for making money in near future
wouldn't getting a math degree in a good uni still make me eligible for getting an interview in a tech company?

What steps can I take to minimize the amount of "silly mistakes" I make?

>>16055116
> (2,4) and (4,2) are the only such pairs.
Thanks; how do we prove this?

>>16055191
[math]a^{b}=b^{a}[/math] implies [math]a^{\frac{1}{a}} = b^{\frac{1}{b}}[/math] by raising both sides to [math]\frac{1}{ab}[/math]
[math]x^\frac{1}{x}[/math] has a maximum at [math]x=e[/math] and decreases monotonically after that, so if it shares a value for any two positive integral inputs, one must be less than e, i.e. x=1 or x=2
it's simple enough to test both of these cases

>>16055266
Oh that's clever, thanks anon

>>16055053
CS is far from being dead

>>16055481
not in 3 years

Have any of you niggas read this?

>>16055146
Plagiarize everything.

>>16054887
>Is there a mathematical way to calculate the total number of different filled sudoku grids without having to use brute force by a computer as help?
It's 6,670,903,752,021,072,936,960.

>originally CS major
>thinking about switching to a dual/combined cs and math thing because I want to prove that I can do slightly beyond basic math

upsides: feels like it would stand out more than pure CS given how atrociously overinflated this field is becoming
downsides: i am a natural ball of stress and anxiety three inches from a nervous breakdown and do not know if I could handle the increased study time requirements

>>16049110
find me narodnik on libera #math

Can a finitely generated group act arc-transitively on the countable random digraph?

>>16055648
Yes but it was calculated partly by using a computer.

>>16055643
not possible on a test

Are all uniformly continuous functions locally Lipschitz?

>>16056311
The cube root is uniformly continuous on any bounded interval (as is any continuous function), but it is not locally lipschitz near 0

>>16055146
Force yourself to write the neatest, most visually appealing, girliest solutions you can
If you're a natural chickenscratch person like me it genuinely makes a big difference just to stop scribbling all over the page and pay attention to what you're writing

>>16054583
Quals are not meant to be deep bro, you can go look at ivy quals online and they're not substantially different from quals at shitty unis
The purpose of the qual is just to make sure you aren't grossly ignorant about fundamental topics

>>16056454
>and pay attention to what you're writing
yeah thats the problem
having a standard procedure to check everything at the end would probably help too

Is there a limit L such that a^b is always greater than b^a given that b > a and that a > L and b > L?

Is there a borderline L on the numberline such that a^b is always greater than b^a given that b > a and that a > L and b > L?

>>16056600
Yes, [math]L = e[/math].
https://en.wikipedia.org/wiki/e_(mathematical_constant)

In Winternitz One-Time Signature why can't any signature be forged when you know the "base" of the chain? I understand the complementary nature of the signature and the checksum in regards to the preimage for example in the middle of the chain (https://crypto.stackexchange.com/questions/63819/why-does-wots-need-a-checksum-during-signing-operation)) but how does it work in the starting end? i.e. if you send a message with all the largest values to be signed by an oracle won't you get the whole chain and can then make any signature?

>>16044914
The complex plane is just R^2 with a weird multiplication and the obvious addition

What are some non-obvious things that are usefully modeled as graphs, where insights are to be had when the tools of graph theory are brought to bear? Is there a book or page that collects obscure or surprising applications of graphs?

All these 30-year-old motivational examples in standard graph theory texts are so uninspiring. I'm looking for more single step applications like, the fact that paths on Cayley graphs are finite alphabet sequences, CPU register allocation, de Brujin graphs in genome assembly; rather than something elaborate such as e.g. Deepmind's "learning to simulate complex physics with graph networks." A perfect example is the Knuth–Plass line-breaking algorithm as the min-cost path on a weighted graph.

>>16056836
Nevermind, I was confused about how the checksum works, the secret key has indexes for signing the checksum, and large values for the message to be signed mean the checksum values are from the chain end ie you can't construct the checksum even if you can construct a signature for the message itself?

>>16055639
Yes. It is really short, just read it

>>16035182
Please tell me a lot of books about doing Linear Algebra with a Computer. I want books where using a Computer is expected because I like it when I get to use a Computer.

>>16057360
>open libgen
>search linear algebra
>seach matrix algebra
>sort by most recent in both cases
>save files with computer or programming related titles or descriptions
>profit

>>16054303
What does G' of group G mean?

>>16057810
Derived subgroup.

