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/sci/ - Science & Math

Previous >>10616540

/mg/ - mathematics general

cube roots edition
>>
Can someone explain to me why you would rather (in a non-expository setting, ie: in your private study) use a chalkboard or a whiteboard over just pen and paper? Not only is it dusty and messy and way more expensive, requires more care, etc, but you are required to erase all of it after you're done, or even before you're done because it doesn't fit.

With pen and paper, you can work endlessly, paper is extremely cheap, bic pens are extremely cheap, and you can store all your rough work so that in the future you might go back into something you did. I've done that a couple of times - look through my stack of rough work to find some resolution I found some time ago.

I don't get it
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>>10637621
get a remarkable
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>>10637626
a remarkable what
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>>10637621
You can erase/rewrite stuff as you go.
There's more room than on a single page.
Uses different muscles so it doesn't hurt my injured hand.
You can photograph anything meaningful you come up with anyway.
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>>10637621
Have you ever worked on a long hard proof and had to rewrite it a billion times?
I have to use LaTeX to write long proofs because if I do it on paper, I keep having rewrite stuff and it gets very very messy.
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>>10637691
Is latex good for note taking in general? I have only used word before but it doesn’t seem as good as latex
>>
>>10637792
It's great for incorporating lots of math in a document.
Word is terrible at that.
So it really depends on what kind of science you do.
>>
Do CW complexes require you to have at least a 0-cell?

I'm asking because I just saw a question to compute the cohomology of CP^n \ CP^r (set minus, not quotient), which feels a bit wierd.
>>
>>10637812
P^n = A^n + A^n-1 + A^n-2 + ... + A^1 + A^0
so you can always find a CP^r taking the last r+1 things
>>
https://discord.gg/VMwGpSF
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>>10637606
on /lit/ I saw a post saying if you want to understand the world then study mathematics, obviously I know maths is involved with everything but he seemed to imply learning maths changes the way you interpret and experience the world

does anyone know what he meant?
>>
>>10637792
There's Lyx which is a mix of Word and Latex.
People should use that instead desu.
>>
>>10637871
Once you cut truly deep into pure math, you can never go back to seeing things normally. The rigour, the clarity, the logic; it's inescapable - your world view will change, not because math gives you any more inherent knowledge of the world (although it might do), but because you can better abstract and reason your ideas to achieve a new level of clarity.
>>
>>10637952
is this something the average person can achieve at home?
>>
>>10637952
How deep is truly deep?
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>>10637956
Harder, but yes. Ultimately it would be best for someone to critique your work, since otherwise it would be impossible to know if what you're doing is right or wrong. I say this, because there is a classic saying - mathematics is not a spectator sport. You cannot just read a book in mathematics and absorb its information as if it were a novel - you need to attempt the proofs and the exercises. However, most exercises do not come with a solution, and here is where the 'expert' would come in to tell you you're wrong. Luckily, in the information age, you can have experts at a distance doing it for you, whether it be here, on /sci/, or more professional sites like Math Stackexchange/Overflow.

I find it rather funny; MSE is probably one of the few stack exchanges where people will solve your problems for its own sake, not for your benefit necessarily. As such, it's a great resource, since for example, in the Physics SE, they will always find a way to say your question is irrelevant.

Deliberation aside, yes, you can do it at home. In fact, there are several meme lists that could pave your way, but be wary - this will not be easy, or short. It will take time and effort.
>>
>>10637964
>>
>>10637973
cheers pal, even if I never change how I view things at least I'll be improving a very practical skill
>>
File: physical maths.png (1.8 MB, 1202x910)
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Threadly reminder to work with physicists.
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>>10637871
Mathematics did not make me understand the world in the philosophical sense.
It did make me much more openminded, intellectually honest (with myself) and objective. That is a required attitude if you want to rigorously proof theorems and it is also useful if you want to discern facts and truth in a world full of lies.
>>
>>10637621

This
>>10637666

When you're scrolling through a long webpage on your computer, wouldn't you rather have a huge screen so you don't have to scroll and your mind can more intuitively remember where something is at based on position? Same goes for a whiteboard vs paper. If I'm working on something that requires a drawing, the drawing could continuously get bigger as I work on it more thoroughly which requires a huge contiguous workspace. I do have 3'x3' sheets of graph paper but laying that across a table and hunching over those to work is killer on the back, not to mention totally inconvenient. The whiteboard you just move to the spot you want to work. EZPZ
>>
Combinatorics q. Say you have a set $A$. How would you go about counting the number of sets of subsets of $A$ (so the number of $X \in P(P(A))$) such that no element is a subset of another ($\forall_{x, y \in X} x \neq y \implies x \not\subseteq y$)?
For example, for $n = 2$ we have 5 such sets:
$\{\{\}\}, \{\{1\}\}, \{\{2\}\}, \{\{1\}, \{2\}\}, \{\{1, 2\}\}$
not counted is for example $\{\{1\}, \{1, 2\}\}$ since the first element is a subset of the second.
>>
What is $\mathbb Q[\sqrt2]\otimes_{\mathbb Q} \mathbb Q[\sqrt2]$? I know it's a 4 dimensional vector space, but how can I describe it as a ring? I know it's not a field, since the canonical map into $\mathbb Q[\sqrt2]$ contains in the kernel $(\sqrt2\otimes 1-1\otimes \sqrt2)$
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>>10638154
So the number of partitions, plus the number of partitions of all subsets of A?
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>>10638154
It seems like there is only one valid countable value of {1,2} for any number of subsets A, then. And if N is some number, then it seems that we can divide by 1 to get a countable value (i.e. 1+N). But to use it, the first element must be a subset of the second.
For example, suppose we have 5 sets:
{{1},{1},{1},{1}}
Then (1+3) = 5, or {{2,3,4}} = 2, or 2+5=1, 2 = 1+2+3+.
In this particular example, the value of (2,3,4)=(x,y) is the same; but when we multiply by 1+3/2 , then the value of x is changed for the second argument and the value of y is changed for the first arguments of each of our formulas. In all cases, the change is additive; that is, x + y * x is the same as x + y + y . In addition, (x,y) must be a nonnegative power (that is, an integer divisible by 2 ), otherwise we could have x = (4,3) .
>>
Rate the following math subjects

Category Theory
Knot Theory
Game Theory
Set Theory(specifically Large Cardinals)
Model Theory
>>
>>10638331
>Rate the following math subjects
>Category Theory
>Knot Theory
>Game Theory
>Set Theory(specifically Large Cardinals)
>Model Theory
Half of those aren't even maths.
>>
>>10638331
Knot>Game>Set>Model>Category.
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>>10638349
They're all math. Stop being silly
>>10638356
Why is knot the best?
>>
>>10638298
No t quite. $\{\{1, 2\}, \{2, 3\}\}$ would be counted in the 3-element case, since neither is a subset of the other.
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>>10638356

Ridiculous.

