/mg/ - maths general

[Anonymous No.11300167]

Previously >>11286076
Fuck frogposters edition.

[Anonymous No.11300188]

Threadly reminder to work with physicists.

[Anonymous No.11300197]

>>11300167
I'll be using this thread.

[Anonymous No.11300216]

Anyone know went the real Hardy space is complete? Been bugging me for a bit now

[Anonymous No.11300531]

Let G be a group.

Prove G is isomorphic to a subgroup of Sn, for a large enough n.

[Anonymous No.11300543]

>>11300531

Simply construct the faithful group action where G acts on itself.

in other words, if you write down the multiplication table of G, every row will be a permutation of G

[Anonymous No.11300553]

>>11300531
i assume you meant to assume G is finite

[Anonymous No.11300573]

>complex analysis is a subset of linear algebra

[Anonymous No.11300593]

>>11300553
>i assume you meant to assume G is finite
>for large enough n

[Anonymous No.11300633]

>>11300593
what are you saying? the statement is false as written.

[Anonymous No.11300639]

>>11300633
>what are you saying? the statement is false as written.
Provide one (1) counterexample.

[Anonymous No.11300658]

>>11300639
[math] (\mathbb{Z}, +) [/math]

[Anonymous No.11300673]

>>11300188
Freed agrees.

[Anonymous No.11300750]

>>11300658
Let [math]n = |\mathbb{Z}|[/math].

[Anonymous No.11300788]

>>11300750
OK smart guy how about we let [math] G = (\mathbf{R},+) [/math]

[Anonymous No.11300790]

>>11300788
Let [math] n = |\mathbb{R}| [/math].

[Anonymous No.11300792]

>>11300790
Yeah but what's the subgroup dummy?

[Anonymous No.11300797]

>>11300792
>Yeah but what's the subgroup dummy?
Take the subgroup of translations.

[Anonymous No.11300817]

>>11300797
Well played my devilishly minded friend, you win this one.

[Anonymous No.11300818]

But what is math even?

[Anonymous No.11300882]

Connes' embedding conjecture disproven by quantum information theorists.
https://arxiv.org/abs/2001.04383

Threadly reminder to work with physicists.

[Anonymous No.11300896]

>>11300882

[Anonymous No.11301397]

>mods delete my link to this thread from the other thread
>people are still posting in the other thread
/sci/ mods are fuckings hacks. migration to /lit/ when?

[Anonymous No.11301464]

>>11300882
Nice.
>>11301397
>expecting anything else from the janny thread

[Anonymous No.11301468]

>>11300818
the study of what our axioms do

[Anonymous No.11301620]

>>11300896

nice avatar faggot

[Anonymous No.11301918]

I'm participating in the belgian maths olympiads tomorrow, wish me luck!
Don't think I'll even be qualified, but I don't mind, I'm doing it for fun. Gotta get more than 3 hours of sleep tonight though!

Will check back tomorrow and say if I got qualified!

[Anonymous No.11301920]

>>11301918
good luck and have fun

[Anonymous No.11301933]

Can some one recommend me a good statistics book?

[Anonymous No.11301942]

>>11301933
>good statistics book
Doesn't exist.

[Anonymous No.11301953]

>>11301942
I'm not fan of those either, but i need it for machine learning techniques. Can't remember too much from uni.

[Anonymous No.11301985]

>>11301953

go teach your toaster to give you a blowjob. CS roach.

[Anonymous No.11301993]

Can someone explain to me what the fuck an homology or cohomology is? I've taken algebraic topology but can't figure out the intuition behind it

[Anonymous No.11302006]

>>11301993

it tells you how many holes there are

coffee cup = a donut

[Anonymous No.11302037]

>>11302006
Isn't that genus?

[Anonymous No.11302067]

>>11301993
It's a categorification of the Euler characteristic

[Anonymous No.11302154]

>>11302037

The betti numbers tell you the holes and their dimensions

[Anonymous No.11302175]

>>11302006
>>11302067
>>11302154
>muh holes
But we can do all this from Betti numbers.

What do the non-free parts of the homology groups correspond to? What the hell does cohomology correspond to?

[Anonymous No.11302278]

>>11301993
Homology and Cohomology measure "failure to lift" in an appropriate sense given by context.
In singular homology, you have a failure of lifting of [math]f : \Delta _n \rightarrow X[/math] where the map doesn't lift to a map from [math]\Delta _{n+1}[/math].
In de Rham cohomology, the local inverses of the [math]d[/math] operator don't lift to a global solution.
Similarly, in several complex variables, you'll mess around with Cousin problems, which are also situations of local to global lifting.
When you work over derived functors, this intuition becomes extremely explicit, and you're essentially measuring an object's failure to be injective or projective, which literally means they lift maps.

This is my intuition, feel free to insult it.

[Anonymous No.11302328]

>>11302278
Based and intuitionpilled.

[Anonymous No.11302343]

>>11302278
My bad, it's [math]\partial \Delta_n[/math] to [math]\Delta_n[/math] in singular homology.

[Anonymous No.11302446]

Can math prove true love exist?

