/sci/ - Science & Math

CS tard here with a Discrete Math question.
If i have to write a proof for the statement If (X or Y) then Z, would the proof Assume X, show Z be enough to prove the entire statement true?

Similarly would Assume Y, show Z work? Basically how do you write a proof for If P Then Q when your P has a logical or in it.

>>11019162
im not entirely certain what it is youre asking but if youre asking what i think you are then youd have to show that in each scenario where your (X or Y) is true that Z follows
so youd need to show X arrow Z, Y arrow Z, (X and Y) arrow Z.
If you just assumed X like you said, or just Y, all youre doing is showing X arrow Z or Y arrow Z

>>11019175
to hopefully clarify, the statement i need to prove is:
(X or Y) arrow Z

and since "X or Y" can be true as long as X or Y is true then i figured "X arrow Z" would be enough to show that "(X or Y) arrow Z" because Y can be false and "(X or Y) arrow Z" is still true so long as X is true.

Hope that makes more sense.

>>11019191
yeah I just wasn't sure if you wanted to show the validity of the actual statement form or just if you had some hypothetical statement, how you'd show it was true. I'm sure now it was the latter because I didn't read the discrete math part

Could you guys recommend a structured path of math materials to self-study math from the start of high school through to the end of a math undergrad?
I'd prefer textbooks, but I'll take whatever I can get.
This could also be something that can be included int the /mg/ OP in the future.

>>11019191
>>11019200
in which case you need to show all 3 possibilities, because (X or Y) arrow Z doesn't follow from X arrow Z

>>11019208
When you phrase it like that then it makes sense. Thanks for your help!

>>11019205
>officialmgcurriculum.jpg

>>11019139
Why are the departments so run down in 98? Like, how did they manage to keep it clean 30 years earlier? They can't even clean up the spilled coffee.

Also, daily reminder to study Lie algebras - whatever you do.

>>11019205
For high school: read Paul's Online Math Notes and take notes. All of it. It includes ODEs and PDEs by the way.
Undergraduate: Read Rudin's Principles of Mathematical Analysis, Dummit and Foote's Abstract Algebra, and Munkres' Topology.

Also pick up a book on Complex Analysis somewhere. And a book on PDEs to go into more depth though thos is optional at this level if you finished Paul's Online Math Notes. That's it. That's unironically the entire degree without leaving anything out.

If you want to get an edge on graduate studies, read Rudin's Real and Complex Analysis and Rudin's Functional Analysis. There are better books on those topics though but it's nice to read all 3 from the same author if you're self-studying. Then go back and read a real graduate-level book on PDEs.

>>11019205
>Could you guys recommend a structured path of math materials to self-study math from the start of high school through to the end of a math undergrad?

>>11019219
they're better now 21 years later?

>>11019223
>Undergraduate: Read Rudin's Principles of Mathematical Analysis, Dummit and Foote's Abstract Algebra, and Munkres' Topology.
Rudin is a meme.

>>11019245
it really, really isn't.

ive got to give a 3 minute presentation on any topic related to math or math education, does anyone have any novel ideas?
i was thinking something to do with crypto but that seems kind of sweaty and gay

>>11019139
i will divide by zero and nobody can stop me

>>11019219
theyre obviously so distraught by what they dont understand that theyve completely neglected to clean their workspace

>>11019304
Some infinities are bigger than others. I remember when I heard first that, out of context, it seemed really strange and I didn't really believe it. Depending on who your audience is they might find it interesting, and it would really only take a brief explanation of 1-to-1 correspondence to understand it.

this infinity talk reminds me
can anyone give me a brainlet explanation of why cantor's diagonal argument doesn't also apply to the rationals? i sort of get that the new number you construct can be irrational but is this always the case? if so how do you prove it

actually now that I think about it why doesn't it work for the naturals? any number you construct using his method would be in the naturals

>>11019367
>is this always the case?
Yes, because it is constructed in such a way that it is different from every rational (recall the construction).
The reason why this yields a contradiction in the case of the reals is that the process would produce a real number between 0 and 1 that would be different from every real number between 0 and 1.
For the rationals, you just produce a real number between 0 and 1 that is different from every rational between 0 and 1: not a contradiction, there are plenty of those.

>>11019367
>i sort of get that the new number you construct can be irrational but is this always the case?
yes, it's always irrational. probably the easiest proof is indirect and feels circular, but it's not: you know that Q is enumerable (you prove this easily by constructing an explicit enumeration). therefore if someone gives you an enumeration of Q and a number which is not in the list, it's automatically not rational, because otherwise the situation would contradict a result which we know is true.

for the diagonal argument to be invalid you don't really need to know that the number constructed is always irrational (even if it's true). it's sufficient to know that it's not rational necessarily, because than you cannot apply the argument in the general situation: remember that you want to prove that Q is uncountable by deriving contradiction from EVERY possible enumeration, and so in order to apply the diagonal argument correcty, it needs to be valid in ALL cases. it should be easy to construct a specific sequence of rational numbers such that the diagonal argument (which is algorithmic) produces an irrational number. once you do this, it doesn't say anything about whether Q is or isn't countable, but it does say that the diagonal argument cannot be used.

>>11019367
>how do you prove it
You use the fact that the rationals are countable.

>>11019375
>>11019429
>>11019448
thank you frens

>stare at expression for 15 minutes trying to make sense of it
>realize they are using the prime as a transpose operator (as opposed to differentiation, and again as opposed to using T)
who the fuck thought this was a good idea

>>11019573
there are only so many recognizable symbols, watter u gonna do

>>11019573
>who the fuck thought this was a good idea
Blame M*thworks, I think it is pretty likely that they are the ones who caused your suffering.

>>11019139
Lads, I fucked up the exam couldn't even remember that W^{m,p} is seperable for p=1 but not reflexive for p=1 and completely forgot the statement of Rellich-Kondrachov...

>>11019573
>professor is too lazy to actually write exam questions so he just copies them from several textbooks with different notation
im actually going to GUN

What does /mg/ recommend for books on the philosophical foundations of mathematics

>>11019205
Id say start off with Set theory (any author as long its rigorous)
Basic mathematics - Lang
Linear algebra ( axler)
Real analysis ( Tao )
And then pick what u want later.

>>11019632
a firm smack on the back of the head and a reminder to stop being a faggot

>>11019725
not a book

>>11019765
You can do the smack with a book.

>>11019632
Having a strong father figure in your life.

>>11019632
Godel, Escher, Bach
Lakoff- where mathematics comes from

>>11019802
Why the homophobia I just want my choice of axioms to be philosophically justified for maximum rigour/autism
>>11019819
Thank you for the recommendations

>>11019632
>foundations on mathematics
For a start, it won't done using mathematics (unless you're OK with foundations with infinite regress), so you might have better luck in a philosophy thread (like /lit/) than a math one.

Nonetheless, if you're mathematically literate enough to browse /mg/ you can also consider:

-The Language of Mathematics: A Linguistic and Philosophical Investigation (Ganesalingam)
-What is Mathematics, Really? (Hersh)

Ganesalingam is in the spirit of analytical philosophy, Hersh is more continental.