Model > Category > Set > Game > Knot

Take your Kirby calculus and go home.
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>>10638364

>Why is knot the best?
It's knot.
>>
>>10638440
mild teehee
>>
>>10638364
Because it's the only good subject in the list.
>>10638439

>ridiculous
>>
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Are there closed form solutions for the arc length of all 3rd degree polynomials or only for 2nd degree polynomials?
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>>10638535
Yeah, weierstrass $\wp$ function
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>>10638314
Really not getting what you mean. What do you mean by
>there is only one valid countable value of {1,2} for any number of subsets
>>
>>10638560
>>10638535
you know you've made it when there's a latex command specifically for your p function
>>
when i get the sex op, all /mg/ niggas gettin some pussy
>>
>>10638331
>Knot theory
Trash
>Game Theory
Based
>Category Theory
No opinion
>Set Theory
Based
>Model Theory
Based
>>
>>10638930

oh my
>>
What are the best books for graduate level studies in the following?

Complex Analysis
Number Theory
Cryptography ( I know not exactly math but it's an interest of mine)
>>
>>10639049
>Number Theory

>Cryptography ( I know not exactly math but it's an interest of mine)
Koblitz
>>
Excuse me for being brain damaged, but I cannot find a proof that, if A and B are infinite sets, f is an injection from A to B, g is an injection from B to A, then there exists a bijection from A to B.
Another way to say it is, why does the existence of an injection define an order relation between cardinalities?
>>
>>10639110
>Excuse me for being brain damaged, but I cannot find a proof that, if A and B are infinite sets, f is an injection from A to B, g is an injection from B to A, then there exists a bijection from A to B.
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem#Proof
>>
>>10639113
thanks
>>
>>10639049
>What are the best books for graduate level studies in the following?
>Complex Analysis
Ahlfors
>Cryptography ( I know not exactly math but it's an interest of mine)
Seconding Koblitz
>>
set-let here, Löwenheim–Skolem says that there is a countable model of ZFC where in-model uncountable sets are in fact countable but lack the necessary bijection N -> S to make it countable. But how can we say that the in-model uncountable set is "really countable" without fixing some sort of outer model? And if so why fix any model in particular as the "real" model?
>>
>>10638154
Check this:
https://math.stackexchange.com/questions/129267/find-maximal-number-of-subsets-of-the-set-u-such-that-no-one-is-a-subset-of-an

It discusses sets of even cardinality, but for odd cardinality 2n+1 you still work with subsets of cardinality n.
>>
>>10639368
I honestly solved the one in your link two minutes after reading the title, but am still stuck on the one anon posted.
Fucking combinatorics and its difficulty scaling.
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>>10638259
Pretty sure this is just $\mathbb{Q}[x,y]/(x^2-2,y^2-2)$.
>>
>>10637013
Yeah I'm sure you've done a lot of engineering during your two times as a glorified coffee fetcher.
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>>10638066

Yes to say that maths will let you understand the world in a metaphysical sense is a very brainlet thing to think. Maths is the language of nature though.
>>
Lads does anyone know any good books or resources to understand maths rather than just memorising things?
>>
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>>10639987
>Lads does anyone know any good books or resources to understand maths rather than just memorising things?
>>
>>10640006

God there's loads how do I proceed with this
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>>10640012
>God there's loads how do I proceed with this
top to bottom
>>
>>10638331
>Category Theory
Cool autism
>Knot Theory
Gimmick
>Game Theory
Cool but kinda dumb
>Set Theory(specifically Large Cardinals)
Autism
>Model Theory
Autism
>>
>>10640012
It's a dumb meme, If you wan't a better understanding go to your foundations, If you don't know anything read "Book of Proof" by Hammack (Free online). Here you will learn the basics of logic and set theory, choose the one you like the most (both will allow you to understand most math) and just get a book on it and read it.

Once you get to the second step Enderton has good intro books on both topics.
>>
>>10639987
Do the excercises.
Every time you read a theorem engage with it: Why is each assumption necessary? Is the converse true? Can you prove a weaker theorem if you relax the assumptions? Can you prove a stronger/more general theorem? What does the theorem tell you? What is the point of the theorem?
Before an exam you should understand every proof given in lecture. By this I mean that you get the key ideas from which the proof could be replicated. Don't memorize a calculation like an autist just know the hard parts.

>>10640012
Ignore him it's a meme.
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>>10639686
oh fuck, obviously
im dumb
>>
>>10639264
>But how can we say that the in-model uncountable set is "really countable" without fixing some sort of outer model?
It's a usual trick of model theory: you're defining a theory (ie an encoding of math), but you define and study it using "normal math", which is called the meta-theory. It seems like circular reasoning, so you should take care of not mixing the meta-theoretical sets with the theoretical sets.
It's like defining a language: you define it by writing sentences, therefore you define a language by using a previously existing language. You cannot really define languages (or math) ex nihilo, so you have to define math using math, which is why formal logic is sometimes seen as a meme.
Formal systems are in three parts: the theory (what you want, defined as a set of sentences), the meta-theory (the tools you have, without having defined them), and the interpretation function (which relates theoretical sentences to meta-theoretical objects).

>And if so why fix any model in particular as the "real" model?
We don't say "real" model but standard model. The standard model is the mathematical object (ie the object used by non-logician mathematicians) that intuitively corresponds to what you are formalizing, but obviously that is philosophically weak. It's like talking about the standard order relation on integers, it is only standard because everyone say so, but mathematically speaking there is no reason to say one arbitrary order relation on integers is "more real" than another.
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>>10639496
>>10638154
I know how to count the number of pairs X, Y in P(P(A)) such that X is strictly included in Y, but not how to proceed. If A has cardinality N and X has cardinality m<N, any superset of X is equal to X plus some subset of A\X, and there are $2^{N-m}$ of those, so there are $2^{N-m}$ supersets of X, minus one if you exclude X itself.
Now, A has one subset of cardinality 0, N of cardinality 1, ${N\choose 2}$ of cardinality 2, etc... So the number of contaminated pairs is
[eqn]\sum_{j=0}^{N-1} {N\choose j}\cdot(2^{N-j}-1)[/eqn]
Call this number q.