[Anonymous No.11302481]

What does each basis in the Hilbert space represent? If we are talking about orthogonal sines, is each basis a complete sine function or is it just one value of the sine function? In other words, if it is a 10-dimensional space, does it mean we can add up to 10 sines together or does it mean that each sine consists of 10 points?

[Anonymous No.11302506]

>>11302481
>basis
I think you mean basis element/vector.
In the case of sine functions each element of your basis will be an entire sine function.

[Anonymous No.11302579]

>>11302506
Yeah not sure about the terminology.
Say, in 3-D space , there are three component vectors that make up the basis:
[0,0,1]
[0,1,0]
[1,0,0]
we can call them xyz or x1 x2 x3
they are all orthogonal, so the dot product of any pair of them is 0. so i was wondering if the basis in the hilbert space is exactly the same concept except each component is a set of points sin(x) such as [0, 0.2, 0.5 etc] . And that's what confuses me: does the dimension of this space define the number of points in each component? or does it define the number of components, or both.
What if my sine component is represented by 10 points but I only have 2 components. Or maybe I have 1000. Or do I have to have 10. Aaaa my brain hurts.

[Anonymous No.11302621]

>>11302579
You might want to link a picture of what exactly you're talking about.

>a set of points sin(x) such as [0, 0.2, 0.5 etc]
When you write a vector like [0,0,1] you are implicitly assuming some kind of basis for your space. Really what you mean by [0,0,1] is [math] 0x_1 + 0 x_2 +x_3 [/math] With real 3-D space this makes sense because there is a standard choice of basis where the three components correspond to the x,y,z directions.

In an arbitrary Hilbert space it doesn't make sense to say something like [1,0,0] without first choosing a basis. Say your basis is [math] \{ sin(x), sin(x/2)\} [/math], then the vector [1,3] represents [math]sin(x) + 3 sin(x/2) [/math].

>does the dimension of this space define the number of points in each component?
>or does it define the number of components
If you're asking the number of components in the vector (ie [1,2,3,4,5] has 5 components) then the answer is yes. This is because what I said above (each component corresponds to a basis vector).

In general we don't really use this notation though. This is because the interesting Hilbert space(s) have infinite dimension and so the vector [ ] could be of arbitrary size.

The other confusing thing is that there are two different notions of a basis when talking about a Hilbert space. There is the notion of a basis which allows you to approximate any element of your space, and then there is the standard linear algebra basis (which must be much bigger, and indeed only exists if you allow the AoC).

[Anonymous No.11302627]

>>11302481
>>11302579
You are too stupid for maths, please give up.

[Anonymous No.11302680]

>>11302278
Obstructions to lifts are the homotopies. Co/Homologies are abelian representation of topological spaces, namely a symmetric monoidal functor [math]{\bf Top}\rightarrow {\bf AbGrp}[/math] (yes, homology is a TQFT) satisfying some regularity axioms. It essentially turns topology into algebra.
Then again (stable) homotopy is a cohomoloy theory so I'm just busting your balls.

[Anonymous No.11302688]

>>11301985
based. im gonna use this one.

[Anonymous No.11302712]

>>11302621
Not sure what picture to link.
The context is the Fourier series. [math]f(x) = \frac{2}{\pi}\sum_{1}^{N=5}sin(nx) \int_0^{\pi}f(x)sin(x)dx[/math]
So the integral on the right is really b_n, and I let N=5 just for simplicity, meaning this is a "5-D space" i.e. we mix 5 harmonics sin(x) all with the same frequency and phase but with different coefficients (amplitudes) b_n. So the basis in this case will be {x1,x2,x3,x4,x5}, i.e. {sin,sin,sin,sin,sin}, correct? Which corresponds to the summation from 1 to 5. But notice how each coefficient b_n is an integral which is another summation, from 0 to PI. What if we pretend it is a discrete sum from 0 to PI with a certain step dx. Would that mean that each of the five components can now be represented as a vector
[ 0, 0.2, 0.4, 0.7 ... 1] that would contain the range of sin(x) values from 0 to PI? And the length of this "vector" has nothing to do with the number of components in our basis {sin,sin,sin,sin,sin}, correct ? I know this may be contrived but just trying to visualize this stuff in my mind.

>interesting Hilbert space(s) have infinite dimension and so the vector [ ] could be of arbitrary size.
See, this is what confuses me. My understanding is the dimension in my example is how many orthogonal sin functions (harmonics) aka components we mix together to represent a signal. I let N=5 But vector [ ] for each component seems to be something else: it is a set of sin(x) values from 0 to PI. Does it make sense?

[Anonymous No.11302754]

>>11302712
>What if we pretend it is a discrete sum from 0 to PI with a certain step dx.
Why are you doing this?

>each of the five components can now be represented as a vector
What do you mean by component here? Do you mean [math] sin(nx) b_n [/math]? You could choose to represent each basis vector [math]sin(nx)[/math] as a finite vector of [math]sin(nx)[/math] values as long as you included enough values so that your basis was still linearly independent. I don't see what this has to do with the integral though; treating that as a discrete sum is a separate issue.

>And the length of this "vector" has nothing to do with the number of components in our basis
This isn't true. Your representation still needs to be independent so you need at least as many components as you have elements in your basis (ie at least 5).