>>11019632
>philosophical foundations of mathematics
Wrong thread. Wrong board even.

>>11019819
These are quite the reads. But be careful anon, your worldview might get shaken up drastically after you finish them. I personally separate my life into pre-GEB and post-GEB periods, since I got enlightened enough to drop the more conventional "maths" and study the pure side (philosophy of maths) almost exclusively (I was in grad school at that time).

>linear algebra
absolute kino, bros

>>11019833
>I just want my choice of axioms to be philosophically justified
What's your mathematical justification for wanting this?

>>11019911
It's really not. Finite dimensional is all trivial and infinite dimensional is only worth studying when there's a topology.

>>11019632
Maddy Believing the Axioms

>>11019917
it's trivial but it's comfy as fuck

>>11019917
>finite dimension is all trivial
Then go and solve all those open conjectures about SL(n) lad.
Literally finite dimensional linear algebra.

>>11019950
Those are more group theoretic than they are about linear algebra desu

>>11019139

So what's the BIGGEST number?

>>11020031
9999999999999999999999999999

>>11020031
+infty

>Study diff geo because you like geometry and applied analysis
>End up with all this algebraic shit
Pls explain.

>>11020031
Tree(2)

>>11020045
can you be more concrete ?

>>11020097
It just seems everything becomes clever manipulation of notation. The geometric arguments are never directly present in the construction of the objects and i the proofs.

>>11020133
can you be more concrete ? (honest question)

c-can someone here help a brainlet understand systems of linear equations using memes and 4chan lingo? ill pay you via paypal when i can start solving problems on my own with a little guidance from you. add me on discord and we’ll talk business
Chrew#7757

>>11020045
I don't exactly know what you're getting at - but worth pointing out that the cotangent space - that is to say, differential forms - end up forming a graded ring and so there a lot of algebra proper emerges.

What're some good quick reads on basic real analysis?

>>11020045
DG is PDEs on topologically more complicated objects. The algebra parts are pure tools to describe what's going on. If you want an example of a very analytically flavoured nice paper, check out https://core.ac.uk/download/pdf/82500356.pdf

>>11020133
That's because differential geometry is high malleable, the problems usually reduce to the lightness real analysis.
If you want to study rigid stuff, go for complex analytic and algebraic geometry.

>>11019632
les nombres (entiers) me semlent constituer comme un monde de réalités qui existent en dehors de nous avec la même caractere d' absolute nécessité que les réaliteés de la nature dont la connaissance nous est donnée par nos sens

>taking a mathematics education paper
>lectures are 3 hours long and the last 3 this semester are on equity and social justice in mathematics
end me lads
this is one of the papers we had to read

>>11020725
what kind of shitty communist school do you go to?

>>11020733
the best university in my country

>>11020744
Burgerland or leaf? If it's anything else I know what I'm going to bait /int/ and /pol/ with for the next few weeks.

>>11020769
his data connection says new zealand in the top left corner

>>11020784
Thanks, didn't notice. NZ is properly fucked nowadays.

>>11020725
>Imagine Social justice change 99.99% Math objects name because begin from cis hetero men.

>>11019833
>I just want my choice of axioms to be philosophically justified for maximum rigour/autism


"There remains the final reflection, how shallow, puny, and imperfect are efforts to sound the depths in the nature of things. In philosophical discussion, the merest hint of dogmatic certainty as to finality of statement is an exhibition of folly." -- My Boy Whitehead

>>11020209
if you have questions you can post them in /sqt/ and people will answer them but I very much doubt anyone in this general is cringe enough to have a
>then you BTFO row two with a based and redpilled multiple of row one
conversation with you, you gigantic fucking sperg

>>11020879
>then you BTFO row two with a based and redpilled multiple of row one
holy shit, I'm tempted to use this line in one of my exercise classes (as a teacher), but I have the feeling that this might either get me into trouble or make the students think that I'm a total nutjob

I have a test this week and it's the first time I might fail a class, because I'm a retard who can't muster the willpower to actually study for once in his life. To make things worse it's also the last exam I would need to take for this degree. If I fail I need to retake the course and waste a couple of months on doing it again early next year.

The only thing that makes this math related is that it's a class on convex analysis. That is all.

Is the left method "allowed" (right side is how it's taught in my Uni book)? I'm not actually sure on the rules when it's not an equation, but I'm more comfortable solving it using quotient rules

>>11020879
>then you BTFO row two with a based and redpilled multiple of row one
>>11020893
Back in the day, a professor of mine would explain stuff like "and then the singularity goes kaboom shlubgrubla and then we've resolved it".
I thought he was autistic, but the rest of the class seemed to have some honest laughter.
How autistic is the class you're teaching?

Is there any sort of generic surjection function kind of like inclusion is to injection?

>>11021286
One answer is partitions (where each element of the domain is mapped to its equivalence class), since every function factorizes canonically into the composition of a partition, a bijection, and an inclusion.

Reminder that NONE of you are real mathematicians until you understand math ethnic studies.

https://www.k12.wa.us/sites/default/files/public/socialstudies/pubdocs/Math%20SDS%20ES%20Framework.pdf

>>11020031
69

>>11019215
>>11019223
>>11019240
>>11019717
Thank you all for the response.
Time for me to get to work.

>>11020031
0 is pretty plump

i can show that there is a multiplicative identity, but i cant get the rest, can anyone help

>>11019139
Never understood this image.

>>11022200
That's why you should work with physicists

>>11021033
Left is "okay" if you would actually manipulate an entire equation, but the terms you have aren't all equal.
You are essentially doing a totally unnecessary step of taking it to the power of e and then taking the ln, which means you still end up with the right thing, obviously.

On the right all terms are equal and it is doing things just in a slightly different order.

>>11020733
Capitalist' butthurt

>>11019950
oh you mean the lie group? the one with a topology? lol
ANYTHING IN LINEAR ALGEBRA WITHOUT A TOPOLOGY IS TRIVIAL.

>>11022194
If there's a multiplicative identity in R, where is it also?

>>11022285
i still do not get it
even the solution manual for the book says its trivial

>>11022266
>ANYTHING IN LINEAR ALGEBRA WITHOUT A TOPOLOGY IS TRIVIAL.
that's not what you originally said anon

>>11022335
Math is not a valid tool same as computers that's why nobody ever uses them

>>11019304
For who? What level of math are they at?

>>11019304
hairy ball theorem

>>11022366
Bus stop vol pint edition

Is it normal to completely understand calculus and be good at it, but not understand linear algebra AT ALL? I'm having large difficulties understanding the point of LA and nobody seems to be explaining it right. Heck, i can visualize everything from calculus in my head easily, but i just can't get LA. Help me out here.

>>11019139
Retard here:
If

u''(x)-V(x-c)u(x)=0,

then u(x-c) is a solution of the equation

u''(x)-V(x)u(x)=0, right?

Mathematica (I have to use it for this shitty project) seems to disagree with me and I don't know who the brainlet is here.