>>16057069
Check out Euclidean minimum spanning trees

>>16057582
PSA do not respond or in any way encourage the "person"you are responding to, he is a nuisance that thrives off of attention

If [math]g : \Omega \to \Omega'[/math]is [math](\mathcal F, \mathcal F')[/math]measurabe,
and [math]f[/math]is a map from [math]\Omega[/math] to [math]\mathbb R[/math] that is [math](\mathcal F, \mathcal B_{ \mathbb R})[/math] measurable.
If [math] \mathcal F = \sigma(g)[/math], there exits a map [math]h : \Omega' \to \mathbb R[/math] that is [math](\mathcal F', \mathcal B_{ \mathbb R})[/math] measurable such that [math]f = h \circ g[/math].
Anyone know what this theorem is called, or can refer to a book which has this theorem?

>>16057948
Doob-Dynkin lemma

Would some kind anon recommend me an article about connections between Euler numbers and Bernoulli numbers?

>>16057582
Thank you for telling me. What is libgen and how do I access it?

>>16057937
Why is the CIA like this? If your people hadn't ruined TV there is a good chance I would be watching TV instead of using advanced Mathematics and Computer Science research.

Is there any technical term for a removal of complex numbers from an expression?

>>16058101
Can you give an example of what you mean, and also why?

Just got rejected from my absolute dream school. This is the last of my reach schools to reject me. I expected it to hurt but with no exaggeration I want to kill myself. I want to be dead right now. I don't want a PhD anymore. If I didn't have people who love me I would literally step off my balcony and kill myself right at this moment.

>>16058973
none of this shit matters bro, you'll have plenty of fun at a big state school math department

>>16058993
I just feel so hopeless. I see people say that if you're really driven you can outwork a lower-ranking PhD but every math department website faculty page is just "PhD MIT" "PhD Harvard" "PhD Chicago" etc etc. And I know I'm being immature about it I just really don't know how I'll rally enough willpower to push through a PhD with such shitty prospects somewhere I don't feel proud to be going. I feel so pathetic, both for getting rejected here and for being upset about going to a "worse" school. I should be grateful I got any opportunities at all.

>>16058993
Whether anon's depression is justified or not, it's almost surely going to put a damper on his (their?) mathematical productivity.
Personally, I can only really conjecture and chase implications when I'm in one of two possible emotional states:
>excited
or
>paranoid (and only for nonconstructive results)
Whenever I've tried doing math while feeling down, it always turns out insipid and I end up deleting it from my notebook upon subsequent consolidation.

>>16059032
This is basically exactly how I feel and precisely what I'm afraid of. I'm stressed about working myself to the bone for 6 years to graduate from somewhere with no prospects for its grad students, and I don't know how I'll manage to overcome that fear well enough to focus on math.

0 = number of kids you will have

>>16035182
so I just need to master these eight books?

How do I compute the degree of the map from S^n -> S^n that sends (x_0, x_1, ... x_(n-1), x_n) to (x_n, x_(n-1), .... x_1, x_0), i.e the "reversal map"? I tried to express it in terms of compositions of reflections etc but it was never fruitful

>>16053939
One of my professors literally majored in English during undergrad and didn't do any math past ODEs and now she's a tenured professor (went straight into a PhD despite knowing nothing). You'll be fine. Also, consider that if you learn more than they do, then you'll eventually catch up.

>>16053939
focus on what you can control

>>16057069
Chu-Liu-Edmonds algorithm is used for dependency grammar parsing
Graph combinatorics is used in statistical mechanics and field theory to determine coefficients of series expansions

>>16059713
He... He blew me

>>16046213
like the other anon said take QM and QFT courses and do some undergrad research with theoretical physicists at your uni
QFT profs actually prefer to hire math undergrads so you'll be good

>>16059143
Just do it simplicially.

>>16057360
see https://web.evanchen.cc/notes/Harvard-55a.pdf

>>16057360
this is what /sqt/ is for, fag

>>16058101
decomplexification

>>16050815
the army wont accept pedophiles, sorry

>>16038554
dont spend too much time on precalc garbage. Those books are bloated to the brim with useless geometry. Get into proofs as quickly as possible

>>16057360
Just write your proofs in metamath for each problem in your textbook

>>16059953
This sounds nice. Is there a book about doing this?