>My understanding is the dimension in my example is how many orthogonal sin functions (harmonics) aka components we mix together to represent a signal
It depends what you mean. The Hilbert space spanned by [math] \{ sin(x), sin(2x), \ldots, sin(5x)\} [/math] has dimension 5 because it is spanned by 5 independent vectors. The Hilbert space containing [math] sin(nx) [/math] for all n has infinite dimension however.

>I let N=5 But vector [ ] for each component seems to be something else: it is a set of sin(x) values from 0 to PI.
Are you doing this with a computer program or something?

[Anonymous No.11302765]

>>11302278
>mess around with Cousin
nice

[Anonymous No.11302784]

>>11301953
Bishop - Pattern Learning and Pattern Recognition

[Anonymous No.11302786]

>>11302754
>Why are you doing this?
Numeric methods.
>Are you doing this with a computer program or something?
Yes, I write some code. I am a CS major. At the same time trying to understand what I am really doing

>What do you mean by component here? Do you mean sin(nx)bn?
Yes. each component is a harmonic, and bn is the amplitude of each harmonic.
> The Hilbert space spanned by {sin(x),sin(2x),…,sin(5x)} has dimension 5 because it is spanned by 5 independent vectors.
Yes, cool, we are on the same page regarding this one.

>The Hilbert space containing sin(nx) for all n has infinite dimension however.
Yeah I get that, I just let n=5.

Thanks, I know I am doing something unmathematical by lettin n!=infinity but I just wanted to understand whats going on. I think I am like 90% clear.

[Anonymous No.11302853]

>>11302786
There is nothing wrong with finite dimensional Hilbert spaces and they can be indeed discrete, for example due to discretization of spacetime since our universe is discrete (both space and time are in fact discrete). Why? The Planck length. Our universe is a Hilbert space that lives on a spacetime lattice with spacing = Lp. And of course the dimensionality of our Hilbert space is itself finite, equal to the number of degrees of freedom. So N can be 5 or even 4.

[Anonymous No.11302870]

>>11302853
>for example due to discretization of spacetime since our universe is discrete (both space and time are in fact discrete). Why? The Planck length.
kys pseud faggot
>>11302786
get the fuck out of our thread

[Anonymous No.11302990]

Let f be a homomorphism between G and H, both groups.

1. Prove that for every g in G the order of f(g) divides the order of g.

2. Using only the definitions of group and homomorphism, prove that f(e) =f(e’), where e is the neutral element of G and e’ the neutral element of H.

[Anonymous No.11303077]

>>11302990
>Reminder: /sci/ is for discussing topics pertaining to science and mathematics, not for helping you with your homework

[Anonymous No.11303087]

>>11302853
>for example due to discretization of spacetime since our universe is discrete (both space and time are in fact discrete). Why? The Planck length.

Yeah, that's BS. There's nothing inherent in either mainstream quantum physics or mainstream general relativity that implies that space-time is discretized at the Planck length. Neither theories have anything special going on at those lenghts either.

[Anonymous No.11303097]

>>11302853
>discretization of spacetime
You do know that has nothing to do with the Hilbert space of states in QM being finite right? Regardless whether if you're on a lattice or not you cannot use finite-dimensional Hilbert spaces as a rep space of the Heisenberg algebra, end of story.

[Anonymous No.11303098]

>>11300673
Lol are you referencing Dan Freed? I took his Analysis on Manifolds class, never thought I’d see him mentioned on /sci/

[Anonymous No.11303113]

>>11303098
Yep. My research heavily intersects his and Hopkins's result on the classification of invertible TOs so I know quite a bit about him.

[Anonymous No.11303191]

>>11303113
Not the anon you replied to, but what are your thoughts on Freed’s notes for his “Analysis on Manifolds” class? I’m taking a differential topology class this spring and am wondering if they’d be good preparation (the notes seem kinda funky tho with all the affine spaces and horned spaces stuff)

[Anonymous No.11303202]

>>11302446
>Can math prove true love exist?
What have you tried?

[Anonymous No.11303338]

>>11302990
hahhahaha fuck off stupid faggot. genetic trash. if you have problems with something as simple as that, im sorry to be the one to say this, but math is not right for you.

[Anonymous No.11303816]

>>11302680
> [math]AbGrp[/math] instead of [math]Mod_R[/math]
>his intuition only immediately works for topological spaces, and needs to be extended to other cases by analogy and classifying spaces
>in particular, doesn't tackle the immediate homology of chain complexes
Certainly you could do better.

[Anonymous No.11303820]

Please stop posting anime.

[Anonymous No.11303877]

>anime in math thread
why, brothers? does this happen in real life? mathematicians watch anime?

[Anonymous No.11303885]

>>11303877
The kind of math students who feel they need to have an avatar on an anonymous imageboard all watch anime, yes.

[Anonymous No.11303897]

Any books like reed and simmon that dont skip half of the proofs? Maybe I should just take on rudin.

[Anonymous No.11303941]

>>11302990
dude

[Anonymous No.11303956]

>>11303097
not him, but whats wrong with discrete finite dimensional Hilbert spaces? if thats not quite Hilbert, lets call it Anon space.

>captcha asked to select motorcycles
>no motorcycles in the picture
>lied and clicked on some shadow
>picked traffic lights on the next image correctly
>passed
>now google will never be able to find motorcycles
i do it all the time and encourage you to do the same. we can beat AI together.