>>11022539
> then u(x-c) is a solution of the equation
> u''(x)-V(x)u(x)=0, right?
What do you even mean by that?

>>11022574
Sorry anon, too tired and got sloppy. I should've written
if f(x)=u(x) is a solution of the equation
f''(x)-V(x-c)f(x)=0,
then f(x)=u(x-c) is a solution of the equation
f''(x)-V(x)f(x)=0
I was sure this is true by a simple linear substitution but computers repeatedly insist they can't do this and now I'm doubting it.

>>11022530
Differential calculus is literally about transforming non-linear problems into linear algebra. Your post screams that you haven't seen functions of several variables yet.
Maybe your LA professor is shit, I don't know, but work on it anon. LA is absolutely essential and not just in calculus.

How does [math]\mathcal{P}(A\times B)[/math] dominate [math]B^{A}[/math] ?

>>11022622
What have you tried?

>>11022637
I know that [math]X\to \mathcal{P}(X)[/math] by [math]x\mapsto \{x\}[/math] is an injection. Is it as simple as associating to each function in [math]B^{A}[/math] the pairs it contains in [math]\mathcal{P}(A\times B)[/math] ?

>>11022329
It is trivial because then 1 must belong to every aR, ie. for any a in R, there must exist b such that ab=1

Now that I think about it, the basis theorem isn't constructive, but it does show that any attempt at constructing a basis by randomly adding in elements that aren't in the span eventually succeeds.

>get 70% on an assignment
>find out later that all the problems were from a text with solutions online
grr

>>11022694
so what exactly are you angry about ? that you could copy the solutions and get 100% ?

>>11022674
No fucking way, dude. You might want to get this deep result published.

WHY IS HE BRAGGING ABOUT BEING AT CAMBRIDGE?

>>11019162
Look into "proof by cases." If you want to prove M or N implies P, you prove both M implies P and N implies P.

>>11022594
Try f(x)=u(x+c)
f''(x)-V(x)f(x)=0
=> u''(x+c)-V(x)u(x+c)=0
=> u''(y)-V(y-c)u(y)=0 where y=x+c

>>11021171
>How autistic
Still to be determined, classes didn't start yet at my place. It's pure math, but still beginner level algebra-related stuff, so probably not as autistic as, say, a differential geometry or topology class, but far more autistic than a statistics or physics related class. Probably gonna make 'em angery anyway by telling them that category theory is just a language.

>>11022335
No, it's not what I originally said, but it's still true. Any purely algebraic fact is worthless and immediate on its own, especially when you have just the right amount of structure as with vector spaces.

>>11022530
Linear algebra is a different way of thinking, but it's also braindead enough that a 6th grader could learn it, so stop getting wrapped up in calculus stuff and just go with it.

>>11022654
Yes.

Hey guys, I'm on IMPA right now, gonna go meet Misha, will he welcome me? We're fellow /mg/ posters after all.

>>11023138
Ask him why does his curriculum have such absolutely barebones analysis.

>>11023138
The library here is pretty comfy

>>11022530
It's all about solving system of linear equations, I don't know why would you want to solve them unlike in analysis where limits are easily visualised and dereivative is just a tangent and integral is area. And you jsut study them but I don't know what's the motivation for solving linear equations unfortunately. Anyway that problem of solving linear equations is solved by a Gaussian Algorithm so everything like rank of a matrix, its determinant, vector spaces etc it's all about gaussian algorithm in the end. So get it and then you will get anything else. There is also a chapter on eigenvalues but I have not studied them yet so I can't say anything about them.

>>11020725
so basically you are studying applied maths lol

>>11022622
the fuck do you mean by dominate? no one uses that goddamn word, goddammit! If you mean is B^A subset of P(A times B) then the answer is yes because you're taking any element f of B^A, that means that f:A->B, but by definition of a function f from A to B f is also a subset of A times B thus f in P(A times B). If by dominate you mean is Cardinality of B^A is stricly less than the cardinality of P(A times B) then the answer is no because you can take A = B = the empty set and then cardinalty (0^0) = 1 and cardinality of P(0 times 0) = P(0) = {0} = 1 so they have the same cardinalities.

>>11021286
Projection on the quotient?

>>11022194
For all nonzero a in R, there's b such that ab = e, because e is in R and for all nonzero a, aR = R. So each element has a multiplicative inverse, no?

>>11022194
>can't get the rest
What rest? Does ring not imply the existence of a unit?
If so, we have ab=b, for any and some a. If there is another c such that cb=b, then (a-c)b=0, but (a-c)R=R, and thus (a-c)d=b. There's an e such that be=d, since Rb=R, then (a-c)be=b, 0e=b and b=0.
Similarly, we have that cb=ccb=(c^2)b=b, and thus c^2=c. But if cc=dc=c, then d=c. So there's a unit.
Checking if I haven't made retarded mistakes is left as an exercise.

>>11023548
Wait, nevermind, the existence of the identity was what he'd succeeded at proving.
I am deeply confused.
Ah, well, if he claims to have done the hard part while struggling at half a line he probably did it wrong.

>>11019139
>the Russians have a top-tier math school which they can attend for FREE
>upon graduation you can go for a PhD in Harvard
What the FUCK Russia?
https://en.wikipedia.org/wiki/Independent_University_of_Moscow

Dumb nigger. I didn't push myself throughout high school and took babby tier math classes. Got to college and had to take precalculus 3 times. I have yet to get passed precalculus and am studying James Stewart's book. Apparently this should be completed in 2 weeks if you're not an absolute retard. 1-2 chapters a day with the 7th being a review day. Is this realistic for a mathlet? I am trying to go at my own pace but feel soon whenever I pick up the book. Any advice? I have an average iq ~115

I'm in CS and have learned just about everything you can without being proficient in math. Basically IT janitor.

>>11023570
In Austria etc. you can also just go to lecture (albeit not take exams - I also don't know how that Russian school would get the stuff from to do the overhead).
Protip, non-US humans also don't let the citizens that vote pay individually for their education

>>11023648
In their school you pass three exams after the first semester and if you do well enough they take you aboard.

>>11022123
Listen to >>11019717 or >>11019223
>>11019240 is a meme. The /sci/ wikia is p good too : https://sites.google.com/site/scienceandmathguide/

thoughts?

>>11023958
There's only Real Analysis, Complex Analysis, Functional Analysis and Harmonic Analysis, the rest are subareas of those 4, that image is just a meme, don't take it seriously.

>>11023570
>https://en.wikipedia.org/wiki/Independent_University_of_Moscow
That school is based, especially the cafeteria

>>11023646
>1-2 chapters a day with the 7th being a review day. Is this realistic for a mathlet?
Maybe if it's literally all you do all day. You should probably be doing at least half of the exercises in the book (at minimum do the odd ones where you can check your work) and I'd estimate a chapter of Stewart contains like 250 exercises. If you can get through every problem in 2 minutes, that's 8 hours on solving the exercises alone.

