/mg/ - maths general

[Anonymous No.10356155]

Talk maths
Talkn't things that aren't maths

[Anonymous No.10356180]

>>10356155
>https://en.wikipedia.org/wiki/Kontsevich_quantization_formula
How do I learn about this shit?

[Anonymous No.10356198]

>another math thread

[Anonymous No.10356200]

>>10356180
This: https://ambry.pw/item/detail/id/1292179?id=1292179
Was the only book I could find on the subject in libgen.
I might be wrong (in fact, I probably am, since I'm not a physicist), but:
>calc
>topology
>smooth manifolds
>lie groups and lie algebras
>linear algebra
>functional analysis
>operator theory
Should cover you. The physics seems to be self-contained.

[Anonymous No.10356802]

I'm teaching myself group theory!

[Anonymous No.10356810]

complex analysis or PDE I can't decide what to attend next semester.
I didn't exactly like the bit of ODE we did in a previous analysis lecture but PDE seems so extremely useful and applicable.

[Anonymous No.10356815]

>>10356810
Go for complex analysis.

[Anonymous No.10356821]

I want to do my PhD in a field that is very mathematical and chances are that my supervisor may be a mathematician.
My background isn't in maths and I'd have to do a fair bit of self study, but I still have a couple of years time to get there so that's fine.

How are mathematicians typically assessed for their suitability as a phd student?
Is it all about your previous university, your taken modules and your grades?

[Anonymous No.10356824]

Why does algebra make sense to me but not topology? I suck at topology...maybe I'm just not seeing the motivations?

[Anonymous No.10356940]

>>10356200
not that poster. How do I work my way up to topology, lie groups and lie algebras, and smooth manifolds. I also never learned about proofs and feel I need to rectify that immediately.

[Anonymous No.10356946]

The problem the stumped the comp sci fags...BEHOLD

[Anonymous No.10356953]

>>10356946
Wrong

[Anonymous No.10356964]

>>10356946
>cube

[Anonymous No.10356968]

>>10356964
English was not their first language, give it a rest

[Anonymous No.10356974]

>>10356968
But it's two different questions depending on if it's a cuboid or a cube. Knowing what a cube is isn't language specific either, the mathematical shapes still have different names. Stop giving him the benefit of the doubt

[Anonymous No.10357014]

>>10356940
If you're a physics student or grad, a lot of the prereqs for these kind of things are probably already developed in the area they're used (I'm guessing string theory) so you're wasting time if you're doing proofs or reading math books on your own, or choosing the topics yourself, since the books on physics will direct you where you need most likely

[Anonymous No.10357035]

>>10357014
I misread your post, I though you were asking other thing. Those things you mentioned are widespread, get a book like group theory for physicists for the group shit (and possibly a bit of topological theories?) and manifolds is in differential geometry so get a book on GR

[Anonymous No.10357052]

I have to do a project in math and CS about projective geometry, do you have any litterature or an idea of applications of projective geometry in CS?

[Anonymous No.10357175]

>>10357014
>>10357035
I'm afraid I'm an engineer. Tried to understand a thing on mechanism kinematics that had an introduction to lie groups and lie algebras, but I didn't understand it. There was something about manifolds too. All I know is that I'm missing the prereqs to understand it.

[Anonymous No.10357261]

>>10356824
Topology is like having a whore of a wife known as "geometric intuition", that'll have sex with the mailman if you blink, and a perfect 2D waifu, known as formalism, that'll never look at another man or lead you astray, but doesn't exist.
The crux of the matter is exploiting your geometric intuition but never actually trusting it. Take the gag out of its slut mouth, ask it if it has any idea why Hausdorff spaces do that, thank it for the suggestion, and try to see if your 2D waifu can work it out properly from that.
>>10356940
>topology
Essentially no requirements. Analysis, depending on the book.
Smooth manifolds and Lie groups require only topology and analysis.
Proof books are for fags.
>>10357175
That's because smooth manifolds don't actually make sense if you don't know topology, you'd have to work with either embedded manifolds, which is horrible, or work with charts over a metric space, which is worse.

[Anonymous No.10357419]

>>10356802
good for you, anon

[Anonymous No.10357791]

>[math]M[/math] is a skew-symmetric square matrix with coefficients -1, 0 or 1
>[math]\mathbf P[/math] is the set of all column matrices with nonnegative coordinates and sum of coordinates equals 1
>Define binary relation over [math]\mathbf P[/math] with [math]P\;\leqslant\;Q \quad\Leftrightarrow\quad P^\top\,M\,Q\;\leqslant\;0[/math]
>Is [math]\leqslant[/math] a preorder? A partial order?
Any idea? I can only do reflexivity.

[Anonymous No.10357798]

The wiki is confusing. What should I learn after calculus?

[Anonymous No.10357806]

>>10357791
Partial order, since M could be just zeroes. Prove transitivity.
>>10357798
Multivariable calculus.

[Anonymous No.10357810]

>>10357791
>Any idea?
What have you tried?

[Anonymous No.10357811]

>>10357806
>give the reason it's a preorder
>call it a partial order
I should go to sleep.

[Anonymous No.10357831]

>>10357806
>Multivariable calculus
I studied this too

[Anonymous No.10357837]

>>10357831
>>Multivariable calculus
>I studied this too
Several complex variables

[Anonymous No.10357849]

>>10357798
Freshman Physics

[Anonymous No.10357852]

>>10357831
Abstract algebra, point-set topology, basic number theory, real analysis, complex analysis or ordinary differential equations.

[Anonymous No.10357914]

>>10356180
Start by learning QM to understand the entire motivation behind deformation quantization, then read about geometric quantization to get a sense of how (rigorous) quantization works on Poisson manifolds.

[Anonymous No.10357938]

>>10357798
Math stops being a linear progression after precalculus. These are the essentials for a well educated STEM major, pick something you have the prerequisites for and learn it; otherwise learn the prerequisites.

Arithmetic (Prereq: Breathing)
Algebra (Prereq: Arithmetic, Basic Reading)
Precalculus (Prereq: Algebra, General Reading)
Matrix Algebra (Prereq: Precalculus | Potentially useful: Calculus)
Proofs (Prereq: Precalculus | Potentially useful: Calculus)
Calculus (Prereq: Precalculus | Potentially useful: Proofs)
Vector Calculus (Prereq: Calculus | Potentially useful: Matrix Algebra)
Fourier Methods (Prereq: Calculus | Potentially useful: Matrix Algebra)
Ordinary Differential Equations (Prereq: Calculus, Matrix Algebra | Potentially useful: Vector Calculus, Fourier Methods)
Probability and Random Processes (Prereq: Vector Calculus | Potentially useful: Proofs, Fourier Methods)
Numerical Analysis (Prereq: Vector Calculus, Matrix Algebra | Potentially useful: Proofs, ODEs)
Group Theory (Prereq: Proofs, Matrix Algebra | Potentially useful: Linear Algebra)
Linear Algebra (Prereq: Calculus, Proofs, Matrix Algebra | Potentially useful: Group Theory)
Complex Variables (Prereq: Vector Calculus | Potentially useful: Proofs, ODEs)
Partial Differential Equations (Prereq: Vector Calculus, ODEs | Potentially useful: Proofs, Analysis, Complex Variables)
Differential Geometry of Curves and Surfaces (Prereq: Vector Calculus, Matrix Algebra | Potentially useful: Proofs, Analysis, Complex Variables, ODEs)

[Anonymous No.10358139]

>>10356155
Am I 10 iq or does this prove collatz?