[Anonymous No.11303962]

>>11303897
>Maybe I should just take on rudin.

[Anonymous No.11303992]

>>11303956
finite dimensional "discrete" spaces (with an inner product) are certainly hilbert, but they are no good for representing the heisenberg algebra.

[Anonymous No.11303996]

>>11303820
>>11303877
Touhou isn't anime. And in any case, it's /mg/ culture.

[Anonymous No.11304438]

Is there a good book about collineation groups of (finite) projective planes? More generally books about group theory applied to finite projective geometry

[Anonymous No.11304439]

>>11303897
do it, it's worth it.
t. 1st year

[Anonymous No.11304477]

>>11302278
could you elaborate on the part about singular homology? pwease uwu

[Anonymous No.11304501]

>>11304439
Im talking bout grandpa rudin boy.

[Anonymous No.11304502]

>>11304477
look up the definition of "singular homology"
it's trivial

[Anonymous No.11304506]

>>11304501
t. first year

[Anonymous No.11304509]

>>11303996
Touhou is retarded /jp/ trash for literal autists that belongs nowhere. It's ridiculous, the games are not even fun in a meme sense, they are worthless.

[Anonymous No.11304516]

>>11304477
There's quite a bit of convoluted algebra to it, but singular homology is essentially supposed to keep track and describe times when it's impossible to fill the dotted red line in.
>>11304501
>Ah, you mean elementary school Rudin, the sequel to kindergarten Rudin?

[Anonymous No.11304522]

>>11304501
unrelated out of Conway, Kreyszig, Rudin, Lax, Stein&Shakarchi what is the best first book on Functional Analysis? Looking for something concise but self contained if at all possible.

[Anonymous No.11304524]

>>11304516
>>11304506
You tool func anal in your first year? Of what graduate school?

[Anonymous No.11304567]

>>11304516
isn't the condition you stated equivalent to the homotopy groups not being trivial?
>n-simplex is homeomorphic to the n-disc
>boundary of the n-simplex is homeomorphic to the (n-1)-sphere

[Anonymous No.11304582]

>>11304567
I said it was supposed to, I didn't say it successfully did that.
Homology and homotopy have a bunch of particularities to their construction which leads to extremely strange behavior.
Remember that before the Hopf fibration the lads imagined that homology was just commutative homotopy.

[Anonymous No.11304632]

Rate my galois theory test

[Anonymous No.11304635]

>>11304632
rotated

[Anonymous No.11304670]

>>11304635
>question three
[math]f=2x+1[/math] because [math]x^p[/math], the Frobenius automorphism, is just the identity in [math]F_p[/math].
I can't solve the rest because I don't recall the tricks for doing actual computations and also because I don't care.

[Anonymous No.11304682]

>>11304635
>60% of the test is cOmPuTe Le GaLoIs GrOuP
I wish test-writers were less obsessed with these questions. I get that they're easy things to throw on an exam but they're fucking boring. Shit's almost as bad as the Sylow theorems.

[Anonymous No.11304688]

>>11304524
huh? second year of undergrad of course.

[Anonymous No.11304704]

>>11304635
daily reminder that Galois figured this shit out when he was about 15 years old

[Anonymous No.11304806]

>>11304670
you can't do that. a polynomial is more than just the values you get when you plug elements in.
x^p = x for all x in Fp, but X^p =/= X as polynomials

[Anonymous No.11304854]

>>11304522
Lax.
>>11304524
Fun anal in grad school?
I learned fun anal in kindergarten.

[Anonymous No.11304899]

What is the probability that an erdos-renyi random graph avoids diamonds

[Anonymous No.11304914]

>>11304854
Based and kindergartenpilled

[Anonymous No.11305082]

I took Calc 1 in the summer and got a 69.49 (which my brofessor round up to a C-). I just started this semester taking Calc 2 and Jesus fucking Christ have mercy. It is literally only and solely integration. We literally just pick up right where Calc 1 left off. God damn, I actually fell physically sick from the amount of integration I have had to do. Tell me it gets better/easier. BTW I'm on the Calc1,2,3=>Linear Algebra track for my major. I never thought math would make me regret being born.

[Anonymous No.11305087]

>>11305082
you may want to reconsider spending the next four years doing what you're doing.

[Anonymous No.11305098]

>>11304854
thank you

[Anonymous No.11305124]

>>11305082
plug and chug integration is not math. it is a soul crushing drudgery. what is your major? probably not math otherwise you'd be doing anal now. calc-1-2-3-linear algebra sounds like an engineering track. and btw calc-3 is easier than calc-2.

[Anonymous No.11305188]

>>11305124
>what is your major?
Theoretical Mathematics w/ CIS minor. I chose this because I didn't know what I wanted to major in while I finished literally every one of my GEs at community college and I found out how quickly I could sequence these. After Calc 3, I can take 4 or 5 core major classes a semester. I kinda regret not transferring sooner and also not just taking a math course every semester.