So we have q pairs {X, Y} in P(P(A)) as above. Call those $C_1, C_2, \dots, C_q$. For each C_i we can compute how many elements of P(P(A)) contain C_i ("supersets"), as we did for elements of P(A). We can actually compute the number of sets in P(P(A)) that contain at least one of those pairs and thus get the solution, the problem is that it is a huge summation: first you add up the following:
+ supersets of C1
+ supersets of C2
...
+ supersers of Cq
then you remove all elements that got counted twice:
- supersets of $C_1 \cup C_2$
- supersets of $C_1 \cup C_3$
...
then you have to add back all the elements that got counted twice in the subtraction:
+ supersets of $C_1 \cup C_2 \cup C_3$
....
And so on. If you have an alternate solution or a way to simplify this summation please tell.
>>
>>10640085
>If you don't know anything read "Book of Proof" by Hammack
proofbooks = memebooks
>>
>>10640090
>Ignore him it's a meme.
I'm not a "him".
>>
>>10640456
my b
>>
Can you do modern math if you don't like geometry?
I really don't like geometry. I find it tedious to learn and work with. Those 600 page tomes of introductory diff. geometry really put me off.
What kind of math should I look into if I am to avoid having to interact with Geometry?
I like Logic, "pure" Group Theory, Algebraic Topology (Knot Theory related stuff mostly), Complexity Theory.
>>
>>10640516
>can you really do modern math if you dislike geometry?
Is it even possible to dislike geometry nowadays? There's modern geometry, projective geometry, analytic geometry, algebraic geometry, differential geometry, symplectic geometry, complex geometry, computational geometry, fractal geometry, convex geometry, metric geometry, combinatorial geometry and probably some others I don't remember. Is it actually possible to dislike every single one of them?
>>
>>10640516
Maybe you could like algebraic geometry LOL. It's very abstract and it eschews calculus.

Any topic between category theory and probability theory that isn't directly related to geomantry could be done with little to no geometry. But what is it exactly that you don't like? Since you like algebraic topology.
>>
>>10640531
Riemannian geometry, inter-universal geometry, diophantine geometry, Finsler geometry ,Tropical geometry, hyperbolic geometry and elliptic geometry.
>>
>>10640531
>>10640535
Basically I'm super insecure about my fundamentals and the idea of having to read and work through 2000 pages of stuff to get a solid grip terrifies me.
>>
>>10640531
>>10640537
Is there some basic description of each of these topics?
I know alg. and diff. but the rest I have no clue about.
>>
>>10640537
probabilistic geometry, infinitesimal geometry, phisical geometry, optical geometry, geometrical geometry, hardcore geometry, softcore geometry, holy geometry, qabalistic geometry, post-industrial geometry, anime geometry, medical geometry, theological geometry, imaginary geometry, italian geometry, anal geometry, literary geometry, AI-simulated geometry, animalistic geometry, postmodern geometry, contemporary geometry, biblical geometry, deep-learning geometry, carbonara geometry, architectural geometry, origami geometry, cow geometry, septic geometry, biological gometry, transubstantial geometry, biomedical geometry, courtship geometry, women geometry, gender neutral geometry, complicated geometry, stupid geometry, geometry for dummies, python geometry, gentoo geometry, autistic geometry, existential geometry, xenogeometry, tibetan geometry, tsunami geometry, single-point geometry
>>
>>10640552
Differential geometry is no more obscure and tough than most undergrad subjects. You probably tackled it by following the meme recommendation charts and found it incredibly long and dull. You should give diff geo (and other geos) a second chance desu. Geometry is a fascinating subject and it intertwines with other areas of mathematics in often surprising ways.
>>
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>>10640537
>Tropical geometry
Sometimes I think people just do research to come up with funny names for things.
>>
>>10640571
>>10640552
For instance I'm taking differential geometry now and even though it's a complete meme it's quite comfy and nothing takes any more knowledge than basic introductory analysis and multivariable calculus to get. Only when you push a good bit later into these books do you start needing some extra stuff.
If you'd like a place to start, we've been using Millman and it's really not long or wordy. It is kind of shit toward the end though. Very fun and readable book.
>>
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Bros, I need advice on whether or not I should take this new Calculus 2 class. My college decided to add 10 'sections' on top of the original Calculus 2 class (while removing the original), and I'm not sure if I should take it or not over the summer.
Here is the New Calculus 2:
-The Entirety of Calculus 2
-Three Dimensional Coordinate Systems,
-Vectors,
-Dot Product,
-Cross Product,
-Lines and Planes in Space,
-Functions of Several Variables,
-Limits and Continuity in Higher Dimensions,
-Partial Derivatives,
-The Chain Rule.
Summary: Techniques of integration, improper integrals, infinite sequences and series, parametric equations, vectors and the geometry of space, functions of several variables and partial differentiation.
Should I take this class over the summer if I want an A?
>(Summer class = roughly 7 weeks with 4 midterms and 1 final)
or should I just take it over Fall and prepare over summer because this class will make me want to off myself and start posting regularly on /r9k/?
>>
>>10640904
It's all piss easy.
You shouldn't have a problem.
>>
>>10640904
>-Three Dimensional Coordinate Systems,
>-Vectors,
>-Dot Product,
>-Cross Product,
>-Lines and Planes in Space,
This is very basic pre-calculus stuff

>-Functions of Several Variables,
>-Limits and Continuity in Higher Dimensions,
>-Partial Derivatives,
>-The Chain Rule.
And these are the first topics usuals covered in calc 2 courses.

How could they "add" these to calc 2?
>>
>>10640912
I have no idea, I was told this by the Calc 1 prof before we register, maybe you can find some Calc 3 stuff here. This is the sections covered in C2
>>
>>10640915
It's all average stuff. Don't worry anon.
>>
f(f) = f