[Anonymous No.10358140]

>>10358139
Illegible.

[Anonymous No.10358164]

>>10358139
If you have to ask whether or not a proof is correct, it's not.

[Anonymous No.10358210]

What are quarternions mathematically ?

[Anonymous No.10358250]

>>10357810
Well I already have these:
[eqn]P^\top\,M\,Q \;=\; \sum_{\left(i,\;j\right)\;\in\;I^2}p_i\,m_{i,\;j}\,q_j \;=\; \sum_{\substack{\left(i,\;j\right)\;\in\;I^2\\m_{i,\;j}\;=\;1}} \left(p_i\,q_j\,-\,q_i\,p_j\right)[/eqn]
but that doesn't seem like enough to prove transitivity. I also tried to start from
[eqn]P^\top\,M\,R \;=\; \sum_{\substack{\left(i,\;j\right)\;\in\;I^2\\m_{i,\;j}\;=\;1}} \left(p_i\,r_j \,-\, r_i\,p_j \,+\, p_i\,q_j \,-\, p_i\,q_j \,+\, q_i\,p_j \,-\, q_i\,p_j \,+\, q_i\,r_j \,-\, q_i\,r_j \,+\, r_i\,q_j \,-\, r_i\,q_j\right)[/eqn]
but it didn't look like I could get something out of it. I get a [math]P^\top\,M\,Q \,+\, Q^\top\,M\,R[/math] but with a residue.

[Anonymous No.10358257]

>>10358250
dude that is one fucked up post

[Anonymous No.10358259]

>>10358250
[eqn]\bf \color{#ff0000}H \color{#ff3f00}i \color{#ff8300}r \color{#ffc300}o \; \color{#b2ff00}i \color{#72ff00}s \; \color{#00ff15}a \; \color{#00ff98}f \color{#00ffdc}u \color{#00e1ff}c \color{#009dff}k \color{#0059ff}i \color{#0019ff}n \color{#2a00ff}g \; \color{#ae00ff}n \color{#f200ff}i \color{#ff00c7}g \color{#ff0087}g \color{#ff0044}e \color{#ff0000}r[/eqn]
I'll try again.

>>10357810
Well I already have these:
[eqn]P^\top\,M\,Q \;=\; \sum_{\left(i,\;j\right)\;\in\;I^2}p_i\,m_{i,\;j}\,q_j \;=\; \sum_{\substack{\left(i,\;j\right)\;\in\;I^2\\m_{i,\;j}\;=\;1}} \left(p_i\,q_j \,-\, q_i\,p_j\right)[/eqn]
but that doesn't seem like enough to prove transitivity. I also tried to start from
[eqn]P^\top\,M\,R \;=\; \sum_{\substack{\left(i,\;j\right)\;\in\;I^2\\m_{i,\;j}\;=\;1}} \left(p_i\,r_j \,-\, r_i\,p_j \,+\, p_i\,q_j \,-\, p_i\,q_j \,+\, q_i\,p_j \,-\, q_i\,p_j \,+\, q_i\,r_j \,-\, q_i\,r_j \,+\, r_i\,q_j \,-\, r_i\,q_j\right)[/eqn]
but it didn't look like I could get something out of it. I get a [math]P^\top\,M\,Q \,+\, Q^\top\,M\,R[/math] but with a residue.

[Anonymous No.10358312]

So I was told to come here.
How do I get a better understanding of large cardinals such as the Mahlo Cardinal? Should I think of them as limits of larger and larger inaccessible cardinals?

[Anonymous No.10358315]

WHY IS CALC 3 SO HARD AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

[Anonymous No.10358386]

>>10358315
it's all trivial if you know calc 1 and 2

[Anonymous No.10358462]

>>10356155
How do I make money off maths?
Currently I run multi phase fluid simulations for a petroleum company as a consultant.
I feel like I need to do something more profitable than being an employee. I need to make something that people want.

What do people want?

[Anonymous No.10358478]

>>10358139
By your logic, if you can decompose n into different numbers, you can decompose n+1 -> 1+1 =2 -> 1 for all n? Won't that be easier?

[Anonymous No.10358704]

>>10356815
why

[Anonymous No.10358831]

>>10358462
just prove the riemann and become a millionaire bro

[Anonymous No.10358911]

can i get a quick rundown on infinite dimensional projective space_

[Anonymous No.10358927]

I was really thinking about dropping PDEs but this is actually turning out to be fun. It's nice to have a physics course again.

[Anonymous No.10359026]

Does it need updating?

[Anonymous No.10359040]

What's the best internet website to teach me post-high school maths for free?

[Anonymous No.10359061]

>>10359040

khan academy then freetechbooks .com

[Anonymous No.10359065]

>>10356802
good !

[Anonymous No.10359070]

>>10356946
well why not b

[Anonymous No.10359112]

>>10357791
Not true, consider
M = [[ 0, 1, -1],
[-1, 0, 1],
[ 1, -1, 0]]
and
p = [0.42, 0.35, 0.23]
q = [0.29, 0.48, 0.23]
r = [0.38, 0.14, 0.48]
then p<q and q<r but not p<r

[Anonymous No.10359288]

>>10359026
Damn

[Anonymous No.10359475]

>>10358911
Which one?

[Anonymous No.10359479]

>>10358911
Which infinity are we talking about here?

[Anonymous No.10359539]

>>10358478
i just realized that it isn't legal to decompose it, ignore it

[Anonymous No.10359633]

>>10359475
In general

[Anonymous No.10359683]

>>10359633
Well same as finite dimensional projective spaces, they still parameterize lines in a vector space. They will be manifolds if the original space is, let's say, Banach.
Probably not very interesting manifolds though since what we like about finite projective spaces is that they are compact (eg. as quotients of spheres) and by extension small, have nice intersection properties on subspaces etc.
Infinite dimensional spheres are not compact, so neither are infinite dimensional projective spaces hence we cannot hope for very nice properties.

[Anonymous No.10359802]

>>10359683
Interesting. Are their any books which discuss this that you know of?

[Anonymous No.10359891]

I had a dream last night that someone made anime girl versions of the ZFC axioms and it was the funniest shit ever

[Anonymous No.10359987]

>>10359891
oh lord
a fucking math idol show
don't even fucking start

[Anonymous No.10360012]

>>10357261
thanks!

[Anonymous No.10360202]

>>10359891
Who was best girl?I bet the Axiom of Choice is a hoe

[Anonymous No.10360218]

>>10360202
Axiom of Regularity.

[Anonymous No.10360355]

>>10359802
I don't really have a book about this, I was kinda thinking "out loud". I had never really thought about this. I guess another interesting feature is that quantum states naturally live in a projective Hilbert space rather than a mere Hilbert space. That might be a reason to be interested in them.
Another thing is that countable-dimensional projective space are interesting from an algebro-topological pov: https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space#Examples
Anyway, from a quick google scholar search, I don't think it is a very popular topic (but maybe it could be interesting !)

[Anonymous No.10360385]

>>10359891
You know what you must do now, anon.