[Anonymous No.11305214]

>>11305188
that has to be the worst program i've ever seen in my life. oh, and by the way. if you arne't liking math now, you're going to fucking hate it when you have to do ANALysis and algebra,

[Anonymous No.11305223]

>>11305188
>linear algebra specifically in the theoretical track
Are there really places out there that will you let you complete an applied maths degree without knowing linear algebra?

[Anonymous No.11305235]

Is there any use for the Walsh/Hadamard orthogonal functions nowadays? All the references I can find date back to like 50 years ago. I bet you guys haven't even heard of them.

[Anonymous No.11305249]

>>11305082
>It is literally only and solely integration.
you're taking a brainlet-tier course, quit crying and cheat the damn thing if it pisses you off so much. the chinks do it and no one cares about moralfaggotry nowadays, so why don't you?

[Anonymous No.11305258]

>>11305235
Applications of the Walsh functions can be found wherever digit representations are used, including speech recognition, medical and biological image processing, and digital holography.

[Anonymous No.11305263]

>>11304682
Well, it's about the only practical aspect of Galois theory; same goes for Sylow. You gotta make sure a student leaves the course knowing at the very least that. Also it appears in qual exams.

[Anonymous No.11305296]

>>11305258
Thats cool but I've noticed that some things in science and technology and math just get out of vogue. Yeah just like that. So when I google some "exotic" orthogonal transformations in the analog/digital domain I mostly find articles from the 1960's -1970's. Another such example is Laguerre polynomials. You'd think why not. But they seem to be completely forgotten. I want to come up with my own set of orthogonal functions, preferably non-polynomial so they could be implemented in analog, which is why I am curious about these trends.

[Anonymous No.11305330]

>>11304704
Are there anny good books/notes that develop Galois theory with only the algebra Galois had at hand? I just get lost in the amount of definitions and I can't see how in his time he managed to do it.

[Anonymous No.11305390]

https://arxiv.org/pdf/2001.04976.pdf
>Two different scenarios when the Collatz Conjecture fails
>Maya Mohsin Ahmed
>(Submitted on 13 Jan 2020)

>In this article, we give two different proofs of why the Collatz Conjecture is false.

[Anonymous No.11305515]

>>11305390
Holy fuck, the madman actually did it! Erdos better cough up that 500 dollars pronto.

[Anonymous No.11305525]

>>11305390
This for real or just schizo ramblings?

[Anonymous No.11305529]

>>11305525
No, it's not.

[Anonymous No.11305558]

>>11303816
Cohomologies with Abelian [math]\mathbb{Z}[/math]-modules as coefficients are completely determined by [math]H^\ast(\bullet,\mathbb{Z})[/math] by the universal coefficient theorem honey. Besides, [math]H^\ast(\bullet,\mathbb{Z})[/math] is in general also a [math]H^0(\bullet,\mathbb{Z})[/math]-module for cohomology theories satisfying the dimensional regularity axiom in addition to the standard EM axioms. Once we know the algebraic structure of [math]H^0[/math], we can pass to any arbitrary [math]R[/math]-module-valued coefficients.
>cases by analogy and classifying spaces
You do know that group cohomologies are just [math]G[/math]-equivariant cohomologies of principal bundles over a point, right? Topology and geometry are fundamental, all other abstract wank isn't.
Especially if the coefficients is a field of characteristic zero; try finding a generalized cohomology theory that isn't isomorphic to the standard singular one, I'll wait.
>>11303956
Try taking the trace of the Heisenberg relation [math][a,a^\dagger] = 1[/math] on a finite dimensional Hilbert space. If [math]\operatorname{dim}\mathcal{H} = N[/math] then [eqn]\operatorname{tr}([a,a^\dagger]) = \operatorname{tr}1 = N,[/eqn] but [math]\operatorname{tr}([a,a^\dagger]) = 0[/math] due to the cyclicity of the trace so [math]\operatorname{dim}\mathcal{H}=0[/math]. There's no QM here, sweetie.

[Anonymous No.11305563]

>>11305558
>honey
>sweetie

[Anonymous No.11305569]

Is measure the same thing as norm? I understand norm. It follows from the Euclidean distance. But measure? Is it the same as metric aka norm?

[Anonymous No.11305584]

>>11305569
No. A measure is an idea of volume rather than distance. A measure takes a subset of our space and tells you how "big" it is.

[Anonymous No.11306007]

>>11305569
Two basic measures you'll need to know:
-Lebesgue measure, which has a version in every dimension (Lebesgue 1 is length, Lebesgue 2 is area, Lebesgue 3 is volume, Lebesgue >= 4 are hypervolumes). The determinant of a matrix is precisely the change in Lebesgue measure of the parallelepiped formed by the basis vectors of your vector space. Ie, take an orthonormal basis {(1,0) (0,1)} of R^2, it gives you a square of Lebesgue 2 measure that is equal to 1 (ie, 1 m^2 for an analogy with physics if your vectors of norm 1 correspond to 1 meter). Pass it through your matrix A and for your new basis you'll obtain a parallelogram of Lebesgue 2 measure equal to det(A). Also, det(A) = 0 iff you've lost a dimension: a flat parallelogram has no area. This analogy extends to higher dimensions.