>>16059978
https://us.metamath.org/downloads/metamath.pdf
Read this in parallel to any proof-based linear algebra book

>>16059978
in case you dislike metamath, there are a bunch of other proof assistants too, like lean or coq https://en.wikipedia.org/wiki/Proof_assistant

>>16059978
>>16059990
for lean: https://hrmacbeth.github.io/math2001/

>>16059978
>>16059990
>>16059996
see also
https://lean-lang.org/lean4/doc/
https://leanprover-community.github.io/
https://lftcm2023.github.io/
https://ncatlab.org/nlab/show/Lean
https://youtu.be/AayZuuDDKP0
https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/
https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/

in case you want to combine metamath and lean
https://github.com/digama0/mm0

For Coq
https://ncatlab.org/nlab/show/Coq#learning_coq

>>16059980
>>16059990
>>16059996
>>16060027
Thank you for showing me all of this. I will give it a closer examination. At a glance I like both how Lean looks and how MetaMath looks. I think I like how MetaMath looks more so I will study it first and see can I use the ideas?

>>16035182
I'm learning math for the sake of it, maybe in the future i'll get a degree but for now I just like it.
I'm a bit lost though, got any book recommendations or any tips?
I'm like halfway thru calculus and got some okay-ish intuition of linear algebra and matrices, mostly thanks to youtube videos, but i'm really starting to notice that I need a more complete base rather than understanding things because "they make sense in the given example".

>>16060311
I should have been clearer on that last bit.
I'm not in a rush to get through calculus, algebra, etc in order to pass a test, I want to really understand why math works because i'm doing it for fun.

>>16060342
>>16060311
>I want to really understand why math works because i'm doing it for fun.
Calculus - Lev Tarasov
Calculus for the Ambitious - T. W. Korner
The Hitchiker Guide to Calculus - Michael Spivak
Intutive Infinitesimal Calculus - Viktor Blasjo
A Primer of Infinitesimal Analysis - John L. Bell
Square Matrices of Order 2 - Vasile Pop, Ovidiu Furdui
Proofs and Refutations - Imre Lakatos

>>16060370
Thank you.

Does anyone know how Apollonius problem translates to higher dimensions in particular the 3 (outward tangent) circles case? Are solutions still guaranteed, I think so.

Would the outward tangent circle still have the biggest possible radius for non intersecting, also yes right?

>>16060370
It feels very weird seeing one of my professors on that list kek.

>>16059953
Do you personally recommend metamath over other options? Why that choice?

>>16059746
Like, look at the local degree or what do you mean?

>>16036889
Probably whatever textbook your course uses, but you also could do problems from Tenenbaum and Pollard (which is super easy to find a PDF of).

My guess is that there's some fundamental building block that's missing in your understanding of the problem setup.

>>16035823
This was the case for me. I went through my real analysis series just fine but got absolutely obliterated in my algebra series. Now I'm in graduate school for comparative literature

>>16060781
Like write down a simplicial complex homeomorphic to the sphere, and choose it so that your map acts nicely on it. Extend to chain groups and compute the degree from the definition.

>>16059003
Don't give up fren. I got rejected from all my reach schools and went to a relatively mediocre public PhD program and ended up with a really lovely post-doc and already confidently building up for my second position. Not gonna say it's not gonna be harder than if you had an Ivy League degree, but it's not impossible -- just make sure you network the hell out of your field while you're still a PhD. If this 4chan autist can do it, you can too

>>16060944
Thank you, anon. I'll work really hard wherever I wind up, it's worth it.

>>16060311
Amann Escher's Analysis series is gold

>>16057360
You always ask for shit but then never do anything with it. I distinctly remember you asking for introductory linear algebra books 2 years ago and you're still at it. Just start reading something already

>>16059143
literally sign of determinant of the linear map

Let functions [math]f_{n}:X\to\mathbb{R},n\in\mathbb{N}[/math] be such that the set [math]\{f_{n}(x):n\in \mathbb{N},x\in X\}[/math] is bounded. Prove that [math]\\ \varliminf_{n \to \infty}\left( \inf_{x\in X} f_{n}(x)\right) \leq \inf_{x \in X} \left(\varliminf_{n \to \infty} f_{n}(x)\right)[/math]

>>16060370
>5 books on school level Calculus

>>16061063
Fuck off.

>>16035182
Does doing math lower your heart rate? i just wore a heart rate monitoring watch while doing math and my heart rate went down to 48 which as low as it is while I’m sleeping.

>>16061090
I use books a way that is different I think. It is uncommon for me to read books from start to finish. When I need to build something that uses an idea from a book, I read that part of the book and try to build it. Sometimes this can't happen because that idea depends on other ideas and I have to build those first. Sometimes this requires looking in more books.