[Anonymous No.10360421]

>>10358315
It really isn't, most of what your doibg is regurgitated/ built on top of stuff you've already learned from calculus 1 and 2. It's all linked pal

[Anonymous No.10360479]

>>10358315
curious, what people consider calc 2 and calc 3?
the two biggest unis here have them swapped between them.

[Anonymous No.10360485]

>>10360479
Calc 2 integral calculus

3 multivariable

[Anonymous No.10360492]

>>10360485
isnt calc 1 limit, diff and integrals, though?

2 should be multi variable or series+diff equations

[Anonymous No.10360504]

>>10359026
>Complex analysis
Gotta be Kodaira for maximum shitposting.
By the way, can anyone tell me whether this meme or a PhD is more demanding?

[Anonymous No.10360511]

>>10360492
I'm from a shithole so calc one was axioms , functions,limits ,derivative .

Calc 2 was integration methods ,numerical approximation and series

[Anonymous No.10360795]

>>10359891
>tfw inner product space waifu will never induce a norm on me

[Anonymous No.10360884]

>>10357261
Hearty kek. Mind if I pasta this?

[Anonymous No.10360942]

>>10356946
A. This is less of a "longest distance" problem (in which case, of course B is the right answer) and more of a "gotcha" problem. Bugs can scatter about the cube and reach every point possible but not return to the same originating location.

[Anonymous No.10360986]

Books on computational algebraic number theory?

[Anonymous No.10361083]

>>10358911
Take the contractible infinite-sphere [math]S^\infty[/math] as the space of unit-norm wavefunctions [math]\psi[/math] in a Hilbert space [math]\mathcal{H}[/math], then chiral symmetry [math]\Gamma \psi = -\psi[/math] identifies diametrically-opposite wavefunctions such that the physical Hilbert space [math]\mathcal{H}_\text{phys} = S^\infty/\mathbb{Z}^2 = \mathbb{C}P^\infty[/math] is exactly the infinite-dimensional complex projective space [math]\mathbb{C}P^\infty[/math], where [math]\mathbb{Z}_2 = \{1,\Gamma\}[/math] is the chiral symmetry groud.

[Anonymous No.10361104]

>>10360884
Go ahead.
>>10361083
>the infinite sphere is contractible
The more you know.

[Anonymous No.10361108]

>>10361104
Yes. Topology is beautiful and no amount of slander will change that.

[Anonymous No.10361134]

*Ahem*
FUCK MATHS

t. got 23/31 in watered down high school level Calc 1 exam for bio course and was happy about it

[Anonymous No.10361173]

>>10361083
based geometer/mathematical physicist/manga I don't recognize poster

[Anonymous No.10361197]

>>10361134
>Calc 1
This is a thread for mathematics.

[Anonymous No.10361210]

>>10361197
Weird way to spell loser, LOSER! ahah

[Anonymous No.10361213]

>>10361173
>not knowing what touhou is
anon... you're even worse than a quaternary!

[Anonymous No.10361224]

>>10361173
>touhou_description.jpg poster
I've started calling him that once I recognized his filenames on /jp/.

[Anonymous No.10361952]

>>10357914
These comics are great.
Do you have more?

[Anonymous No.10362507]

What do you do in Calc 1? In germany the first thing you do in uni is analysis 1, is that the same?

[Anonymous No.10362517]

>>10362507
yes it's the same, many euro countries just call it analysis from the start

[Anonymous No.10362520]

>>10362507
I'm from chile and we have a shit secondary education,so calculus 1 is axioms, inequalities,limits ,derivatives and aplications.

Analysis is an advanced course, you had to take cal 1 ,2,3 and d.e to take analysis

[Anonymous No.10362521]

>>10361224
Kek

[Anonymous No.10362636]

>>10361224
Haha you should visit my field!

[Anonymous No.10362663]

>>10356155
An ordinal and it's successor ordinal
are equinumerous. How doesn't the successor have an extra element?

[Anonymous No.10362672]

>>10362663
Ordinals aren't cardinals

[Anonymous No.10362689]

>>10362672
But if S(a)=aU{a} the {a} is an extra element well i guess it isn't but why?

[Anonymous No.10362877]

>>10362689
Ordinals aren't about size, they're more about rank. When you ask if two ordinals are equinumerous you're comparing their cardinals. For example omega and omega + 1 are equal because they have the same cardinal N. The integers have way more numbers than the naturals, but they are equinumerous. But Omega+1 only has 1 more element than Omega, so it would be easy to pair one off with the other

[Anonymous No.10362942]

Why don't we teach group theory (up to Sylow's theorems) sooner? Modular arithmetic with prime modules is a nice and easy first encounter with "different operations" while symmetries/permutations are extremely intuitive. Combine these two facts and group theory becomes ideal for a first "serious" math course, rendering "intro to proofs" classes obsolete and, I would suspect, substantially raising the grades in classes like real analysis (at baby Rudin level), since most people failing are doing so because it's the first time they've been required to be rigorous and not because the material is hard. Furthermore, in more advanced classes algebra, we could look at everything in terms of groups and reduce proving a lot of elemental properties about rings, fields and vector spaces to saying your structure is a group under each of its operations.

tl;dr teach group theory instead of "introduction to proofs" and save yourself a lot of trouble

[Anonymous No.10363102]

>>10362942
It's literally too hard for most students. A few would be OK with it but most would struggle and then get hopelessly behind on the material. My group theory class didn't technically require any sort of "intro to proofs" prerequisite but the people who had never done proofs before utterly failed at, for example, showing that coset membership is an equivalence relation.

[Anonymous No.10363104]

>what? of course I'm a mathematician, I'm studying to be an actuary!

[Anonymous No.10363106]

Redpill me on some good resources to learn more about Hawke's processes

[Anonymous No.10363122]

>>10363106
I'll skin you alive and bathe in your blood.

[Anonymous No.10363136]

>>10357052
Anything with 3d graphics

[Anonymous No.10363147]

>>10357052
Go the other way and write a program to find the number of isomorphism classes for some interesting family of projective spaces

[Anonymous No.10363236]

>>10356824
You need a good, visual teacher to understand topology. Preferably face to face in a small class.

Algebra you can figure out yourself. Topology you need someone to show you the connections before you can see them yourself.

[Anonymous No.10363940]

>>10362517
But my uni only has analysis I and II, so i guess that would also include calc3.

[Anonymous No.10363953]

>>10363940
Are you sure about that? Most euro countries would just call that course multidimensional analysis. Or they could be retarded like my uni that actually had two pairs of classes called analysis I and II, where the second ones corresponded to what americans call analysis. Although it was always funny when some random 1st-years showed up at the wrong class

[Anonymous No.10363955]

>>10363122
How about you fuck off?

[Anonymous No.10363965]

>>10363953
Well there is "messure and integration theor"y after that, so that's maybe calc3/ the euro equivalent?

[Anonymous No.10364016]

>>10362517
I wouldnt call it the same. At least in the NL I know we purposely don't do calculus for mathematicians. The idea is that it's better to take it slow and do all thr proofs.
Physicists do have calculus courses because they dont give a fuck about proofs anyway

[Anonymous No.10364042]

>>10361952
Source is my diary

[Anonymous No.10364337]

>>10356198
Optimized.

[Anonymous No.10364339]

>>10357849
Optimized.

[Anonymous No.10364342]

>>10359026
Optimized.

[Anonymous No.10364351]

>>10364337
>>10364339
>>10364342
Based

[Anonymous No.10364377]

>>10359065
Optimized.