-Probability measures. Basically like Lebesgue measure, except the measure of your "whole space" is always equal to 1 (rather than say R^n which has infinite Lebesgue n-measure). Take an equiprobable dice experiment: draw a long rectangle and divide it into six equal areas. These correspond to your atomic events, and their area relative to the whole space is precisely their probability (1/6). This correspondence between measures/integration/probability is the foundation of modern probability theory, and the reason why you can calculate volumes or things like raytracing through Monte-Carlo simulation.

Measures are the "n-dim rulers" of abstract math.

[Anonymous No.11306035]

>>11305263
Your average working mathematician is going to need to compute some random-ass Galois group about as often as he's going to need to know whether or not a group of order 1998 is solvable (i.e. exactly one time, when it's asked on his qualifying exam).
It's only practical in the sense that you can't pass your quals unless you know how to do it; the theoretical parts are the only thing you'll ever see again once you finish your quals.

They don't show up on every exam because they're good questions or even important ones, they show up because you could write them up after a half bottle of vodka and there's a virtually neverending supply of tiny variations.

[Anonymous No.11306055]

>>11300216

Check Proposition 6.4.10 of Grafakos' "Modern Fourier Analysis"

[Anonymous No.11306139]

CS major here. I come here for ideas that I can implement and something useful for everyday person. This implementation hide all the abstract concepts of mathematical laws.

Which is the hottest math idea as of now. Category theory? Type theory?

[Anonymous No.11306144]

>>11306139
Several complex variables.

[Anonymous No.11306158]

>>11306144
What about it? "Several complex variable" is not an idea. You can call it terminology of some abstract object satisfying some rules. May be you can call it static idea. An idea that is moving is true idea. Or a group of static idea become motive in some dynamic system. That is true idea.

[Anonymous No.11306430]

>>11304704
Daily reminder that he had Vandermonde's and Cauchy's papers to get his ideas from. And, hell, if he knew Italian and read Ruffini, he could've just as well figured it out at 14.

[Anonymous No.11306434]

>>11305330
>and I can't see how in his time he managed to do it.
Vandermonde, Lagrange, Ruffini, Cauchy, Abel. All, except Ruffini (which is in Italian) in French, untranslated, GOOD. LUCK.

[Anonymous No.11306440]

>>11306434
P.S. I think I saw Cauchy's 1815 paper on permutations translated to English in some blog. WHATEVER.

[Anonymous No.11306471]

>>11306434
galois was french so what's the problem?

[Anonymous No.11306479]

>>11306471
Well, I was replying to a person seemingly eyeing the possibility to repeat his steps to an extent.

By the way, Galois' writings DO have an English translation. I think his entire writings, actually. Google the book yourself, if need be.

[Anonymous No.11306531]

What are the most and the least trendy fields of mathematics in 2020? I bet all the cool kids want to learn cohomology and category theory. And what is the most boring boomery uncool stuff?

[Anonymous No.11306564]

>>11306531
"Trendy": Algebra

"Boomer" Analysis. Guess I'm a Boomer.

[Anonymous No.11306580]

/mg/ lads:
How do we save the rest of /sci/?

[Anonymous No.11306581]

>>11305390
Why doesnt this dude provide an example of a number that doesn't converge to 1?
He supposedly disproved the conjecture right?

[Anonymous No.11306583]

>>11306580
we don't

[Anonymous No.11306586]

>>11305525
Tip: the CNN anon always makes sure to give us his name.

[Anonymous No.11306588]

>>11305529
So its not schizo ramblings or is it not for real?

[Anonymous No.11306590]

>>11306580
Calculus captcha

[Anonymous No.11306595]

>>11306531
perceptions based on what i see at my third-rate university:
anything connected to diff eqns or real analysis - dead as fuck
complex analysis, everything in functional analysis - uncool, although not as terrible as the previous ones
dynamical systems, ergodic theory - somewhat trendy, mainly the parts that have connection to number theory and combinatorial stuff
algebraic geometry - hot and trendy
category theory is mostly a meme but i've spotted the occasional retard talking about categories

[Anonymous No.11306608]

>>11306531
>Uncool
College algebra, trigonometry and geometry

>Meh
Pre-calc, statistics for social sciences

>Hot and trendy
Calculus, fluid mechanics, linear algebra, differential equations, statistics for emgineers and advanced calculus for engineers.
I hear calc 3 is specially cool, as you learn how to calculate triple integrals and those are really hard.

[Anonymous No.11306642]

RIP Christopher Tolkien. He majored in mathematics if you guys didn't know.

[Anonymous No.11306644]

>>11306144
>>11306158
SCV is not at all a 'hot' trend in mathematics nowadays. Dynamical Systems, Algebraic Geometry and Category Theory are all much more trendy.

[Anonymous No.11306676]

Is there any research being made about fractals? What's their importance?

[Anonymous No.11306706]

>>11300167
Why does B blow up to zero? Isn’t e^(inf) = inf not zero?

[Anonymous No.11306726]

>>11306706
Do you mean [math]\lim_{x \rightarrow \infty} e^{-x} = 0[/math] ?

Also, theres a stupid questions thread

[Anonymous No.11306750]

>>11306726
>that limit
heh reminds me of an analysis test on which I fucked it up. a good lesson which I remember till now.

[Anonymous No.11306786]

will i need a supplementary category theory book to accompany aluffi's algebra chatper 0 or is it self contained enough to give the gist of the theory.