Solve for f.
>>
>>10640931
f=z or 1/z or something, depends on the spaces
>>
pls help
how do you prove that
[eqn]A^* \cdot {}^* A = (A)[/eqn]
where
[eqn]A^*=A+A)+A))+A)))+\dots[/eqn]
[eqn]{}^*A= A +(A+((A+(((A\dots[/eqn]
>>
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Serious question, assuming that I hardcore 9am-5pm and then 8pm-12am study for 3 months, can I plausibly clear calc 2 and calc 3?
>>
>>10640931
literally any projection
>>
>>10640996
hmm, perhaps... but just barely scrape by. You require an IQ of at least 180 to get an A in calcs 1-3, but with a full schedule of study such as yours, perhaps you can get a B or more.
>>
>>10640904
>this class will make me want to off myself and start posting regularly on /r9k/?
only if you're a total brainlet. this stuff is trivial
>>
>>10640941
>>
>>10640912
Thats calc 3 at most schools.
Calc 2 is typically tougher integrals, applications of integration, sequences and series, taylor series, and basic differential equations.
>>
>>10640996
You'd probably clear each in about a week with that schedule, no joke.
>>
>>10640931
not unique
>>
>>10641051
Do you think I can take that class over the summer and pass with an A?
>>
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>>10641052
Wait, you serious? Like I'm actually planning on studying like an autistic savant throughout the summer from 9 to sleep because I failed pretty bad
>>
>>10640996
You can pass Calc I-Calc III with about 5 hours of hard work each.
>>
>>10641077
Each day? I was planning on studying for twelve with a four hour gap in the middle but I might just split it into one for every four.
>>
>>10641152
No, in total, lol.
But to be honest it really depends on who you are and your background.
>>
>>10640941
wut
>>
>>10641052
Yup.
>>10641077
Mildly weaker yup.
>>10641152
No, in total. You have to remember that teachers unironically expect students to study everything in the last day, and difficulty could be at least three times higher in normal schools.
Until you get to grad school, there it evens out.
>>
>>10637812
D^n has no 0-cell.
>>
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does anyone have math maymays
>>
>>10641312
analysis
>>
>>10641315
KOWALSKI
>>
>>10641312
>be me
>working on this super hard problem
>find a proof after years of hard work
>algebrist arrives
>invents an algebraic variant of the problem
>solves it in three steps
>he now gets all the credit
>>
>>10641312
>>
>>10640552
I'm on the same boat, bruh (depending on what you mean by "fundamentals"). It seems that there's no other way but to swallow the pride and tank through the books.
>>
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>>10641420
>>
>>10640552
I know precisely what you feel. Like, you'll read in a book some argument that comes out of nowhere and it turns out to be a standard technique somewhere else, but ultimately boils down to intuitive arguments based on the geometry of the situation. If you want to mitigate that sort of stuff while still doing something interesting, I suggest doing algebraic stuff, like algebraic geometry or topology, or even autismo style ring theory and group theory. I'd suggest algebraic number theory but that's the queen of ad hoc, outta nowhere arguments
>>
>>10641064
I don't know anything about you, but I can say with confidence that the average 11 year old could.
>>
>>10641069
Dude these classes are literally just remembering a bunch of tricks and formulas, then practicing them.
If you want to do the class with Spivak or Apostol, then you might take your time and that schedule might make more sense.
I mean you're in one of the few threads on the internet which is full of people who took those classes when they were 16 and found them fairly straightforward, so I wouldn't necessarily recommend our advice. However if you really work at it, maybe pick some online lectures to go along (I highly recommend MIT OCW), you should be okay.
>>
>be double major in math and engineering
>get in
>realize I don't know how to do engineering because I spent 90% of my time grappling with my math classes in undergrad (engineering was piss easy still passed with a higher mark)
>feel at home
>I don't know why I'm here
>but at least I know the process of doing mathematics
Laugh at me. I deserve it.
Only posting as a cautionary tale to other double majors.
>>
>>10640552
If you know some algebra and analysis then get baby Hartshorne: https://www.springer.com/us/book/9780387986500

It will give you some nice ways to relate it to both, and even explain it as a theory on itself with Hilbert Axioms. Geometry is p cool.
>>
>>10641483
So am I overestimating the amount of time I need to get a good score? Like should I decrease from 12 to 5 like what everyone is saying? Sorry, I'm a bit dumb which is why I thought about going 12 hours
>>
>>10641051
I don't get why people keep asking what to expect from calc n when clearly programmes change from university to university
>>
>>10641573
No. Study as much as you can until you feel confident. Don't plan X hours a week that's stupid.
>>
>>10637871
Essentially what your other replies have specified, but there’s a certain (and yes I know this sounds pretentious and bare with me) transcendality about maths that had an effect on me.

By its nature of being objective, and the closest thing we have to ‘truth’, it puts how stupid some things are in the world into perspective. Can people stop acknowledging terror attacks if they want to stop them? Why do people lie to themselves about what they want and who they are? Why can’t all people try and dismantle their own opinions to build up better ones?

I’m quite high, sorry if that was rambley, unremarkable and pompous but I get a real sense of perspective from maths
>>
>>10641168
no

>>10641029
i've tried simply multiplying the two series but something doesn't work. there's no way to simplify the addenda. you ned to pull of some trick but i dont know how.
>>
>>10641607
>the closest thing we have to ‘truth’
???
>>
>>10641573
No, you shouldn't take anons so seriously.
Some people out there, i.e. half the people here, don't have a gap between learning a theorem and learning how to apply it to solve problems. It's something that happens when you develop "mathematical maturity", or autism as it's more coloquially called.
You could memorize all the theorems in five hours, but you'll probably have to grind a bit to apply them.
>>
>>10641630
>half
like at most 2%
>>
>>10639368
Don't think that's the same thing. Here you want the maximal cardinality of one element of P(P(A)) that satisfies the condition. I'm asking for the cardinality of the set of all elements that satisfy the condition.
>>
>>10641610
Is this language theory?
Is this algebra?
Why are your parentheses all wrong?
>>
>>10642211
Where did you find the question, anon?
>>
>>10642512
why would they be wrong?
>>
>>10642524
$A + A) + A)) + A))) + \ldots \cdot A + (A + ((A + (((A\ldots = (A)$
A closed parenthesis should match a previous open parenthesis, and vice versa, which is obviously not the case here. Your formula does not make any sense.
>>
>>10642539
its actually
[eqn]A\cdot A +A) \cdot A +A \cdot (A + \dots =(A)[/eqn]
>>
>>10642539
parentheses are here used as left right operator. we use {} as delimiters
>>
>>10638331
>Category Theory
Actively studying. I like it
>Knot Theory
Seems like a meme
>Game Theory
No opinion
>Set Theory(specifically Large Cardinals)
Too exotic for my tastes
>Model Theory
Important
>>
Are there mathematical koans, i.e. problems that have no solution that one investigates for enlightenment or fun?
>>
>>10637606
This would make a great mario kart track
>>
>>10642715
Riemann hypothesis
>>
>>10642715
finding solutions to diophantine equations
>>
Listen up faggots. Knot Theory is not a meme. It actually has interesting applications in physics.
>>
>>10642964
dubs say you're wrong
>>
>>10642964
physics is a meme in itself, so I dont get your point
>>
>>10642517
I didn't find it directly, but I think it's the answer to a generalization of an exercise in Sierpiński's General Topology.
He defines Frechet (V)Spaces (which I think are different from https://en.wikipedia.org/wiki/Fr%C3%A9chet_space). A (V)space is just a set K such that for each element $a \in K$ there is an associated set of subsets of K called a's neighborhoods. So like a generalization of topological spaces with zero structure.
In a (V)space, we say that p is a limit element of a subset $E \subseteq K$ if every neighborhood of p has a non-empty intersection with $E \backslash \{p\}$.
We call the set of all limit points of E the derived set of E, denoted by E'.
Finally, we call two (V)spaces on the same set K topologically equivalent if the derived set of each subset in one space is the same as in the other.
In the book, there's an exercise that asks you to show that there are $19^4$ equivalence classes of topological equivalence of 4-element (V)spaces.
I think by answering my combinatorics question, you can get the number of equivalence classes of n-element finite (V)spaces.
Call the answer to my earlier question $M_n$. I believe the number of equivalence classes is $M_{n - 1}^n$.
>>
>>10643065
Well, consider that Mn might just have no nice expression and you can just write Mn. I couldn't find it at least.
>>
Does anyone have that really cool one line proof of the uncountability of R?