[Anonymous No.10364381]

>>10359112
Optimized.

[Anonymous No.10364383]

>>10361083
Optimized.

[Anonymous No.10364388]

>>10361108
Optimized.

[Anonymous No.10364391]

>>10363122
Optimized.

[Anonymous No.10364397]

>>10364042
Optimized.

[Anonymous No.10364438]

>>10364339
Where would group theory go on this. I stole a book on it a while ago idk y even though back then i was in algebra 2 lol

[Anonymous No.10364457]

anyone have a bigger library?

[Anonymous No.10364478]

>>10364438
Abstract algebra.
>even though back then i was in algebra 2
Where I'm from algebra 1 is group theory

[Anonymous No.10364491]

>>10364457

That's like 30 grand worth of books, isn't it?

[Anonymous No.10364727]

[Anonymous No.10364731]

>>10364727
https://terrytao.wordpress.com/2019/02/05/request-for-comments-from-the-icm-structure-committee

[Anonymous No.10364943]

>>10364727
And?

[Anonymous No.10365150]

Lang's Linear Algebra book or Strang's Introduction to Linear Algebra?

I'm about to start my second year, and this will be my second subject on linear algebra

[Anonymous No.10365152]

>>10364727
fucking stupid cunt. may god smite her to the darkest depth of hell where she gets raped by a pack of infernal niggers

[Anonymous No.10365156]

>>10356155
How do you guys take math notes? Do you take any at all? I was thinking of following Jordan Petersons advice of just listening to lectures.

[Anonymous No.10365164]

>>10365156
I used to write down basically everything the lecturer said. Now my hands are broken and I don't write anything down. I didn't notice a significant decrease in my marks.

[Anonymous No.10365177]

>>10365150
leaning more at pure math : lang
leaning more at application : strang

[Anonymous No.10365181]

>>10365156
I'm copying everything the prof writes down, and listen to him at the same time.
Works for now at least

[Anonymous No.10365184]

>>10365164
How do you study for exams then with no reference? Do you just read the textbook and do problems?

[Anonymous No.10365190]

>>10365184
I mostly reread the textbook or rewatch core lectures. I do some problems. I find assignments force me to have a good grasp of the content before I get to an exam anyway.

[Anonymous No.10365264]

>>10365156
I never took any. I graduated with the highest grade.

[Anonymous No.10365274]

>>10365156
I don't really remember things when I write them down. Writing is just too distracting for me to focus on what the lecturer is saying and impedes me when it comes to properly understanding what is being said.

Most of my maths modules also had excellent lecture notes/slides which would have made taking any notes pointless.

[Anonymous No.10365344]

>>10364397
Optimize this please

[Anonymous No.10365373]

>>10365344
Optimized.

[Anonymous No.10365462]

>>10365150
>Lang's Linear Algebra book or Strang's Introduction to Linear Algebra?
Lang is a meme.

[Anonymous No.10365567]

>>10365373
*headpat*

[Anonymous No.10365712]

>>10364727
>My one concrete suggestion: dedicate some time during the main banquet or opening ceremony to celebrate the lives and mathematics of prior ICM prize winners who have died since the prior ICM.
I'm actually okay with that, it'd be nice to celebrate those that the community lost.
>>10365156
I basically try to read the material ahead of time, take notes during that time, and when I get to the actual lecture I just listen and ask clarification questions when the time comes. If there's any material I didn't see I try writing down the broad strokes. I find trying to listen and takes notes at the same time reduces you ability to do either. That said, if the teacher agrees, you may want to try recording the lecture.

[Anonymous No.10365742]

>>10365344
Optimized.

[Anonymous No.10365900]

>>10365150
Both memes. Use Hoffman and Kunze or Friedberg.

[Anonymous No.10365919]

>>10365184
rereading notes over and over is a stupid-ass way to study for exams.
Exams test you on your ability to solve problems. If you want to solve problems well you should practice by doing problems, not by passively reading a chain of definition-example-proof.
On the odd occasion you are actually required to regurgitate a serious proof verbatim out of the notes, you're almost always warned in advance by the prof anyway.

[Anonymous No.10365939]

>tfw trying to decide between great program with crappy (but livable) funding and high-ranked but less interesting program with literally double the stipend
this sucks
would have been so much easier if one of them had just rejected me

[Anonymous No.10365985]

>>10365742
Gee anon that is a lot of optimization

[Anonymous No.10366028]

Underfriend here. I was taking a music theory class, have any of you applied group theory to music intervals or any other interesting things it can be applied to?

[Anonymous No.10366239]

>>10365939
better than both rejecting you
what are the general locations? you might be spending the extra stipend just on living costs if the high ranked program is in an expensive area.

[Anonymous No.10366572]

>>10360942
>This is less of a "longest distance" problem (in which case, of course B is the right answer)
No, it honestly isn't B. It isn't a gotcha problem.

[Anonymous No.10366584]

>>10362942
So, Europe?

[Anonymous No.10366613]

>>10365462
>>10365900
Why does everyone say Lang is a meme? And even if he is, does that necessarily mean he's bad?

[Anonymous No.10366645]

>>10360218
Regularity is the worst axiom

S={S} is a cute set and she should have more appearances,
but Regu's too much of a bitch to her

[Anonymous No.10366652]

>>10366613
Recommending Lang's 900 pages Algebra to undergrads taking Algebra I.
But his other texts are fine.
>>10362942
People here take ring theory then group theory.
>>10364457
You really like Springer.
>>10365184
If I know all theorems and definitions, I'm alright.
But if I was actually fucking fucked on something and absolutely needed a 10:
>memorize all definitions and theorems
>decompose all proofs and examples given in the text into methodic proof techniques (i.e. "how to infinite descent for dummies")
>solve exercises while explicitly using these

[Anonymous No.10366778]

>>10364388
Based

[Anonymous No.10367100]

>>10358704
Not the same Anon, but in my PDEs course, it was basically how do we turn PDEs into ODEs and then use methods from ODEs to solve PDEs. There's some other interesting stuff but there wasn't a deep dive into it. Go for complex OP. It's more fulfilling and it's mad wizardry

[Anonymous No.10367115]

>>10364438
What book? If it's highschool level then right after or during proofs. If it's graduate level then during or after abstract algebra.

[Anonymous No.10367118]

>>10365150
Linear Algebra by Shilov (Dover Books)
Finite Dimensional Vector Spaces by Halmos (Dover Books)
Linear Algebra by Friedberg, Insel, and Spence (not to be confused with their Elementary Linear Algebra)
Linear Algebra by Hoffman and Kunze

or

Linear Algebra and Its Applications by Strang
Matrix Analysis and Applied Linear Algebra by Meyer

[Anonymous No.10367154]

Randomly pick n vectors in the space [math] \mathbb{Z}^{k} [/math] where for each vector [math] \vec{x} [/math], [math] \left | x_i \right | < d, d \in \mathbb{N} [/math]. What is the probability that the vectors will be linearly independent?

[Anonymous No.10367319]

>>10366239
It's the opposite, actually. Average cost of an apartment in the poverty-tier uni is $50-100 less than in the well-funded one because it's in a big metropolitan city whereas the well-funded one is in midwestern shitsville
I'm mostly torn because the volume of interesting shit going on around the cheaper one (mathematically and otherwise) is way higher. Of course there's some interesting stuff at the other one too or I wouldn't have applied, but it's not the same level.