[Anonymous No.11306798]

>>11306750
Its a pretty limit.
Im an analysis noob, so at this point the only """limit""" ive proved is that given [eqn]A \subseteq \mathbb{R}, A = {1/n : n \in mathbb{N}}[/eqn]
Then by the arquimedian property, [math] infA = 0[/math]

[Anonymous No.11306801]

>>11306798
Shit, fucked up my tex

[eqn]A = \{ 1/n: n \in \mathbb{N} \}[/eqn]

[Anonymous No.11306804]

>>11306786
Hahahaha, Alluffi's Algebra is an algebra book written for freshmen just starting to pick up algebra, get right into it friend.

[Anonymous No.11306813]

>>11306798
Actually that isn't true anon.
Supposing that by N you mean N+ (as otherwise 1/0), then [math]inf A = 1 = min A[/math]

It would be true however if you were to take n from the set of (positive) reals or even (positive) rationals

[Anonymous No.11306816]

>>11306813
my bad confused inf with sup - whoopsie!
you are correct anon

[Anonymous No.11306820]

>>11301918
Hi, how did it went?

[Anonymous No.11306858]

>>11306676
>DeMarco & Lindsey: Convex Shapes & Harmonic Caps, 2016
>Wareham & Lasenby: Generating Fractals Using Geometric Algebra

Problem with fractals is that there are many different classes of them, depending on how you generate them; and the term "fractal" for this reason is pretty ambiguous. Because of this diversity/Ambiguity, there isn't a really a category theoretic approach to fractals yet, which would help classify and understand their links to the rest of mathematics better. I'm only distantly aware of research that touches Escape-Time Fractals.

There are some links of ETF-type fractals to chaos theory, interestingly. Dendrite/vascular/root like structures tend to be analyzed with fractal-based math. But I'm a bit a out of my depth on the subject.

Most ambitious thing I've seen with fractals is trying to develop a form a calculus over them. See the work of J. Harrison in geometric measure theory.

Fractals that aren't ETF-type fractals, for example the Cantor Set or the Sierpinsky triangle, have been found in topology, cellular automata, number theory (eg, the Pascal triangle on GF2)...

[Anonymous No.11306868]

>>11306804
>T. mathstackexchange
>Need a linear algebra book. Try Hoffman & Kunze for a basic class Artin for well prepared freshmen.

[Anonymous No.11306904]

>>11301918
post problems pls:)

[Anonymous No.11306916]

>>11302990
you literally just have to plug in the definitions, come on

[Anonymous No.11306966]

Hi, /mg/. Do you think I can learn fourier series and half range fourier series in 4-6 hours?

[Anonymous No.11306968]

>>11306966
yes if you close /mg/ and start now

[Anonymous No.11306970]

>>11306968
Ok. Any resources you'd recommend quickly? My lecturers resources suck so I've had to pretty much learn the whole module myself in the last 2 days.

[Anonymous No.11307004]

>>11306816
No worries i literally made the same mistake and confidently answered "1!" When the professor asked the class what we think the infinum is.

[Anonymous No.11307035]

>>11306608
> fluid mechanics
is that even math?

>advanced calculus for engineers
sounds like oxymoron. if it is calc with proofs it is called anal. engineers don't do anal.
> triple integrals
not harder than double integrals, just a lot more steps, so just more tedious.

[Anonymous No.11307040]

>>11301918
why belgian? did you get to pick a country? pretty cool.

[Anonymous No.11307062]

>>11306858
Holy fucking shit that's fucking awesome, thanks for the thorough explanation lad, I got really interested in all that now.

[Anonymous No.11307135]

>>11307035
>so just more tedious.
Spherical coordinates are nigh impossible, I never understood them very well when I did vector calculus.

[Anonymous No.11307155]

>>11307062

Glad I can share the love. There's definitely a lot to research concerning fractals, but because it's very complex and messy, you probably won't find much formal help in said research. The guy who "solves" fractals will probably have an extremely diverse and profound understanding of mathematics. That said, even a CS nerd can probably contribute because of that absence of real direction for research, as well. The fact that many fractals seem to pop up in various parts of mathematics is quite intriguing, and the reason why I say this.

If you're the CS+Math+Artsy kind, I suggest you implement your own fractal renderers, either in 2D over complex polynomials (generalize the Julia protocol from z^2+c to any polynomial) or more general classes of complex functions (cos^2(z) + c). Consider various ETF protocols (not just Julia) and some Mariani-Silver algorithm goodness (pic related).

Do note that basically any ring/algebra can have an associated polynomial ring (and many have some definition of exp/cos/sin/etc through Taylor series) and can lead to various spaces and types of fractals. Though for n-dimensional fractals of the kind shown by Lasenby and Wareham, you'll probably have to learn geometric algebra (definitely worth the look, check Alan MacDonald on youtube for a basic gist) and raytracing as well (raytracing on fractals in particular can sometimes use techniques specific to fractal rendering).