Also would appreciate similar proofs of anything, that is short clever and elegant
>>
>>10643080
>one line proof of the uncountability of R
"Why don't you try counting it yourself and find out?"
>>
>>10643077
Might not. But then again, it might!
>>
>>10643099
This would the uncountability of any infinite set
>>
>>10643126
depends on your dedication I guess
>>
>>10637798
>Word is terrible at that.
you can use a simple form a latex syntax in word
>>
>>10637606
is it possible to find the inverse of a floor function?

lets say I have
y= x - floor(x)

which I guess is equivalent to y = mod(x, 1).
shouldn't it be possible, to return at least one correct answer. I don't need all of them since there are infinitely many, but couldn't we treat it like the unit circle and return one correct answer?
>>
>>10643142
In general, yes. Because it's not bijective
>>
>>10641331
Someone post the greentext. The one about Algebraic topology
>>
>>10643142
> is it possible to find the inverse of a floor function?
floor() is neither injective nor surjective, so it can't have an inverse. If you restrict the domain to the integers, then floor() is just the identity function, which is its own inverse.
>>
>>10643080
I think you're confusing it with the uncountability of the permutation group $\mathfrak S(\mathbb N)$.
Consider a conditionally convergent series $A$. By Riemann rearrangement theorem, we can rearrange the terms to get any real number. Each rearrangement corresponds to a permutation of the indices, so there is an injection $\mathbb R\to \mathfrak S(\mathbb N)$
>>
>helplessly stuck on a question
>lurk internet to find solution
>nothing
>bite the bullet and write the question on stackexchange, tex it up, explain what I tried, what went wrong
>click
>inspiration comes out of nowhere
I've posted this same scenario on /mg/ so much I might as well just post questions on MSE before I even attempt them...
>>
>>10641443
could this be an obscure loss?
>>
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>>10637606
>$\prod_{n=1}^\infty \mathbb{Q}$ is a free $\mathbb{Q}$-module, but $\prod_{n=1}^\infty \mathbb{Z}$ isn't a free $\mathbb{Z}$-module
>>
why don't the textbooks ever talk about analysis over finite fields.
>>
>>10643347
What analysis?
Lad, every function on a finite field is a polynomial, since we can just interpolate a finite amount of steps. For a combinatorial argument, the smallest function that zeroes everything has degree n, and there are n^2 polynomials of degree between 0 and n-1, but that's also the number of functions from the field to itself.
You can just take formal derivatives, but they aren't worth anything.
Weil randomly figured out some actually interesting results, but that shit's rare.
>>
>>10643373
wildberger said we should investigate it. I think he has a point
>>
>>10643248
winrar
>>
Thomas’ Calculus: Early Transcendentals or James Stewart's Calculus: Early Transcendentals for self-studying over the summer hardcore?
>>
>>10643432
>>10643248
it says it in the filename you dumbfucks
>>
>>10643250
Every vector space (ie module over field) is free, trivially.
The latter is non-trivial
https://mathoverflow.net/questions/46475/infinite-direct-product-of-the-integers-not-a-free-module-over-the-integers
>>
>>10643444
Use whatever your class uses. Duh.
>>
>>10643444
Neither - Apostol Calculus
>>
tfw don't understand forcing
>>
i need to pass my CompGraphics assignment, what are some simple but effective topological objects i can model in webgl?
>>
>>10643853
Riemann surface of cube root function
>>
>>10642715
Collatz
>>
>>10637871
Plato claimed you should study mathematics before beginning with philosophy.
>>
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>>10643853
I once showed a 3d projection of a 4d surface rotating in R4 for a similar course, but it ended up being just random deformation of solids
>>
>>10643465
I guess what I'm asking is what is wrong with the ring $\Z$ that this fails to be free. Or more generally put, what conditions on $R$ will make $\prod_{n \in \mathbb{N}} R$ a free (or not free) $R$-module?
>>
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>tfw failing my analytic ring theory class
fuck bros
>>
>>10644384
I failed my calc 2 + analytic geometry class because it was a 'test' for the math department, retaking it next semes, dont feel bad
>>
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>tfw failing Riemannian combinatorics again
>>
>>10643465
>Every vector space (ie module over field) is free, trivially.
This is only true if you assume it to be true.
>>
>>10644415
that's why i said trivially
>>
>>10644352
I've been thinking for a while, and the best answer I have is "a generating set is uncountable, but it intuitively needs to be countable".
>>
>>10639035
hows calc 1 going
>>
>>10639049
>Complex Analysis

Narasimhan - Complex analysis in one variable

>Number Theory
Hecke - Lectures on the theory of algebraic numbers

>Cryptography
Koblitz
>>
I finished a math degree but won't go to grad school. Is self-study at the graduate level possible or realistic? If it is, how can I begin to develop a (modest) course of study?
>>
>>10644488
>Is self-study at the graduate level possible or realistic?
You have to be a bit more specific about "grad level". Getting your phd involves doing original research and plowing deep into a field. Even before that in order to get a masters you usually have to take some special topics courses that will vary from dept to dept. At the most basic level, the material shared and considered standard for a 1st/2nd year grad student would be
Algebra at the level or Dummit and Foote/Lang/Aluffi
Analysis at the level of Rudin/Folland/Alfhors/Simon
Topology at the level of Munkres/Massey/Bott and Tu/Milnor
Geometry at the level of Lee/Tu
There's a lot of overlap here and you can substitute these books for others, but the material in them is considered the basics. For there you actually start digging in deeper to other fields and honing in on your topic of study. But for the most part, it actually is pretty doable to self study these. A good chunk of people do in order to pass their prelims when entering grad school. I'd say algebra is the easiest for self study, followed by topology, geometry, and then analysis. After that the world's your oyster, you just gotta pick where you wanna head to next.
>>10638331
They are all equivalent because they are all homotopy type theory in various dresses. Except game theory, not yet at least.
>>
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Is there an actual process to finding a perfect difference set or do you just have to try shit until it works?
>>
>>10644696
Wtf you just remove the elements that are in the one set from the other. This isn't your fucking wedding.
>>
>>10644662
>>
>>10644699
I meant with regard to the integers mod some prime power not some venn diagram nonsense.
>>
Sorry I'm new but as a somewhat retarded EE engineer who never was challenged all that much and scraped by statistics, this has been bothering me all night as a basketball foller since the NBA draft happened under its new format of lottery drawing its picks for the teams. I asked in >>10644944 but not sure if I'll get a response so asking here.