[Anonymous No.10367331]

>>10367154
Maybe an easier question you could try to answer is: "what is the probability the vectors will be dependent?"

[Anonymous No.10367333]

>>10367154
Define randomly picking vectors.

[Anonymous No.10367374]

>>10367331
I wasn't trying to solve it, I just thought it was an interesting question
>>10367333
Absolute autism

[Anonymous No.10367664]

How do I remember what the center, normalizer and the centralizer are? I can't stop mixing them up in my head

[Anonymous No.10367673]

>>10367664
it is fairly easy to remember what the normalizer is, because the condition for the _normal_izer is the same condition of a _normal_ subgroup.
you should always be able to remember that, and then the centralizer is just the other one.

it should not be possible to confuse the centre with the other two, it's not really a similar definition at all (it doesn't even have the same number of parameters). If you regularly confuse the centre of a group with the normalizer of a subgroup you may be too much of a brainlet to be saved.

[Anonymous No.10367685]

I'm a blue collar fag grinding through community college to catch up on math, but the material takes a brute force approach to getting me "caught up" I'm going through polynomials now, what books should I buy to supplement my studies to keeps the blinders off. I feel like I'm being herded to a specific mindset.

[Anonymous No.10367692]

>>10367685
Watch lectures on MITOpencourseware.

[Anonymous No.10367716]

>>10367692
Sweet, checking it out now.

[Anonymous No.10367926]

>>10367100
then your pdes course was shit
lol
imagine taking pdes and not doing distribution theory, weak solutions, and tons of relevant inequalities

[Anonymous No.10367976]

>>10367154
>Randomly pick n vectors in the space Zk where for each vector x⃗ , |xi|<d,d∈N.
define "randomly"

[Anonymous No.10368174]

>>10367976
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[Anonymous No.10368189]

>>10358210
R^4 given the structure of a (non-commutative) field.

[Anonymous No.10368200]

>>10365156
I fell for the self-study meme
I usually just write down the book in my own words, complete the proofs if need be and write my own comments in the margin.
Ignore the fact that it's in French lol

[Anonymous No.10368211]

>>10367154
Z^k isn't a vector space
You could reason by cases, it's mostly combinatronics anyway

What norm are we talking about ? The problem is much easier if the norm is the sum of all absolute valued components

[Anonymous No.10368358]

>>10365274
Good for you. I haven’t had a single pure math prof use anything other than chalkboard or whiteboard, which I prefer anyway.

[Anonymous No.10368384]

>>10368211
It is a vector space over Z, which is implied. And the "norm" is absolute value, because the thing inside is a scalar.

[Anonymous No.10368414]

>>10368384
>It is a vector space over Z
Z isn't a field, how does this make sense

[Anonymous No.10368823]

>>10356946
Top face, 3/4 of the way between B and the point above A, i.e. (0.75, 0.75, 2.0), distance = hypot(0.75, 2.75)

[Anonymous No.10368844]

>>10368211
>>10368414
Module.

[Anonymous No.10368869]

>>10368844
Well, Euler has already solved it for Z as a module over itself, so we just need to generalize.
Z^2 is also easy. Two vectors are linearly dependent if and only if one's a multiple of the other. Then the probability becomes a one-dimensional subspace, and rounds down to zero.
Downloading Tao's book on random matrices and will be right back if it actually helps prove the rest, but it's probably zero for dependence.

[Anonymous No.10368915]

>>10368869
>actually confusing \zeta (2) with that
What the fuck brain.

[Anonymous No.10369063]

This is literally what a genius mathematician (Jim Simons) said:
>Interviewer: who's your favorite mathematician besides yourself?
>Simons: Archimedes and Euler

[Anonymous No.10369077]

>>10356946

[Anonymous No.10369081]

>>10356946
better visualization

[Anonymous No.10369304]

>>10367333
>>10367976
Come on guys, you know it means uniformly... Way to not think

[Anonymous No.10369424]

>>10358139
This is obviously false since when you add 2n+4 to n it's not the same number so you're not going to apply to n the same operations which made it become 1 in your supposition

[Anonymous No.10369746]

>>10369063
Why are mathematicians either exceedingly humble or complete assholes? Or literally autistic like Von Neumann.

[Anonymous No.10370072]

> want to apply for math program
> go to their website
> they require two separate letters of recommendation

fuck you. I have an adderall prescription and I can go to the math library. You dont have a monopoly on mathematics. I don't need your bullshit program. Fucking normies. I will slaughter you.

[Anonymous No.10370074]

Does anyone remember how it was proved that f>g implies [math] \int_a ^b f(x)dx> \int_a ^b g(x)dx[/math] back in calc 1?
I can prove it by proving it for the Lebesgue integral (i.e. let [math]M_a={x| f(x)-g(x) \geq a}[/math], exploit countable additivity to force some set with positive measure, prove the difference gives a positive integral) or with the Lebesgue integration lemma (if it's not continuous on a dense set it's not continuous anywhere, the space where it's continuous contains at least one interval, Weierstrass extreme value gives one interval that pushes the integral into being positive), but I have no idea how it was proved back then. I get the impression that you either need a strong statement to prove f-g is larger than some positive value on at least one interval, which sucks, or that it literally wasn't proved.

[Anonymous No.10370096]

Hey I just finished a bs in comp sci, near the end I got really facisnatrd with algorithms, calculus, theoretical computer science. I want to master in applied mathematics now. Am I in for a rough time? The highest math I've done was calc 2. But I have atudyied and really enjoy group theory a bit.

[Anonymous No.10370100]

>>10369063
is that cringe? seems fine to me

[Anonymous No.10370125]

>>10370074
Just write them down as their Riemann sums and notice that for the same "rectangle width", the "height" from f is larger than the "height" from g
You can just also use measure theory memery with the particular case of the Jordan measure

Honestly the cleanest way to prove that with Lesbegue integration theory is also to use the simple functions that converges toward your desired functions and apply dominated convergence theorem

>>10370100
It's a bit too neutral and gives "snob" vibes
Euler and Archimedes work are (mostly) undergrad level, not that they weren't great but it's mostly well-known things now

[Anonymous No.10370175]

>>10370125
>Just write them down as their Riemann sums and notice that for the same "rectangle width", the "height" from f is larger than the "height" from g
Doesn't work because [math]a_n>b_n[/math] for all n doesn't imply [math]lim ~ a_n>lim ~b_n[/math].
Measure theory isn't Calc 1.

[Anonymous No.10370186]

>>10370125
I mean it's like saying Newton is your favorite physicists

[Anonymous No.10370190]

>>10370175
You're right, you're forced to add an equal
My measure theory comment was just trying to find a way that's simplier than yours

Try to evaluate the integral of f - g, see that it's positive, and apply linearity of the integral ?

[Anonymous No.10370205]

>>10370190
>the integral of f-g is positive
That's really the heart of the problem. Consider f(x)=x^2 except f(0)=1. It's strictly larger than zero, but its inf is still zero, and it's integrable, which exemplifies how we need to prove that f(x) has inf>0 on at least one interval to apply a lower sum argument.
>>10370186
Newton is my favorite physicist.