Understanding the new theory conceptually is totally doable since each chapter probably only really has 1-2 new ideas, but conceptual understanding isn't the whole picture. You need to be able to actually compute things, and there's no way to get good at that other than practice a bunch.

>>11023958
Umbral calculus isn't even calculus. It's combinatorics.
It's also not as cool as the name makes it sound (although it's still pretty neat).

why are epsilon delta proofs of nonlinear polynomials so hard

>>11025183
What's a nonlinear polynomial?

>>11025183
What for example?
Polynomials are, analytically, pretty simple objects.

>>11025290
im retarded i dont know why i said that instead of degree >1

>>11025290
A polynomial which isn't linear, I would assume.

>>11025183
They're really easy. All you need to do is
1) Epsilon-delta for constant functions. Trivial
2) E-d for constant functions. Even more trivial
3) E-d for sum of functions. Trivial
4) E-d for a product of functions. A bit less trivial, but still trivial

>>11025301
Oops! meant to say 2) E-d for f(x)=x (identity function)

>>11025301
>ε=5
>ε=δ
Truly the trickiest of demonstrations.

Anybody here do Operations Research. I'm doing a 4th year research style operations research course, pretty interesting stuff.

>>11025527
>""Operations Research""
>/mg/ maths general
Off-topic spamming is frowned upon here.

>>11025527
I do, it's interesting. Can you recommend any good books?

>>11025553
No books, but the following papers were given in class to explain a few interesting techniques:
- https://arxiv.org/pdf/1603.02378.pdf
- https://people.orie.cornell.edu/huseyin/publications/informs_tutorial_formatted.pdf
- https://pdfs.semanticscholar.org/15b1/a89147803bacfd0bf64167207e6442533744.pdf
- http://www.cs.uleth.ca/~benkoczi/OR/read/lagrange-relax-introduct-fisher85.pdf
- https://www2.isye.gatech.edu/people/faculty/Martin_Savelsbergh/publications/or46b.pdf
For our last assesment piece, we need to take a recently published paper and try to improve their formulation for better speed. I began by trying to reformulate as a flow problem, and use delayed column to increase performance. In the end, we got better performance by removing a single variable in the original problem. Big yikes.

>>11025527
Optimal transport and tropical geometry

>>11025531
Begone autist

>>11025527
I did a short class on optimisation. It's interesting and practical but the course definitively lacked rigors and proofs ; we got told that the KKK conditions were "always needed, sufficient in the case of convex functions" without telling us why.
Most of the class was about numerical methods, coding algorithms, and solving problems rather than exploring the mathematical meaning of it.
It was pretty useful though.

>>11019573
>probability class
>AB means the intersection of A and B
>[math] \overline{A} [/math] means the complement of A
>A+B means the union of A and B when they're disjoint
>calculus class
>[math] \overline{A} [/math] and cl(A) both mean the closure of A
>int(A) and A° both mean the interior of A
>the proof writes if as in pic related which always confuses me because from the second row it looks like a fucking intersection with some smudges

So, in finite dimensions, Hahn-Banach theorem reduces to the hyperplane separation theorem. Was the finite-dimensional, geometric version of this theorem known before Hahn-Banach? Does anyone have explicit references?

>>11025963
>KKK conditions
I think you studied a very different kind of optimization than I have. Please share your textbook, I'd love to learn more

>>11025294
you’re more retarded than you realize

>>11026000
I meant KKT
My brain is broken

>>11024086
Thank you, this is a realistic approach. I get what you're saying that conceptual understanding isn't everything. My approach has been to skim chapter, review exercises, read chapter complete the recommended exercises, and then do all concept problems and odd number skills problems. I've always had trouble with sitting down and actually solving problems, I enjoy math but find exercises/drills tedious. I've been working about 2 hours a day and using a comprehensive math review book to study along with. Don't mean to blog. Thanks for the help anon.

>>11025996
The hyperplane separation "theorem" is a triviality.

That shit is outrageous

https://ncatlab.org/nlab/show/Science+of+Logic

I'm glad I discovered Aluffi. I've been looking for a graduate level algebra text to learn from and this categorical perspective is really cool to someone who knows nothing about Category Theory.
Seeing certain things in terms of universal properties is like ascending to a new level of mathematical consciousness.

I'm so close to being fucking done with this subject.

>what's a group? a groupoid with a single object of course. what do you mean you can't give me any marks for that?

>>11026715
What subject, lad.

>>11026604
Can you believe there are people who read/contribute to nLab seriously

>>11019632
George and Velleman's book is nice

>>11022530
In essence, you are studying systems of homogeneous linear equations and their solution sets in K^n, or at least that was the initial motivation.
You can see them internally, via a minimal spanning set, or externally (dually) as solution sets of linear equations. Duality allows you to go from one viewpoint to the other.
You can also think about them geometrically as linear subspaces of K^n (eg. lines, planes, etc. passing through 0) and think of basic linear algebra as studying intersections of such subspaces.
Then, you can study general finite-dimensional vector spaces, which are interesting because a number of things form finite-dimensional vector spaces. These are essentially the same objects as before, except that you might not necessarily have a natural choice of a basis which identifies your space with K^n. Once you fix a basis, you are taken back to linear systems etc.
This gives you the fundamental duality of linear algebra. You should think of all concepts in linear algebras as coming in pairs, one concrete (on K^n), and one abstract (on abstract vector spaces).
For example, given vector spaces V, W of dimension n and m with a choice of bases, you have correspondences:
column vector of size n <-> element of V
row vector of size n <-> element of V*
matrix of size m,n <-> linear map V -> W
transpose of a matrix of size m,n <-> dual linear map: W* -> V*
etc.
Usually, one of the descriptions makes things easier to understand, but there is no general rule for which one it should be. You acquire a flair for this with experience.

>>11026604
>https://ncatlab.org/nlab/show/mysticism
nlab is crazy sometimes.

>>11026842
Once you have this abstract notion of f.d vector space, you can study its endomorphisms (ie. linear maps from it to itself). There are several reasons to study this, many of them coming from the theory of linear differential equations or recurrence sequences.
The case of one endomorphism is completely solved over C (it is called the Jordan normal form).
The simultaneous study of several endomorphisms is more complicated and at the foundation of an area of math called representation theory

>>11023071
Yeah no, Jordan form is already non trivial though purely algebraic, and I would very much like to see you explain trivially how to classify pairs of matrices of size n up to simultaneous conjugacy.
Representations of quivers are not a trivial subject. It contains many open problems even though it essentially amounts to classifying families of f.d. vector spaces and linear maps up to simultaneous conjugacy.

How do I best prepare to barely pass a math exam where I know I won't have enough time to really grasp all of the concepts any more?

I'm thinking just to review assignments and hope to see something familiar.

>>11026909
At that point, it becomes a matter of failure mitigation. You clearly don't understand anything, so you have to resort to what you do know how to do - memorizing what other people did and attempt to reproduce it. I would suggest in class examples, anything marked with 'I leave this as an exercise' (just google the answer, cause lets be real you've been doing this all semester), and of course assignments. Wake up the day of the exam early and warm up at least 2 hours before the exam drilling definitions, proves, problems, etc. Before the exam, take a hit of jenkem and pray.