I'm confused about one particular part of the drafting which is when they draw 4 balls from 1-14 in any order, how do you get the numerical ordering of it after you sort it in ascending order? For example, the combination 4, 7, 12, 13 is combination #754. How do you get that this is the 754th combination?
>>
I have to do a 10 minute presentation. I'm open to any ideas, hopefully in the calculus field. I was thinking of talking about the hairy ball theorem.
>>
>>10644962
>calculus field
what, the real numbers?

10 minutes on the hairy ball sounds painful, what are you gonna do for the other 9?

Perhaps talk about differential forms and stokes theorem, or de rham cohomology, or perhaps line integrals on riemann surfaces and how it solves the logarithm problem on the complex plane, or perhaps on pathological counterexamples - a space filling curve or everywhere continuous, nowhere differentiable functions, or Lie groups, or differential Galois theory. Or if you're really at a baby level, just ordinary complex analysis - how the results in the real case are different to those in the complex case, and why the latter is 'better'.
>>
>>10637606
fwefewf
>>
does anyone know where I can find a proof of the bruck-ryser-chowla theorem? pic related in case anyone knows it by another name, regarding symmetric (b,k,lambda)-designs. im pretty sure it has to do with the incidence matrix of the design but i cant put the pieces together
>>
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Just an anon that doesn't know much about math passing by. Wow you guys are into some brainy stuff!
>>
can anyone set me straight here, is this doable just using the properties of a projective plane or is there some theorem im forgetting that will help
>>
>>10644352
See
The proof that every vector space has a basis fails for abelian groups because Z is not a field (picrel is the key point). Also there are abelian groups with no nonempty independent sets, like finite groups, but the infinite product Z^N is not among these.
>>
>>10637606

Can anyone point me to a good definition of the tensor product as used in this context (algebraic geometry)?

I used to know what it was, I think it has the property

$dim\ A = a,\ dim\ B = b \implies dim(a \otimes b)=a*b$
>>
>>10643853

You can make some pretty cool ones with elementary ray marching

ray marching just means you convert an implicit equation to a function by subtracting the LHS and treating the RHS as a function which you numerically approximate
>>
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>>10644384
>ring
Hey anon I still have my math notes in PDF format. Is this of any interest to you?
>>
Next year will be the year I email a professor and ask about doing a UROP
>>
>>10644696
Look up Paley difference sets.
>>
Minus or take?
>>
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is this list a meme?
>>
>>10645671
Yes
>>
>>10645430
tfw can recognize shafarevich from a mile away.

As for your question, literally any book that deals with commutative algebra. Atiyah Macdonald does it in the second chapter.

From memory, chapter 1.1 from Liu's algebraic geometry and arithmetic curves has that precise example for more general algebras.

However, tensors really aren't that complicated. Just think of them as 'symbols' that satisfy bilinearity in the terms. For vector spaces, as you claim, the dimensions multiply, however in your picture it's dealing with infinite dimensional vector spaces so don't know how that would help - indeed the importance of that is that transcendence degree increases by one. The latter just has to do with the fact that vector spaces are always free.
>>
>>10645439
lol yeah, give me your babby algebra notes thanks
>>
>>10645671
Great, I love Liu but hate fulton
>>
>>10645297
jesus dude you'd think you'd want to know a little better before you start saying these things.

by your 'proof' then $\bigoplus_{i=1}^\infty\mathbb Z$ is not free over $\mathbb Z$
>>
>how to compute them mod k
>what do they even mean?
>>
>>10645142
hmm, you should have mentioned this is finite projective geometry, was a bit confused
>>
I'm taking my first abstract algebra class right now and it's going pretty well and I'm also really enjoying it, but I am noticing my linear algebra skills are not that great... So do you guys have a nice linear algebra book for someone who understands abstract algebra and has done a linear algebra class before, but just wants to hone his skills?
>>
>>10645710
Why? It just proves not all abelian groups are free.
>>
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How much of a meme is the infinitely large napkin book?
>>
>>10641312
>>
>>10645671
Everything before set theory is a meme
>>
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>>10637606
Can I learn differential equations in a month if I have ten hours a week of spare time during my internship? I am hoping to grind through Boyce and DiPrima's book; might limit lectures in favor of reading and exercises
>>
>>10645876
The Linear Algebra a Beginning Graduate Student Ought to Know - Golan
>>
>>10646126
If you only have 10 spare hours in an entire week you're probably doing too much.
>>
>>10646133
I work from 9 to 5 and spend the rest of my free time lifting, cooking, and reading/studying French. I can account for an hour every weekday and do as much as I can on the weekends. If I were to spend more than two hours on Saturday and Sunday studying, I could bring the overall to more than ten.
>>
>>10646145
Never gonna make it.
>studying french
Study french while simultaneously studying maths.
>>
>>10646145
I'd call everything you do outside 9-5 spare time, but okay.

If you actually spend an hour a day you could learn a little bit. I don't know what your goal is.

The real problem I have with this "hour a day" thing is it's very easy to fill an hour and lie to yourself that you've done something. Just do as much in a day as you can until you're satisfied with what you've learned.

If you're doing this for a class, there's a reasonably high chance you'll have to cut into some of the time your other free time.
>>
>>10646145
You should make friends
>>
>>10646273
>implying I'll ever find someone both willing to put up with my neuroticism and that shares my obscure interests
>>
>>10645671
I prefer the official /mg/ curriculum: >>10640006
>>
>>10646496
Meh. You're more average than you think.
>>
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What kinds of dance are allowed in Riemannian manifolds? Does it depend on curvature?
>>
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>>10646273
I have friends.
>>10646496
I'm actually pretty happy with the person I am. Not sure how happy you are when you pretend to paint me as an insecure person for the sake of a joke
>>
can someone tell me how to take a math class the right way. I'm taking my first proof based lin alg class and its taking up so much fucking time. I need to streamline my study habits somehow.
>>
>>10646555
You have to read and understand proofs
>>
>>10646555
just think about math all day. you should be thinking about math in the grocery store. there's no such thing as streamline.
>>
>>10644992
>does anyone know where I can find a proof of the bruck-ryser-chowla theorem?
>>
can i just email a department head of mathematics at a college i want to go to with a standard gmail account, or is that cringey
>>
>>10646643
What else do you have? Use your current school email if you want. Nobody gives a fuck as long as it isn't Yahoo/Hotmail/AOL/other retirement home shit
>>
>>10646643
Gmail isn't cringy, but what are you emailing him for? "Hey, I've akshually studied the entire bachelors curriculum on my own please please please let me in,"
>>
>>10646503
it doesn't feel that way