[Anonymous No.10370254]

>>10370205
But all the stuff he did was simple compared to now. It's his legacy we enjoy. Same with Euclid and Archimedes

[Anonymous No.10370281]

>>10356940
smooth manifolds are very easy without many prerequisites. I never really liked calculus even tho I was p good at it, but loved algebra. You really only need basic multivariable calc and the linear algebra that goes with it, with the tiniest bit of undergraduate level topology. I still don't know much about lie groups/algebras, but the bits I do know are much clearer to me from understanding their motivation from manifolds.

[Anonymous No.10370283]

>>10357052
there are many concepts in algebraic geometry which unite projective spaces and number theory, with the latter already having many applications in CS (and CS having applications to it).

[Anonymous No.10370324]

>>10364478
algebra 1 in US refers to the middle/high school class where you are introduced to basics like low-degree polynomials, complex numbers, simple sequences and series, bivariate linear equations, and exponential functions.

[Anonymous No.10370331]

>>10365150
If you've already done basic LA, jump to artin's Algebra. Ppl shit on it bc it is a very "concrete" intro to abstract algebra, but I found it to be a great stepping stone to higher math; very good at balancing computation, applications, and theory. If you want more of any one of those legs, read it concurrently.

[Anonymous No.10370358]

>>10369304
>Come on guys, you know it means uniformly...
I'm not a "guy".

[Anonymous No.10370460]

>>10370096
pls reply

[Anonymous No.10370469]

just wanted to come in here to tell you guys browsing this thread has made me feel like an actual 90 iq retard, math is my worst subject so seeing this shit literally just boogles my small brain. how do yall do it? did you ever struggle with math in school or have you always been good at it?

[Anonymous No.10370481]

>>10370469
i got Bs and Cs all through highschool and college without studying at all i am only just now interested in it

[Anonymous No.10370495]

>>10370469

i never do well in classes

i just read a lot of math books

[Anonymous No.10370502]

>>10370469
they have higher than average iq and are autistic

[Anonymous No.10370507]

>>10370502
i wonder if most extreme academic people are autistic. seems to be a correlation there.

[Anonymous No.10370791]

can math make me better at touhou? What's the configuration space of touhou look like? I mean if we take each frame of a sucessful run extract out the bullllets then it froms a 3d shapee. Now that makes a pretty picture, but is only slightly informative in planning runs. Player position influence bullet patterns to some degree, but maybe there's a way to model this?

[Anonymous No.10370822]

>>10366028
https://arxiv.org/abs/1204.3216
>>10370072
Mathematics is a social endeavour. If you can't even cop 2 rec letters then you really have no business doing mathematics.

[Anonymous No.10370854]

I need a good textbook on statistical analysis frends

[Anonymous No.10370975]

>>10370469
it's always been easy. i'm probably not even good at it, everyone else is just really really bad.
some people are better than me, and some of them are even pretty good. but very, very few people are actually very good at math.
do you like solving puzzles and such? if you do, that's pretty much what higher math feels like. anything you are familiar with is probably nothing like real math. it's all about proofs and arguments.

[Anonymous No.10370982]

https://www.youtube.com/watch?v=zbSKhv5xmNY

pretty neat for a 14 year old

[Anonymous No.10370986]

>>10370791
touhou games are generally very simple to understand strategically, and only difficult in execution.
some bullets are aimed at you, in which case you stream (move slowly to one side to keep them concentrated but always missing you). some bullets are aimed, but away from you, in which case you stay put. some bullets are randomly aimed or aimed in static configurations, in which case you look ahead of your player's hitbox for spaces which will be empty shortly and move into them. there's a little more to it than that on harder modes, in terms of making sure those spaces have more empty spaces beyond them or making sure you can "switch" streams and change directions, but that's basically the strategy behind touhou.
of course, there's all kinds of scoring optimization to get lives and such, and certain stages have good routing (especially all of UFO).
the tough part is actually doing any of it

[Anonymous No.10370989]

>>10370822
ty friend

[Anonymous No.10370992]

>>10370982
topology is trivial, unimpressive
it's clear he has no idea what he's doing

[Anonymous No.10370995]

>>10370992
i take this back, i clicked on his channel and he's doing some real stuff there
not bad, i'm impressed now

[Anonymous No.10371001]

>>10370995
Yeah my bad, I meant to link the channel just clicked that video randomly

[Anonymous No.10371039]

how do i prove [math](a+b)! \geq a!b![\math] for all integers greater than or equal to 0, i assume i would have to hold either a or b constant and induct on the other to show it is true for all a and b.

[Anonymous No.10371044]

>>10371039
what is (a + b)!/b! equal to? expand it out. how many terms?

[Anonymous No.10371083]

>>10370982
heh watch this:
https://www.youtube.com/watch?v=SnUnkr3shDg

[Anonymous No.10371106]

>>10371083
Absolute madlad
https://www.youtube.com/watch?v=u1qn8MpdO4M

[Anonymous No.10371118]

>>10371083
https://www.youtube.com/watch?v=vY72YVRZVuA

[Anonymous No.10371129]

>>10371083
>>10371106
>>10371118
This is real 140+ IQ

[Anonymous No.10371173]

>>10370982
I think he made a mistake in his picture demonstrating finite intersections, he introduces extra sets for no reason

[Anonymous No.10371220]

>>10370822
Fuck you Erdos.

[Anonymous No.10371231]

>>10370982
>14 years old
>doing mundane topology shit instead of fun math things
holy fuck that's sad

[Anonymous No.10371262]

>>10371044
i see, so dividing both sides by b! i could show that (a+b)(a+b-1)... (b+1) is greater than a!
?

[Anonymous No.10371299]

>>10371262
(a+b)...(b+1) = (b+a)...(b+1) > (a) ...(1) = a!

[Anonymous No.10371551]

>>10370074
I know you're not allowed measure theory, but can you at least use the criterion that a function is Riemann integrable iff it's continuous almost everywhere? I haven't done calc 1 so I wouldn't know what's covered

[Anonymous No.10371555]

>>10370481
Is this meant to be impressive? Lmao

[Anonymous No.10371649]

bros help me how the fuck do you prove that if [eqn]a_i \neq a_j, \forall i \neq j[/eqn], then
[eqn]\sum_{j=0}^{n} \prod_{i=0}^{n} (a_j - a_i)^{-1} = 0[/eqn]

[Anonymous No.10371655]

>>10371649
fuck, forgot to type that every product was supposed to exclude the zero, i.e. be for every [eqn]i \neq j[/eqn]
somebody help this brainlet

[Anonymous No.10371656]

>>10371649
Did you try induction?

[Anonymous No.10371659]

>>10371656
yeah but I wasn't able to get the induction to work
I think it may need some sick hack that I just haven't realized

[Anonymous No.10371666]

>>10371659
That's probably because the formula is written incorrectly.
As is, you're dividing by zero. Rewrite it adequately, and try to force how the terms "cancel out".

[Anonymous No.10371670]

>>10371666
No, even with it written correctly (I mentioned here >>10371655 that it was supposed to skip the zeros factors) the induction failed. But ok, I will try once more, hopefully I didn't just make some basic mistake

[Anonymous No.10371736]

>>10371649
You can convert into a polynomial in a_n and show that that polynomial is identically 0 by demonstrating more roots than the degree of the polynomial.

[Anonymous No.10371737]

>>10370125
Oiler's work is elegant as fuck though.