>>11026858
"hurr durr cyclic vector subspaces" done
and representation theory is for morons, that's the only reason those exercises are "open".

>>11019139
Any hints with this proof? Or just show me how to do it? Working my way though Townsend which has no solutions

>>11026231
Completely untrue, especially in infinite-dimensional locally convex spaces.
>>11027197
Two hints:
1. What group rotates rectilinear states into circular, spinning states?
2. What is the double-cover and representation theory of that group?

>>11027201
Nigga that’s greek to me - speak low IQ

>>11026814
Math itself.

>>11019139
How the fuck do I understand derivatives? This shit is so tedious.

>>11027213
Define "understand".

>>11027197
>verify that the matrix is unitary
HMMM, I SURE WONDER WHAT DID THEY MEAN BY THIS.

>>11027210
Can you please not use the n-word? I find it very racist and disgusting.

Anyways, think about how [math]|{\bf x}\langle[/math] is related to [math]|\pm \hat{\bf z}\rangle[/math]. The z-component tells you the handedness of the photon.
Alternatively if you're in 4D spacetime [math]M[/math], the [math]spinor[/math] bundle [math]S\rightarrow M[/math] allows you to define the unitary, idempotent chirality operator [math]\gamma_5 = i \prod_{\mu=0}^3 \gamma_\mu[/math]. So you just need to write linearly polarized states in terms of states in the eigensectors of [math]\gamma[/math].
Hope this is helpful.

>>11027237
what are you some kind of darkie?

>>11027240
No. It's just very distasteful.

>>11027237
>>11027243
He said nigga and not nigger, you nigger. It's fine.

>>11027243
not really, if its written with an 'a', and it's on four chan, fag.

>>11027237
I meant eigensectors of [math]\gamma_5[/math], of course.
>>11027251
Enough with the f-slur. It's personally offensive.

>>11027265
>eigensectors
What.
Is that physics notation for something I know about?

>>11027265
I've been around homosexuals in my math department and I find that they smell of sulfur pretty generally. Do you?

>>11027283
he's working with bundles and not with spaces, because she's an obscurantist who has little interest in providing actual instruction.

>>11027283
Given an idempotent self-adjoint operator [math]A[/math], its eigenvalues are [math]\pm 1[/math]. The eigenspaces corresponding to these eigenvalues are called "eigensetors", because the spectral theorem (given [math]A[/math] acts on a nice enough Hilbert space [math]\mathcal{H}[/math]) tells you that [math](\operatorname{im}(1-A))^\perp \cong \operatorname{ker}(1+A)[/math]. So the short exact sequence [math]0\rightarrow \operatorname{ker}A \hookrightarrow \mathcal{H} \rightarrow \operatorname{im}A \rightarrow 0[/math] gives rise to an orthogonal decomposition [math]\mathcal{H} = \mathcal{H}_+ \oplus \mathcal{H}_-[/math] into sectors of [math]A[/math], where [math]\mathcal{H}_\pm = \operatorname{ker}(1\mp A)[/math]. In fact, this gives [math]\mathcal{H}[/math] a [math]\mathbb{Z}_2[/math] grading.
As for [math]why[/math] these objects are named "sectors", it's because the physics, described by Heisenberg dynamics of observables on [math]\mathcal{H}[/math], completely decouples with respect to this decomposition. For instance, if [math]A = (-1)^F[/math] is the fermion operator, the eigensectors are the "bosonic" and "fermionic" sectors. If [math]A = C[/math] is the charge conjugation operator, the eigensectors are the "charge" and "neutral" sectors. These idempotents [math]A[/math] can be used, in a sense, to "detect" the degrees of a freedom of a physical theory.

>>11019448
>>11019429
>>11019375
all retard answers

>>11019367
the reason why the diagonal argument doesn't work on the rationals is they don't have an infinite random 'tail' of decimals after it. The decimal tail of a rational number is either 000000..., or a repeating sequence. You can't 'algorithmically' find a rational that does not fit in the enumeration since given any rational number, its tail end is already predetermined by a finite truncation in the decimal expansion.

>>11022674
perhaps u might see it more obviously non-constructive when you see its equivalent to the axiom of choice

>>11027220
be able to do them well enough so as not to fail calculus.

>>11027396
why can't i just replace with things that make up another pattern then, you fucking moron? prove i can't. you are a braindead idiot claiming other people's posts are bad when your post is far worse.
>inb4 le intuitive argument!!!
>>>/lit/ is that way.

>>11027416
okay, remember the rules and just do them
there are like 4 rules for christs sake, and really it's just 2 when you break it down (product and chain)

I KNOW YOU GUYS ARE GOOD AT MATH
SO TELL ME THE ODDS FOR CRAPS SO I CAN WIN

BEST BETS
BEST ROLLS

>>11027418
yeah but then implicit and logmorithic. The rules aren't hard they just combine them in wierd ways that make it hard.

Like
(sqrt(x-1))/(e^(cos(x))(5x+4)^3)

Please someone help me with vectors parallel lines, perpendicular or neither. I'm really confused and don't understand and the textbook is poorly written

Hey guys. Math student here, late undergraduate, so I have taken real analysis and abstract algebra.

I have a real problem with 'learning math'. It is very hard for me to open up a math textbook and learn from it in a A -> B -> C -> D -> ... -> Z manner.

I think its a manifestation of my anxiety to make sure that I go in a sequentially totally ordered fashion, but it seems more natural for me to skip around.

Why does everyone on stackexchange and /r/math recommend reading a math textbook like an autist by doing every exercise and making sure you over understand every sentence?

>>11027450

The basic form is

X = tV + B.
X' = tV' + B'.

X and X' are parallel if V = V'k for some k in the real numbers.

If X and X' are not parallel, then
if X and X' do not intersect then pick neither.
If X and X' do intersect but B dot B' does not equal zero then pick neither
If X and X' do intersect and B dot B' = 0 then pick perpendicular.

>>11027441
Sure, those are the other 2 rules. You use one when x and y are not easily separable and the other when there's a variable in both the exponent and the base. They're also very formulaic. Don't trick yourself into thinking they're complicated or trying to reason with them.
The thing you posted just needs to be broken down. One way I think of it is to name parts: f(x) = sqrt(x -1), g(x) = e^(cos(x)), h(x) = (5x+4)^3. Then I want the derivative of f(x)/(g(x)h(x)). I don't remember quotient, I just use product on f(x)* 1/g(x) * 1/h(x). I do remember that on 3 things, product rule just differentiates each one one at a time and leaves the others. So this is f'(x) * 1/g(x) * 1/h(x) + f(x) * -g'(x)/g(x)^2 * 1/h(x) + f(x) * 1/g(x) * -f'(x)/f(x)^2, where I just chained the reciprocals. Then I just need f', g', and h'. I compute those quickly, plug it all in, and see if anything cancels when I simplify.
It's not a small amount of work, but if you actively break it into pieces you won't get lost in the mess as often.