>>10646511
I was talking about myself, not trying to impersonate you
>>
>>10646650
bs in comp sci but i loved the math part of it, i studied linear algebra on my own and lots of calculus and discrete in my major
>>
>>10646649
yeah i can use my school
>>10646650
as i sad, i want a masters in math at a decent school, not MIT, but not shit, i have 3.2 gpa and am well versed in calculus and discrete, did a bunch of computability theory, and taught myself vector algebra what does the standard math major have 4 years in?
>>
>>10646669
dont even bother applying for masters u should just go straight for fields medal
>>
>>10646682
so no? id figured that a masters in math was at least possible after comp sci
>>
>>10646669
>taught myself vector algebra
Amazing. Truly the next Neumann.
>>
>>10646650
protonmail and pgp
>>
>>10646770
does the average undergrad not know linear algebra?
>>
>>10646776
The average undergrad learns linear algebra in his first semester.
>>
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>>10646682
>>10646770
Stop being a cunt anon
>>
>>10646784
So i cant go from comp sci to math? just give me a damn answer that dosent involve you stroking your ego, and ill get out of this shitty thread
>>
>>10646805
Yes anon. It'll be really fucking hard but you can do it if you try.
>>
>>10646805
You can because introductory courses, good luck.
>>
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>>10637657
>>
>>10646633
>the only thing keeping me from getting the proof myself was forgetting the product of squares is also a square
god damn im a spaz
cool book though, thanks champ
>>
>>10646810
>>10646808
i hate all of you
>>
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Any brainlet friendly textbook on differential geometry?
I need to get started on manifold, convectors, distribution/codistributions, Lie derivative and friends.
>>
>>10646669
>>10646643
Let me tell you what they person is likely to say, something along the lines of "we love enthusiastic applicants, unfortunately we are not taking applications at this time, please submit at application for the next round of selections before December 15th" You don't just email someone and get into a masters program, you apply. Likewise when apply they usually have a set of requirements that will vary from school to school with most wanting you to have taken both the gre and the math gre. Along with that they may have some course requirements.
>>
>>10647360
we're math. doesn't sound like you like math very much does it?
r/mathematics is not math. i will tell you 100% confidently that my relationship with math is very similar to my relationship with /mg/ (in other words a whole lot of bullying both ways)
>>
>>10647410
milman is brainlet friendly but very classical, might not scratch the itch you want it to
>>
>>10647410
tu
>>
>>10646669
>>10646770
Since you have been bullied already, I'll give you the straight answer. No, you are not prepared to take a master's in math. Math builds up on itself ad infinitum. You are at the lowest tier of buildup, the zero level. By the time you've finished a 4 year undergrad, you are expected to have learnt the basics of at least 3-4 subjects, and have built up in one or two courses from those basics. You have not learnt a single 'basic' subject.

Those include basics in: linear algebra (not just matrices, but proper theory), real analysis, complex analysis, multivariable analysis, point-set topology, group theory, ring theory, differential geometry of curves and surfaces, number theory.

Calculus is not a basic because it is irrelevant to mathematical theory.
>>
>>10646805
> So i cant go from comp sci to math?
No. The math you learn in a CS degree is roughly the first semester of a math degree. Or possibly a foundation.

Look online for slides or course notes from the final year of a math degree. It's way beyond what you learnt in CS or what you taught yourself.
>>
>>10647623
>>10647709
Thanks. I guess I'll go through Millman quickly as a warm-up for Tu.
I want to get familiar with non linear control (Currently reading about feedback linearization). I get a wavy hand understanding of the whole manifold stuff and Lie brackets and Lie derivative are like black magic but I can read it through.
End goal is to be able to get comfortable with differential geometry enough to be able to "think in it".
>>
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>>10641018
>>
Lads, in my uni's PhD program's website there's a page called "PhD's written test."
At first I thought it was some test you needed to take to apply for the program, but I couldn't find anything of the sort in the edital, and then I thought it was literal tests of disciplines, but they're essentially stuff like "Algebra" (which is basic cat theory and noncommutative ring theory), "Analysis" (measure and functional), which is mostly either undergrad or masters stuff.
Does anyone have any idea or will I need to literally go there and ask what are those?
>>
>>10648110
send an email?
>>
>>10648112
Makes sense, but I'd rather not ask them at all.
>>
>>10647809
>Look online for slides or course notes from the final year of a math degree
What exactly should i search
>>
>>10648115
Perhaps it's something like qualifiers? In my uni I had to take 3 exams at the end of the first year (phd)
>>
>>10648118
That should be it, thanks.
>>
>>10648122
I'd check just in case you need to pass actual application exams, but sure.
>>
Can someone recommands me a textbook with many calculus/analysis problems. I just want to be better at computing integrals/derivatives. Is the brainlet book Calculus from Stewart a good to train?
>>
>>10648135
yes

most textbooks on real math dont care about solving integral problems
>>
>>10648135
https://www.wolframalpha.com/problem-generator/?scrollTo=Calculus
>>
Let $\beta = \{x^2,x,1\}$ and $\beta' = \{a_2x^2 + a_1 x + a_0, b_2 x^2 + b_1 x + b_0, c_2x^2 + c_1x+c_0\}$ be two ordered basis of $P_2(x) [\math]. How can I find the change of coordinates matrix? I'm struggling with this kind of questions. Any help and intuition behind change of coordinates? >> >>10648210 A matrix represents a linear map from the space of polynomials of degree at most two to itself. This means, the linear sends every element of the domain to its target by means of multiplying a vector on the left by a matrix. In this case, you (implicitly) claim that you have an isomorphism of the spaces, ie: there is an invertible matrix that represents the transformation. This isomorphism is completely determined by its effect on the basis, since every element can be written as a finite sum of elements of the basis. This means that you need only check the effect of the matrix on the basis elements. Now, since the matrix is an isomorphism, you can go either way, ie: from the first basis to the second or vice versa, but really, one is much easier than the other. Write [eqn]\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix}\begin{bmatrix} 1 \\ x \\ x^2 \end{bmatrix}=\begin{bmatrix} a_2x^2 + a_1 x + a_0 \\ b_2 x^2 + b_1 x + b_0 \\ c_2x^2 + c_1x+c_0 \end{bmatrix}[/eqn] Solving this matrix equation will tell you the change of coordinates. From here on, it's pretty easy. >> I go to a shitty CS university where they don't teach us Linear Algebra or Matrix theory but I want to learn them for my own knowledge What is the best resource to learn these by myself online? >> >>10648314 Strang MIT lectures LADR >> >>10648135 if you like proof-based analysis problems, or want some much tougher calculus problems, check out "problems and theorems in analysis" by polya and szego. legitimately some of the most fun i've ever had with math. >> File: strang.jpg (16 KB, 480x360) 16 KB JPG >>10648314 the linear algebra lectures by gilbert strang on MIT OCW should be perfect for you, not so abstract and very fun to watch. check em out. >> >>10648314 My teacher used Nakos' Linear Algebra. Good book for the basics. >> What does [math]\text{Sym}^d$ stand for?