[Anonymous No.10371740]

>>10371736
Amazing, thanks anon. I was already trying to solve it directly by just reordering the elements but this idea seems much better

[Anonymous No.10371745]

>>10371740
Yea it's quite efficient. I remember seeing a similar problem in sqt some weeks ago.

[Anonymous No.10371753]

>>10371649
>>10371649
Notice that if [math]f(x)=(x-a_0)...(x-a_n)=\prod^n_{i=0}(x-a_i)[/math], then you have that the [math](n-1)[/math]-th derivative is [math]f^{(n-1)}(x)=(n-1)!\sum_{j=0}^n(x-a_j)[/math]. In particular:
[eqn]\frac{f^{(n-1)}(x)}{(n-1)!f(x)}=\sum_{j=0}^n\prod_{i\not=j}^n(x-a_i)^{-1}[/eqn]

[Anonymous No.10371774]

>>10370822
>Mathematics is a social endeavour
tell that to grigori and shinichi.

[Anonymous No.10371789]

>>10370469
>>10370975
Everyone else is just horribly bad at math. In highschool I only learned for my final exam in math and got A's since elementary school but that doesnt mean shit. The math you do in school is nothing compared to what comes after. Also I think teachers can't explain the deeper connections and beauty math has, leaving many students bored and uninterested.

[Anonymous No.10371825]

>>10371555
he's saying he didn't get good grades, that grades aren't required to develop an interest later
not being able to tell what people mean by things in context is a sign of autism

[Anonymous No.10371829]

>>10371262
yes, you can. (a + b) >= a. (a + b - 1) >= a - 1. etc.
so what about the product?

[Anonymous No.10371833]

>>10371774
both morons
the real person to reference is wiles, but of course he was being asocial for a reason

[Anonymous No.10371871]

>>10371829
a,b,c,d>0, a>c, b>d => a*b>c*d

[Anonymous No.10371880]

>>10371871
yes, that's my point.

[Anonymous No.10371946]

someone help me solve 2 and 3?please? ty

[Anonymous No.10371954]

>>10371946
i just fucking helped you in the other thread and even told you to go to /sqt/
/mg/ is not for math help, people here will just tell you to go away if you post a precalc level problem

[Anonymous No.10371964]

>>10371954
yea dude i saw your post
searched for sqt but wasn't that active so I came here
I'm newfag to sci btw so my bad

[Anonymous No.10371980]

What's a good book to learn math from scratch?

[Anonymous No.10371986]

>>10371964
>not active
It don't work like that. People obviously aren't going to post in it if no one's asking anything.
And it having less replies than it should is the fault of the absolute retard who forgot to literally add the title and forced me to make a new one so that people could actually find it by searching the catalog.

[Anonymous No.10371987]

>>10371649
I don't know, but I got this

[Anonymous No.10372054]

>>10371986
you are both morons, always remember that :)

[Anonymous No.10372077]

Can someone help me out here? I don't understand how to prove the lemma.

[Anonymous No.10372079]

>>10356155
How do I obtain the phase angle of a transfer function?

[Anonymous No.10372198]

>>10372077
This looks and obviously is randomly generated

[Anonymous No.10372343]

>>10372198
>his undergrad didn't cover contrafunctional measure homology

[Anonymous No.10372404]

>>10372077
Now THIS, is what I call epic!

[Anonymous No.10372419]

>>10372077
>linear and positive
>globally partial
>given a unique
kek

[Anonymous No.10372483]

>>10370096
>I want to master in applied mathematics now. Am I in for a rough time?
Depends on the department and what not. Some grad students at my undergrad uni were also taking senior undergrad courses to learn more foundational material before heading onto grad courses. You should talk to the schools you're applying to or look up at the curriculum. For example. if you have no experience in analysis then you're in for a rough time since measure theory, probability, and geometry of manifolds are gonna be a bitch. It's less about the classes being inherently difficult and more about not being familiar with the prerequisites and style. It also will suck if you don't have any experience with proofs. Then again, this is applied math so you may go a different route. Try finding out what kind of classes are necessary for the degree and if they're more proof or computationally based.

[Anonymous No.10373026]

FUCK

Can someone explain what a regular function on a (quasiprojective) variety is? I feel like I'm being pushed around by Shafarevich with all these different definitions. First, he defines a regular function on an affine algebraic set to be a polynomial map, basically a tuple of elements of the coordinate ring [math]k[X]:=k[x_1,...,x_n]/I(X)[/math]... ok, easy. Then a rational function on an affine algebraic set as a partially defined map of rational functions, a tuple of elements of the field of fractions of the coordinate ring, [math]k(X)[/math]. Again, understandable.

Now come in quasiprojective varieties. A regular function is now locally a rational function of same-degree homogeneous polynomials that has non-vanishing denominator in some open neighbourhood of every point. A map between quasiprojective varieties is regular if there is some affine piece of the codomain, and a neighbourhood of each point mapped there that restricts to a regular function. I can only assume that an isomorphism of varieties is a biregular bijective map.
There is no other indication yet of what (explicitly) [math]k[X][/math] might be in the quasiprojective case. Now there is this exercise:

Show that the variety [math]\mathbb A^2 \ (0,0)[/math] is not isomorphic to an affine variety. I've looked around and it seems everybody keeps saying that its coordinate ring is [math]k[x,y][/math]. Why?

[Anonymous No.10373037]

>>10373026
Go use Qing Liu or Hartshorne.

[Anonymous No.10373050]

>>10373037
Funny you say that... I have both in my possession. But no, I'm already learning schemes on the side, I need to learn fucking classical quasi projective functions!!

[Anonymous No.10373055]

>>10373050
It's the same thing but with ilegible terminology.

[Anonymous No.10373073]

>>10373050
>those books
I'm jealous

[Anonymous No.10373089]

>>10373073
Thats only my AG books, have a few more, it's pretty handy. Miss my copy of atiyah MacDonald though (it's very highly requested)
>>10373055
I need to understand the version with less autism before I understand the one with more

[Anonymous No.10373270]

How do I learn math? I already did everything on Khan Academy and I'm almost finished with Hammack's book of proof, but I'm to the end where I need to know how to derive stuff from first principles and I have no idea how to learn to do this.

Where should I look? Wat do?

[Anonymous No.10373310]

Hey guys, I'm self studying probability theory and got stuck with an exercise, could you give a hint or some intuition on how to solve it?
Let [math]C[/math] be a non-empty class of subsets of a set [math]\Omega[/math] and let [math]a(C)[/math] denote the algebra generated by [math]C[/math]. Show that all the elements of [math]a(C)[/math] can be written as [math]\bigcup_{j=1}^{m} \bigcap_{k=1}^{n_j}A_{j_k}[/math] with [math] m \in \mathbb{N}[/math] and for each [math] j,k A_{j_k} \in C [/math] or [math]A_{j_k}^c \in C[/math] and the [math]m[/math] sets of the form [math]\bigcap_{k=1}^{n_j}A_{j_k} [/math] are pair-wise disjoint.
I've already tried playing with some toy examples but I'm still pretty much lost, any help is appreciated.

[Anonymous No.10373368]

>>10373310
>probability
That's a measure theory problem.
And it's essentially the argument from linear algebra for the smallest subspace being the span, or the argument from topology of the closure being the set of limits.