>>11027451
because stackexchange and r/math are full of undergrad pseuds and people who couldnt make it who spend all day advising others on how to fail at mathematics like them
anyone with a future in math does not need to do every exercise and read every sentence. they read the theorem headers, quickly prove everything in their head, and skim through the exercises looking for any that might present more than a few moments' diversion.

Guys, I think I found a new technique to tackle Collatz mapping (my thesis is on the Collatz conjecture) and Terry Tao, who recently made some pretty significantly progress in this area, is giving a talk at my university next week - but it's on PDEs. Is it appropriate to go to his talk and discuss my ideas with him even though it has nothing to do with his talk?

>>11027525
yes. please amuse him with your drivel you fool.

>>11027417
The poster is asking WHY it doesn't apply to the rationals. And the reason is that precisely if you start with a supposedly rational number that is not enumerated, you will immediately arrive at a contradiction due to the fact that the number is already predetermined by a finite number of algorithmic modifications to the number. The fact that in general, any enumeration does not allow for this repetition, contradicts the initial assumption that the number was a priori rational.

How do you think of the line bundles [math]\mathcal O(d)[/math]?

>>11027525
fucking stupid reddit scum
imagine unironically being that guy
this is why number theorists are a plague

>>11027568
what in the fuck are you on about. stop posting any time.

>>11027568
>the number is already predetermined by a finite number of algorithmic modifications to the number
that totally doesn't prove that the result is not rational

?

https://m.youtube.com/watch?v=v_o-allq8LQ&list=PLDesaqWTN6ESk16YRmzuJ8f6-rnuy0Ry7&index=10&t=0s

Does this guy even know math?

>>11027451
>Why does everyone on stackexchange and /r/math recommend reading a math textbook like an autist by doing every exercise and making sure you over understand every sentence?
don't do this. skim through the whole book. then skim through the chapter. then read the chapter, but focus on understanding the definitions and main theorems and try to get some kind of bigger picture. you can skim through the next 2 chapters to see if and how the stuff is used. then you can read the chapter in details with proofs and exercises if you're interested.

don't read math textbooks line after line. your brain needs time to process. if you don't know anything about a theory, it's better to work through the theory multiple times going into more details each time.

>>11026971
That works for one endomorphism, not for a pair. I don’t even care either way, I am not a representation theorist, but thinking like this is either poor bait or legit retarded

>>11027525
Yeah sure, but do show him that you have at least skimmed his paper and do not tell him that you want to solve it (it’s not clear from your post if you do, but if you do, don’t tell him).
Mathematicians, especially famous ones like Tao are swarmed with crackpots. If you start talking about Collatz (and especially solving Collatz) without some background, I think his crackpot-o-meter will go crazy and he might get annoyed

>>11027433
ANSWER ME NOW

>>11027677
Write P as a span of vectors.

>>11027722
you might be one of the stupidest people i've ever had the disgust to interact with on /mg/. be proud it's not on all of /sci/. go play with your toys somewhere else.

>>11027433
>>11027752
only scum gamble

>>11027742
it's an autistic reddit post you shitter. stop encouraging the trolls.

>>11023337
saying that linear algebra is just "solving systems of linear equations" is literally like saying that calculus (or analysis) is just "applying bunch of random rules like x^2 -> 2x to transform functions into different functions".

>>11027793
linear algebra is quite literally founded on solving the equation Tv = w for a linear operator T and a known w.

>>11027793
Yeah, you're right
Linear algebra as a "tool" is just solving systems of equations
Linear algebra as a discipline gives you a good understanding of vector spaces (vital for Fourier analysis and functional analysis, see Hilbert Spaces), gives you good analogies for the theory of linear operators, and many other things
Not to mention the countless applications

>>11022530
LA is more abstract than calc. You can't just draw potatoes and figure out what's going on by transposing the problem in 1 or 2 dimensions and using your geometric intuition.
On the flip side, the proofs are easier to remember and to do "yourself". Just identify what you want to show, what you already know, try to formalize the problem with a system of equations and you're all set.

For instance, the incomplete basis theorem is harder to visualise than, say, the intermediate value theorem in basic calc, but you can prove it almost immediately by just remembering the definition of a basis.

>>11027819
Linear algebra is a tool to study vector spaces over a field K, which are a powerful structure you see everywhere else in mathematics & in applications.
Its most important idea is the notion of a "basis" and of dimensions. Understanding those ideas is fundamental for any mathematician, engineer, physicist, chemist, whatever needs to do at least a tiny bit of math.

Quantum Mechanics is a direct application of linear algebra (with the wave functions identified as vectors, and the operators being identified as matrices).
The main subject of Functional analysis is the study of topological vector spaces. All theorems in LA apply.
Fourier analysis make heavy use of LA, since its object of study are Hilbert spaces with a particular scalar product.
And so on, and so on.

" Tu = v has a non-trivial solution <=> any non-trivial solution of Tu = v is unique <=> det(T) = 0 " is basically the fundamental theorem of linear algebra but it's already all there is to it.

>>11027838
Let me just add one thing. Babby linear algebra and babby (single variable) calculus feel like completely different subjects. But differential calculus is actually all about replacing non-linear stuff with linear stuff (and taking the limit). This becomes more apparent in multivariable calculus, even more apparent in vector calculus, and in differential geometry (my field of interest) calculus and LA literally become one single subject. And you will be thanking god that LA is as easy as it is.

>>11027838
Whoops, I meant Tu = au, I'm retarded

>>11027844
>calculus and LA literally become one single subject.
Same for real analysis, LA becomes the special case of L^p(|N_{<=n},|R,#).

From top to bottom

Round(x)
Ceiling(x)
Floor(x)

Sorry for shitty quality

>>11027289
>working with bundles and not with spaces
Wouldn't that be an eigensection or an eigenbundle?

Lads, why didn't you tell me Encyclopedia of the Mathematical Sciences books were so comfy?

>>11022530

ALl linear algebra is using row reduced echelon form on a bunch of equations in some number of unknowns

>>11028039
>prove the universal property of the tensor product
>gues I'll use row reduced echelon form on a bunch of equations

Morally, what's the difference between a quotient and a projection?

>>11028068
In what context?

How do you simplify this into x + log2 (3)

>>11028081
log(2^x + 2^(x+1)) = log(2^x (1 + 2))=log(2^x) + log(3) = x + log(3)

>>11028068
what's your definition of projection and quotient ?

>>11028068
A projection should be thought of as a direct sum decomposition, so in some sense a special case of quotient map, which «splits»

>>11028074
Ideally the most general context that distinguishes them, though I'd be happy with a geometric (or at least topological) context -- as a math autodidact I haven't really learned to develop intuition/visualisation for these fields.

My current conceptualization is that a projection is a surjection out of a product P, so is meaningless outside of the product construction (which in turn acquires its meaning from its universal property). On the other hand, I think of the quotient operation as going from a "mere" object P to its factors, by "fiber-ing" along some sub-object N of P. I'm wholly aware that this is not a helpful way to think of them, though.