Use big project books to test ideas. I did this when I wanted to see if MAID-LISP could simulate a 16-bit computer. That failed. People who don't do experiments never realize in terms of progress, failed experiments are actually good, because if you know why it failed you can adjust and iterate over the problem again and either make a better experiment or decide to abandon that line of experimentation. Because of that experiment I saw how the language should get changed.

Computer Science experiments can be done quickly on most computers now, and computers are cheap. They are also powerful. Some of the older people reading here probably have personal computers which would have qualified as world-class supercomputers within their own living memory. If in those days you ever thought "If I had a supercomputer, I'd...", realize now that you have one by the standards of that time and go do it. Computer Science is cheap. Computer Experiments are fast and basically free.

>tl;dr: I don't have to understand what I am doing to do it. If the book gives me an idea, and I give the idea to a Computer, then the Computer can operate the idea and I can operate the Computer.

>>16057069
Here's another cute theorem I found while going through my undergrad notes:
>Every Latin rectangle can be completed to a Latin square.
The proof uses Hall's marriage theorem:
https://web.math.ucsb.edu/~padraic/mathcamp_2012/latin_squares/MC2012_LatinSquares_lecture1.pdf

>>16061341
yet people who just read their linear algebra books know more than you. Civilization is built on the transfer of knowledge between generations. If everyone wasted their time experimenting on solved problems, we'd still be cavement bashing rocks to create fire and acting amazed every time it works

>>16061435
I think it is more like this. Imagine two maids who want to learn how to cook.

The first maid reads cookbooks from cover to cover. She reads all the technical explanations of how food cooks and what the cooking process is doing to the food. She reads about different styles and techniques and seasoning and presentations. She commits these things to memory.

The second maid selects a random cookbook that looks nice, flips around until she finds a recipe that looks nice, and then attempts to cook it. Then attempts to cook a few customized variants of it.

Both of them are studying cooking, just have different styles of doing it.

>>16061474
bad analogy though, since the material in a textbook usually builds on the material from previous pages

>>16061474
I'd more seriously consider your opinion if not for the simple fact that there are hundreds of thousands of people that know linear algebra, and you don't

>>16061492
>there are hundreds of thousands of people that know linear algebra
Source of this bold claim? There might be tens of thousands, but hundreds of thousands?

>>16061568
Linear Algebra is high school material in China.

>>16061568
>>16061492
According to google there are about 50k math majors awarded yearly, which by itself easily makes "hundreds of thousands." If you include CS graduates, it becomes millions of people and this would still be drastically under-counting since most STEM degrees involve a linear algebra class.
>>16057360
He posts here because he gets off on the attention and the (You)s affirm his pathetic larp. Just stop replying and he will go away.

>>16061030
Want to re-iterate what I said about networking -- of course you want to work hard and write the best dissertation you can, but what you REALLY need to do to help your chances post-graduation is get to know the people in your field. Go to as many seminars as you can, even though you won't understand 95% of the material. Ask your department to fund your travel to conferences. If you can, give talks while you're still a student -- intimidating but professional mathematicians are still gonna see you as sort of a kid, so expectations really won't be high, and the point is to get your name out there and make connections.

>>16061255
Ngmi

>>16061474
With the only difference there being that math books are not cook books. They develop an actual theory and are not just a random collection of theorems.

You'll never learn math if you never think about the big picture and refuse to try consolidating your thought.

>>16061665
That's already a big goal of mine as a PhD student. It was a bit of a rude awakening to start applying for PhD programs and realize just how much I had missed out on opportunities as an undergrad by not playing the networking game enough. Better to learn it now while it's not too late to start.

I dropped out of college when I was taking calc 2. It's been about 5 years. Is the keisler elementary calculus book from the wiki good to try to catch back up and finish learning calculus in this situation?

I'm pretty solid on algebra still, but I'd need to go over some trig identities. I think I can do that as I go through the derivatives part. My goal, as a neet, is to just get calculus and linear algebra out of the way so I can stop making really stupid programs and work with electronics more for fun.

>>16061341
youre just lazy, no shame im admitting this. It takes a certain kind of mindset to commit oneself to seriously studying something over a long period of time and youre clearly too scatterbrained and autistic for that.

>>16061611
>>16061574
The question is: how many of those people actually know linear algebra? Knowing about matrices (and using Sarrus rule to compute a determinant of 4×4 matrix because why not) doesn't exactly count as "knowing linear algebra".

New >>16062685

>>16061341
>MAID-LISP
It's not real.