[Anonymous No.10373386]

>>10373368
I know, the notes I'm following start with a bit of measure theory in order to define probability spaces and I just have analysis 1 under my belt so that's the reason I'm struggling a bit
thanks

[Anonymous No.10373438]

>>10373270
pick up Linear Algebra by Hoffman and Kunze or by Axler (if you want the meme approach), or if you've seen Linear Algebra and want something different try an introductory Abstract Algebra book. I recommend Dummit and Foote but you might decide you want a less intense transition in which case Fraleigh isn't a bad choice (and very very readable).

[Anonymous No.10373458]

>>10373386
You don't know linear algebra?
Well, it sort of goes like this:
The set of all elements that can be written like that forms an algebra.
Any algebra that contains C contains all those elements.
Then, it's the smallest algebra.

[Anonymous No.10373494]

hello, total brainlet question here. How'd they simplify this to get the answer?

[Anonymous No.10373507]

>>10373494
there's a typo, second to last should be lim h to 0 16h/h (or possibly lim 16)

[Anonymous No.10373508]

>>10373494
just expand using distributibity and note that 16t - 16t cancels so youre left with 16h/h = 16

[Anonymous No.10373518]

>>10373507
>>10373508
so this is the second derivative, which means what? The instantaneous rate of acceleration, which is the average velocity?

[Anonymous No.10373730]

>>10373518
the second derivative is the instantaneous rate of acceleration, but it's not the average velocity
rather, it's the rate at which the velocity is changing at that moment

[Anonymous No.10373736]

http://www.kurims.kyoto-u.ac.jp/~motizuki/news-english.html
>2019-02-10
>・(Papers) Updated comments concerning:
>The Geometry of Frobenioids I: The General Theory.
>The Geometry of Frobenioids II: Poly-Frobenioids.
>・(Papers) Revised version (list of revisions):
>Inter-universal Teichmuller Theory I: Construction of Hodge Theaters.

[Anonymous No.10373895]

/mg/pill me on godel incompleteness

[Anonymous No.10374082]

>>10364339
Why would you take Point-Set Topology before Real Analysis? many Real Analysis topics show up in Point-Set Topology.

[Anonymous No.10374100]

>>10373895
if you want vague explanations, just watch a popsci video on youtube or read the wikipedia page
overall first order logic and model theory is a big disappointment, the overarching message is that nothing can be done and matter what axioms you pick you're fucked
this line of negative results starts with simple things like lowenheim-skolem theorem, but the godel incompleteness was the last straw, from that point mathematicians begun to lose interest in autistic logic wankery

[Anonymous No.10374161]

>>10374082
I'm dealing with this right now, all the students that took anal already do proofs so much faster than the professor expects so going to office hours always makes me feel dumb because she thought we got it so quick during lecture.

[Anonymous No.10374198]

>>10373026
Well everything you said is correct, now obviously there is no theorem that states that the ring of functions on A^2\{0} is k[x,y], but you can derive it from your definition here:
>A regular function is now locally a rational function of same-degree homogeneous polynomials that has non-vanishing denominator in some open neighbourhood of every point.
To see this, you can start with a regular function f, cover A^2\{0} with (explicit) open affines on which you know how to compute the regular functions and see that f must be polynomial.
Another thing is, you might have found it confusing, in the case of affine varieties, you seemingly have two definitions for regular functions: one is restriction of polynomials and another is locally rational functions such that blabla.
It is a nontrivial fact (but a fact nonetheless) that both these definitions coincide and, if you look at this with your knowledge of schemes, you might see that it amounts to saying that regular functions on an affine variety form a sheaf.

[Anonymous No.10374199]

>>10373050
I hope you did not actually pay for all these

[Anonymous No.10374243]

>>10374161
That's rough. But be persistent. If Real Analysis is one of your first proof courses, it can be tough (it can tough even if it's not). Don't feel dumb :)

[Anonymous No.10374826]

>>10374100
>animeposter
>has no idea what he's talking about
Wow, shocker

[Anonymous No.10374831]

>>10374826
>>has no idea what he's talking about
I'm not a "he".

[Anonymous No.10375068]

>>10374826
please enlighten us

[Anonymous No.10375147]

>>10356155
Is there any area of math as dry as rep theory?

[Anonymous No.10375154]

>>10375147
Literally nothing is drier than pure category theory.

[Anonymous No.10375155]

Has anyone of you nerds ever got a job? If so, how was it and what did you do?
I'm almost finishing my math major and I'm scared shitless of dying of starvation.

[Anonymous No.10375179]

>>10375155
Even if you're a complete retard who did no networking at all and didn't go to grad school, you can get hired at any community college.

[Anonymous No.10375182]

>>10375147
>Is there any area of math as dry as rep theory?
Analysis

[Anonymous No.10375209]

>>10375155
Just b urself bro

[Anonymous No.10375223]

>>10374198
Ok, but now I'm stuck in the following exercise. Prove that any quasi projective variety is open in its projective closure.

What is even the closure of that variety? Is it the closure of its open affine cover?

[Anonymous No.10375303]

>>10375223
By definition, a quasiprojective variety is the intersection of an open set and a closed set in a projective space, hence it comes with an embedding in projective space.
Its projective closure is just its closure when seen as a subset of projective space.

[Anonymous No.10375329]

>>10375303
Then isn't then the proof obvious? If an open set of a variety is an intersection with an open set in projective space, then an open set in the closure (which is a variety since it is closed in projective space) is an intersection with an open set, and since the original variety is open in projective space by definition, then it's direct?

[Anonymous No.10375450]

>>10375155
unironically learn2code. if you didn't you may be fucked

[Anonymous No.10375452]

How do you manage big latex projects? I'm doing my honors thesis and it's already becoming difficult to deal with. I've already subfile'd off each chapter but it's still a mess.

[Anonymous No.10375528]

>>10375147
That's because you're not doing rep theory correctly, i.e. with applications to QM.

[Anonymous No.10375560]

Redpill me on coinduction

[Anonymous No.10375644]

Is it true that if you're slow with freshmen/sophomore maths, then you have no chance as any kind of engineer? I average high Bs and low As in my math courses.

[Anonymous No.10375668]

>>10375644
Why would you ask maths people about your aptitude for engineering?

[Anonymous No.10375721]

>>10370469
I failed algebra 2 and scraped by geometry with a D, I also had no trig experience going to college. All it really took for me to start doing A level work was finding an intrest in the subject once my professor made it intresting. Try to do some self proofs and you might find yourself a lot happier than before.

[Anonymous No.10375729]

>>10375644
See
>>10375721
Im currently in my 4th year and I bombed highschool math.

[Anonymous No.10375733]

>>10375644
No, because skill with physics/engineering and with calculus and linear algebra are separate things.
>>10375668
I'd make an /eng/ to try and contain them a bit, but I'm not really feeling it.

[Anonymous No.10375803]

>>10375329
The original variety is not open in projective space, but yeah it is the argument. Write [math]X = O \cap F[/math] with O open and F closed in [math]\mathbb P^n[/math].
Then, we have [math]X \subset F[/math] and therefore [math]\bar X \subset F[/math] and therefore [math]X = X \cap \bar X = O \cap F \cap \bar X = O \cap \bar X[/math]

[Anonymous No.10375894]

How many different words have mathematicians came up with that mean the same thing as "subset family"

[Anonymous No.10376124]

>>10374831
Who cares? You aren't going to get the same attention you do in /b/