>>11027433
All odds are 50/50 anon. Either you win or you don't

I'm trying to prove that if f is differentiable on R then f' is continuous on R. How would I prove that?

>>11028119
LADS.
I HAVE A TEST SATURDAY SO I CAN GET INTO A PHD PROGRAM.
RECOMMEND ME YOUR BEST LAST MINUTE ANALYSIS BOOKS.
>>11028229
You use that other definition of the differential.
The one with an o, you know the one.

>>11028237
Actually, I read proving f is continuous.
You can't prove that f' is differentiable because it's wrong.

>>11028229
this is not true. f(x) = x^2 sin(1/x)

>>11028242
Give me a counter example then.
>>11028244
The derivative of x^2sin(1/x) does not exists for all x hence not statisfying the hypothesis of what I'm trying to prove (or disprove).

>>11028252
f(x) = x^2 sin(1/x) is differentiable at each point

>>11028252
>The derivative of x^2sin(1/x) does not exists for all x hence not statisfying the hypothesis of what I'm trying to prove (or disprove).
Not that anon, but you can define
f:R->R
by
f(x)={x^2 sin(1/x) for x!=0
0 for x=0}
Then the derivative exists for all real values of x and yet the derivative is not continuous.

>>11028265
(including the obvious extension f(0) = 0. I hope it's clear that we're talking about the extension and not just about the formula which is valid only for non-zero x)

>>11028266
>>11028270
The derivative of this function is indeed not continuous at x=0. Thank you.

Is Differential Geometry interesting to study? This spring will be my last semester of undergrad and I was planning on taking either algebra, number theory or topolology, but I just realized none of them are offered in the spring.

It's really either that, complex analysis (which seems like a lot since I'll be taking it concurrently with real analysis), or probability/statistics (which might be a good fit since I'm dual majoring in CS, but IDK.) I've never had much interest in Geometry but I thought taking a class on it might change that.

>>11028306
>Is Differential Geometry interesting to study
yes

>>11028306
>is diffgeo interesting
Yes.
>I've never had much interest in geometry
Then, no. Just go for complex analysis. It's extremely comfy.
Besides, diffgeo that doesn't require topology has a 500% chance of being dogshit.

>>11028306
DiffGeo is applicable to CS, if plqny on working with computer graphics at least

>>11028083
what did you do here? The way I try to solve is to use the x^(a+b) = x^a * x^b, but I'm not sure how to factor the 2^1

>>11028317
>Besides, diffgeo that doesn't require topology has a 500% chance of being dogshit.
this is true actually

>>11028370
2^x+2^(x+1)
= 2^x+(2^x)*(2^1)
= 2^x+(2^x)*2
= 2^x*(1+2)

>>11028303
why are you thanking them, idiot. they literally just spoiled you. And why have you asked for a counter-example in the first place, you are even more of an idiot than i thought. You still have no intuition whatsoever on why the continuity of a derivative might fail. Not like it's super important or whatever but if you will treat every other example like that you are not going anywhere. Idiot.

>>11028396
are there any rules or properties I should acquaint myself with to solve these kinds of problems? There's very little in my book about simplifying when there's problems in the form of x^(b+c)

>>11028370
>not sure how to factor the 2^1
>>11027611
The line bundle associated to a divisor on a Riemann surface?
Essentially, the holomorphic functions on a Riemann surface are the sections of the trivial holomorphic bundle. We have a divisor, and we want to associate a meromorphic function to it, but we can't, since it isn't a section of the trivial holomorphic bundle, so we apply the oldest trick in the book: we find somewhere where it does exist.
This gives a line bundle.

>>11019139

What is a good book to learn about Stanley-Reisner rings?

Last year I was reading about them, but I forgot the name of the book I was reading :(

>>11028444
x^(b+c)=(x^b)*(x^c)
x^0=1
x^1=x
Other than that, it's just the rules for addition and multiplication.

>>11028481
Anything by Springer or Birkhoff is good.

how do i solve x'(t) = x(t-1) bros...

>>11028651
Discrete time system and interpolation

>>11019139
I am having hard time understanding lambda calculus. Is this maths or comp science?

>>11027677
Which book sir?

>>11028686
That's obviously not a book,r eatdad.

>>11027677
Extremely easy.
the normal vectors are
(1 1 1 1)
(1 1 -1 -1)
you need to find two independent vectors which are perpendicular to them. They will span the plane
One example
A= (1 -1 0 0 )
and
B= (0 0 1 -1)
a point that's in the intersection is
r = (1/2, 0, -1/2, 0)
Hence the vector form is P = {r + aA + bB}

Does PRA, PA or any other system of arithmetic have a finite submodel?

>>11028664
but I want exact solutions, fren...

>>11028755
You can also solve this without any calculation by just thinking geometrically in phase space ; represent the trajectories of your system in phase space (x', x) [that's why I told you to use a discrete time system and interpolate, to get a geometrical picture of what was happening)
These trajectories will probably help you with the problem

>>11026797
based
I've also read a bit of Emily

I'm really struggling with understanding lines in vector column form. This is the only concept I am having a hard time with and there isn't many videos explaining it

>>11028884
Parallel is the directional vector (the one in "t") are proportional
Perpendicular if the scalar product is 0
Any other case is neither

I keep hearing that category theory is not real mathematics
Why is that?

>>11028892
it's the algebraists screeching over the fact that they are no longer the abstractest (and therefore on the biggest brainflex pedestal) in town

>>11028892
It's so abstract and removed from your mathematical intuition
It's still math, just a very abstract category of math
In the same way that study of diff. eqs. is still math even though it's extremely practical

>>11028890
What would I search to learn more about this? I never see examples in column form so it makes it more confusing to me

>>11028892
Similarly to logic, set theory, and theoretical CS (which have been called "not real math" at some point), category theory can be studied as a foundation for mathematics, in which case it ceases to be mathematical. Think of how proofs of the "fundamental theorem of X" (e.g. X = algebra, arithmetic etc.) can't make use of theorems of X, on pain of circular reasoning.

Of course, you can interpret category theory as an extension for some parts of algebraic topology, from which it derives its mathematical legitimacy (at least until you start getting weird ideas like trying to define logical equality in terms of homotopies).

>>11028992
>studying math foundations is not math
are you high?

>>11019162
ok, you're looking for something like modus ponens, like, you have something like this:

A -> B
A
--------------
B

like, if A -> B is true, and A is true, then B must also B true, A and B are not like atomic propositions, but more like subformulas, so you can have, like, A = (p or q) and B = something else and get the same thing without loss of generality. Sorry for bad english, i'm kind learning right now, like, in the web

>>11028892
People who say that are trying to prevent you from becoming pic related.
There's no reason to study cat theory until you actually start needing it for other stuff.

>>11029042
well the only people who end like this were stupid ones to begin with

>>11028992
>theoretical cs
>maths
This meme needs to die.
>>11029042
>not even interested in the fields that category theory was invented to deal with
Kek.