I keep seeing it eg $\mathbb P(\text{Sym}^d(\mathbb C^n))$ in AG texts but can't google it and it's taken as obvious notation.

I was thinking perhaps symmetric matrices, but could also be permutations, or something.
>>
>>10648321
>>10648364
Strang looks kino
>>
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What book can I use to selfstudy real analysis after little Rudin. I am taking a advanced real analysis in the fall which uses foundations of modern analysis by Friedman and I want to learn it in advance.
>>
>>/lit/13126885
>>
fuck me

>>>/lit/13126885
>>
Could someone tell me what $\mathbb{C}[x,y]/\langle x^2+y^2-1\rangle$ has to do with geometry, or point me to a source that explains it?

More importantly I'm asked to come up with a "freaky" topological space but I can only think of generic ones like the Cantor set, Sierpinsky-like fractals, finite field topologies or the Zarinski topology. Does anyone know a really original "freaky" space?
>>
>>10648847
It's a unit circle
>>
>>10648847
How do you even claim to know what the Zariski topology is without knowing what the space you just posted is??
>>
>>10648847
have you tried R mod Q ?
>>
>>10648868
thank you this looks interesting
>>10648850
y tho?
>>
>>10648887
>>10648854
>>
>>10648854
The post doesn't claim the author knows what the Zariski topology is
>>
>>10648847
The union of all circles with center (0, a) and radius a, where a goes through all the rationals.
>>
>>10648902
fine
>>10648850
>>10648847
In classical algebraic geometry, there is an equivalence between algebraic varieties (think: solutions to algebraic equations in several variables) and quotients of polynomial rings like the one you posted.

A relation such as yours, x^2+y^2=1 defines the unit circle in a graph of $\mathbb C^2$. You can look at this circle as a topological space of its own with the Zariski topology. In the equivalence, you take a polynomial ring in two variables like C[x,y], and declare x^2+y^2=1 to be zero, and all it's multiples. Then you find that the Zariski topology (although not needed here, but whatever) essentially says that the closed sets of the circle (since the circle is one-dimensional, unfortunately it's not very exciting) are precisely finite collections of points. In the equivalence, these points correspond precisely to prime ideals in your space (in this particularly unexciting case, they also turn out to be maximals).

So you can always interpret C[x,y]/(x^2+y^2-1) as a circle and closed sets are maximal ideals.
>>
>>10648601
I'd guess it's the symmetric algebra/symmetric powers
https://en.wikipedia.org/wiki/Symmetric_algebra
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is math like an rpg?
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Lets say you have the function f(x) = sin(x).
When you solve for x, meaning when sin(x) = 0, there are infinitely many solutions described by the sequence n*pi where n is an integer.

My question is, what if we're dealing with a sine function where there are infinitely many solutions, but the sequence is unknown. In the standard example there is a linear relationship between each solution (each solution is a multiple of an integer), but what if the relationship between each solution is some exponential sequence that we don't know. Is there any way to find at least one of the infinitely many solutions. I don't need a sequence to describe every solution I just need one.
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>>10637621
At least use a uniball you fucking monster
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>>10649442
It depends function to function.
If your function is continuous you can use computational methods to narrow down a guess.
For an arbitrary function there is no easy way to find roots.
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>>10649442
Can you elaborate more on what you mean? As it stands, it's not entirely comprehensible what you're asking for.

By the way, solving for x does not mean when sin(x)=0
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>>10649458
Ok that's basically what I was wondering. I've always just assumed it was easy to find the roots of any function.
>>10649461
I mean finding the roots for a function with infinite roots.
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>>10649481
any function or a sine function?
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>>10649489
well for a sin function, that's like sin(2^x). But I would also like to know for any function too.
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>>10649481
>I've always assumed it was easy to find the roots of any function.
For example, one of the biggest open problems since the nineteenth century, the Riemann hypothesis, is literally finding the zeroes of a function.
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>>10649497
But you already know the answer for that. the answer is $x=\log_2(n\pi)$ for $n\geq 1$
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>>10649499
True. That is a good example.
>>10649505
oh ok. that makes sense. I guess I just need to think more about what I'm actually trying to ask.
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>>10649522
A hard to answer one is for example $\sin(x)=x$. A harder one to answer is e^x = x.
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>>10649537
sinx = x has solution x=0
e^x = x has no solution
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>>10649703
yes
wrong
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What is the point of the <bra|ket> notation? Is it just an excuse to say bra in an academical context?
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>>10649705
e^x > x for all x
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>>10649715
well yeah in the complex numbers
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>>10649709
>used by mathematicians and all decent people
>masterfully ambiguous, either side can be filled with either a vector or a linear functional without any mistakes being made
>clean
>the virgin bra-ket
>used by physicists
>invented by infinite function man
>doesn't implicitly borrow the reflexiveness of a Hilbert space
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>>10649719
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Can you define a space lie this:
>take a piece of the plane
>make a ton of holes, big enough to leave some space between them and without making the holes overlap or touch
>the small spaces left also have ton of holes between them, but also has some space left
I suppose it eventually will become something like the Sierpinsky Carpet, but I kinda intriged if you can build it in the first place.
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>>10649782
Probably wouldn't be well-defined if the holes are not allowed to touch, because there would be sequences of holes arbitrarily closed to each other in the limit. Otherwise yeah it would be well-defined, why not.
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Why can there be no balanced incomplete block design with $(52,52,r,k,\lambda)$ and $2\leq k\leq v-1$
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Will getting a degree in applied math help me be a better programmer? I already know how to rpogram web apps but i wanna get into AI and VR
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>>10648814
it's easily the best set of math lectures i've ever watched
he just loves basic linear algebra so much
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I might drop out of my PhD program because the prospect of living on 20k a year for the rest of my 20s is fucking depressing. I could work a part time job at McDonalds in NYC and make more, while still having just as much time to do math.
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>>10650219
you should only do a phd in pure mathematics if there is no question in your mind about whether it's more appealing to finish the phd in pure mathematics versus to get a well paying job
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>>10648818
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>>10650219
>he cant live off 20K
the fuck do you use so much money for
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>>10649980
Yes
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>>10650219
I live off 15k AUD; stop consuming.
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>>10645973

This brings back nightmares
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>>10650656
brainlet
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>>10650656
you should kill yourself
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>>10651685
well said

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