>>11019139
>friend struggles to find postdocs despite having a really good dissertation that actually advanced the theory
>departments across the country, ours included, are eager to take on new alg geo postdocs even if they already have two or three
When did you realize that academia is a giant circlejerk, and choosing not to work in a prestigious area is essentially choosing not to be a successful mathematician?

>>11029080
About the 3rd undergrad year, I think

>>11028892
I find it incredibly ironic that there are practicing mathematicians that criticize things for being 'abstract.'

>>11029080
what field?

>>11029080
Tell me more.
>despite having a really good dissertation that actually advanced the theory
Was this what he told you?

how can there be 2 direction vectors? shouldn't 1 always be redundant

why was yukariposter banned

>>11029093
Homotopy theory

>>11029103
I actually went to her defense and talked about it with her and some faculty, so I can confirm it wasn't just some filler paper to get the degree.

How would I prove this? I don’t need a full answer, just a hint. The other direction went fine.

>>11029255
Can you type out what you mean? I can't read it.

>>11029255
My hint is asking middle school questions in /sqt/.

>>11029255
>The other direction went fine
It's symmetric. Both directions are the same. Think about it.

>>11029259
This is from Paul Halmos‘ Naive Set Theory.

>>11029258
B‘ is a subset of A‘ implies that A is a subset of B, both being subsets of some other set E.

how do you remember all the math you've done? do you have a routine or something?

>>11029264
>from Paul Halmos's Naive Set Theory
Yes, Paul Halmos's Naive Set Theory covers middle school shit.

Is A' the complement of A?
Do you know what the contrapositive of a logical implication is?

>>11029278
>Is A' the complement of A?
Depends on the context, but probably yes.

>Do you know what the contrapositive of a logical implication is?
The contrapositive of
P=>Q
is
(not Q) => (not P)
The former implies the latter (and classically the converse is also true)

>>11029274
Just use textbooks or online resources for reference.
I don't think it's reasonable to be able to have all the mathematics you've learned immediately available for recall from memory. But provided you learnt it properly to begin with you should be able to refresh yourself rather quickly.
If you never understood the stuff to begin with then you are a scrub.

>>11029286
Ok, translate those subset statements into their definitions. See if anything jumps out at you.

>>11029299
It wasn’t me you responded to, but thanks.

>>11029274
I have a slow, autistic, but useful way to remember it. I just take a textbook, skim through it to get the "feel" of the book, identify the parts I want to read in full and remember, then I just copy it almost 1- to-1 with my own notation, comments, and the omitted proofs written in full. This way, I "make mine" the textbook, and when I need to refresh my memory, I just read my notes which I understand immediately and with more "context" than just re-reading the textbook (I usually outline what I think is important to remember)

I take too care of making my notes as clean and beautiful as possible, so I get a midly vain pleasure when I re-read them
Pic related is what it looks like

>>11029274
Simple answer: You don't, at least not everything at the same level of detail. But that's not necessary, since the longer you study and later do your own research, the more stuff will be in a way etched into your brain. Still, your own research and in particular the needed technices will always be clearer than the rest.

And as an advice always make sure your research notes are in a readable format, like number the pages on paper, write down real language sentences as to what you do, and so on. Will make remembering stuff easier.

>>11029331
that's pretty neat

what happens if I bomb my first linear algebra midterm? Is my GPA fucked for the rest of the 4 years?

>>11029577
people only really care about your last 10 or so serious courses. don't worry about it.

Guys please help. I need to create an example of a function f : R --> R (R being a real number) such that f(f(f(R))) = f(f(R)) =/= f(R).
I thought f(x)= x^-1 would work but that would make f(f(f(R))) = f(R). What the hell do I need to do?

>>11029274
You don't, but you remember the stuff that you need (even then, maybe not in full detail, but you understand the structure of the theory, where to look up details, and the general sketch)

How do you catch up if you fall behind in math class

>>11029709
>pick a set X
>pick a set Y in X
>set up the double projection trick: f(R)=X, f(X)=Y, f|Y=id.
>>11029790
You study.

im lost

>>11029797
Thanks for answering my question but your answer kind of confused me. Could you please clarify? So when you say set X it could be like X contains (1,2,3,4,5,6,7) and Y contains (2,3,4,5) so Y is a set in X right?
Also, what I should read to further understand the double projection trick you said because f|Y=id makes no sense to me. I am an undergrad if that explains my ignorance.

>>11029806
the solution is

>>11029823
Sure thing, those X and Y work.
f|Y=id means that, for any y in Y, f(y)=y.
There is no thing to read further.

>>11019139
are hyperreals made up? I read a book that used them, have I been pranked?

>>11029869
what in math isn't made up?

you think I can learn the non-checked concepts in 6 hours?

>>11029857
Thanks a lot for your help.

CS Tard here taking Discrete Math. Literally fucked up the proof of this theorem on an exam today. Feels bad man.

>>11029957
If there's one thing programmers should know is how to divide the problem into a number of boring cases

You start out with a Riemann surface S, and you give it the usual metric.
Then, you embed it into the line bundle over S whose sections are holomorphic functions, S x C, with the obvious metric.
A holomorphic function on S is a section of S x C, and trivially a biholomorphism onto its image with the obvious complex structure.
By Gauss's result on local isometries, the curvature of the image needs to be the same as the one in S, or at least a constant multiplie everywhere.
And, thus, the only holomorphic functions in S are constant functions, unless S's curvature is everywhere zero.

This is an extremely retarded proof of a babby result, and I've most likely made a mistake somewhere, but I genuinely can't stop thinking about it.

>>11029999
I actually had 4 cases, and the first 3 were fine. Had a dumb mistake on case 4. Only problem I fucked up on but goddamn it stings.

don't know the latex syntax apologies in advanced nerds

when you start with 0 and add n it's multiplication. But what is there a name for when you start with {x | x != 0} and add x+n?

Hi, ESL here, I have a question, well it might be a grammar problem.
So this question “The quotient of a number and -5 has a result of 2. What is the number?”
So here, i’m not sure who is divided by who, and i gave the answer n=2.5, which was apparently wrong and n was -10.

My question is does the original question imply who is the divident and who is the divisor, or was it a poorly worded question.

>>11030048
question is kind of phrased weird. a more verbose way would be more like "the quotient (result of dividing) a number, x, and -5 is 2. What is the number?"

quotient = 2
a number = x
-5 = -5

x / -5 = 2 (quotient)
(x/-5) * -5 = 2 * (-5) [multiple by -5 on both sides to isolate x]
x=-10

>>11027525
are you that autist from reddit?

>>11029042
that pic lmao

>>11029274
I don't. What remains constantly in my mind are the definitions and some vague notions of what is true about them, together with various proof concepts.

>>11029274
>how do you remember all the math you've done? do you have a routine or something?
Memory palace.

>>11029806
What is the question? Just do the algorithm and plug in, existence is trivial.

>>11029872
>you think I can learn the non-checked concepts in 6 hours?
Why don't you try it and find out?

>>11029274
No, not everything. The point is that you'll be able to go back to those things that you learned already and pick them up again easily.