[Anonymous No.11307187]

>>11301920
Thank you! It was somewhat fun but very frustrating nontheless, all my friends got better scores than I did :(

>>11306820
Hello! It didn't go amazingly well, I scored 82 out of 150. (30 problems, wrong = 0 marks, blank = 2, correct = 5, 1h30) Those were only the qualifiers and I had 0 training plus mediocre sleep, if I make it to the 2nd round (which is possible, you usually need more than 78 to get qualified but it's not based on score, they just take the top X scores), I promised myself I'd at least train a little!

>>11306904
Here are a few of them, some of the most interesting ones! (Keep in mind I still haven't seen all the material required to answer every question, so you may find those easy!)
No calculator allowed, only pen and paper!

1. A date is palindromic, and if it's writing is ddmmyyyy, if we are the 2nd of february 2020 (02022020), in how many days will we have to wait until the next palindromic date? (note : 2020 is 364 days long!)
[integer 0-999]

2. The sum of 3 distinct prime numbers is 40. What is the difference between the 2 largest numbers?
[0 / 12 / 16 / 20 / 24]

3.How many couples of (a, b) exist such that
[(2a-b)^2 - (a-2b)^2]^2 = 225 ?
[0 / 2 / 4 / 8 / An infinity]

4. The equation x^4 - 40x^2 + k = 0 (in unknown x) admits 4 real roots that make an arithmetic progression. What is the value of k?
[integer 0-999]

5. How many triplets (x,y,z) of consecutive integers exist such that x < y < z and xyz=x+y+z
[integer 0-999]

6. What is the expression in (pic related) worth?
[integer 0-999]

>>11307040
No, I live in belgium, I'm in highschool (18 years old but had to retake a year). And every highschool student gets to participate if their school participates.
Do any of you guys have tips to do better at problems? I'm always too scared to really get into maths because I don't want it to become another reason for self loathing because of poor performance, but I can't help and love mathematics...

[Anonymous No.11307362]

>>11306139
nobody cares you fucking genetic trash faggot.
keep your CS bullshit on >>>/lgbt/

[Anonymous No.11307385]

>living with my grandma
>she supports me in every decision I make
>starts a math degree because I love it
>doesn't care about money
>grades are low, but I'll keep persisting
>father comes to live with us
>get into an ugly fight with him, the whole building knows about it and some acquaintances of the family on the street knows as well
>leaves my grandma apartment because I'm ashamed of the people in the building and in the street and because I don't really like my father
>starts living with aunt
>she and her family are pressuring me into getting a slave job
>constantly put me down because I'm not 'normal' enough
>rarely supports me in anything
>It's been 8 months now

Holy shit, what would you guys do in my situation? Would you go back to living with grandma? She always tells me to come back when we talk, but it would be very shameful to face all the neighbors who heard the fight and I would have to live with my father again.

[Anonymous No.11307423]

>>11307385
first step, stop posting with anime pictures. second, fuck them do what you want; so long as it's on your own dime.

[Anonymous No.11307434]

>>11307423
>you own dime
I don't make a penny, lad, it was my grandma that sustained me, now it's my aunt, all I do is go to college. But my grandma always supported me, while my aunt constantly pressures me into slaving myself in a job. I want to go back to my grandma's apartment, but it's hard, people would judge me a lot, that street is full of assholes.

Anime is the essence of 4chan, brah

[Anonymous No.11307437]

>>11307434
if you're living on someone elses dime, what do you really expect?
>Anime is the essence of 4chan, brah
science and math board, not anime board.

[Anonymous No.11307440]

>>11307437
>if you're living on someone elses dime, what do you really expect?
That they would be supportive of my decisions? That they would try to understand me a little bit? I never had any problems with my grandma regarding work.

[Anonymous No.11307447]

>>11307385
it is a difficult situation anon, i don't know what to tell you. you can't really greentext enough context since only you know all the details.

[Anonymous No.11307457]

>>11307440
well, decisions ain't cheap. you think money just poofs into existence? and what if at the end of studying math you don't get anywhere with it? all that money down the drain, they're at a loss, you're still useless and nobody wins.
maybe try getting your own capital before pursing an academic path that has little to no explicit monetary gain.

[Anonymous No.11307489]

>>11307457
So I'm supposed to slave myself in a jpb I don't like which will consume precious study time and probably damage my college attendance just because of 'mah munny', that would be a very sad life for me bro.

[Anonymous No.11307497]

>>11307489
maybe try putting your damn math skills to work and TA/tutor people for money. get an on-campus job or w/e. people have done it before and continue to work+uni . it isn't easy, but boo hoo life isn't supposed to be else we'd all be billionares living the high life.
you realize you'll outlive your grandma and parents you know? what then?

[Anonymous No.11307499]

>>11307035
Cant you recognize shitposting?
Fucking autists, I swear.

[Anonymous No.11307501]

>>11307499
This was meant for:
>>11307035

[Anonymous No.11307506]

>>11307501
Sorry that was meant for: >>11307501

[Anonymous No.11307507]

>>11307497
>tutor people for money.
Doesn't sound so bad, but it's probably not much money at the end of the month
>people have done it before and continue to work+uni .
Not people majoring in pure mathematics, it's the hardest degree
>you realize you'll outlive your grandma and parents you know? what then?
By that point I was expecting myself to already be working as a professor

[Anonymous No.11307510]

>>11307501
I mean swap >>11307501 and this >>11307506
it was for >>11307499

[Anonymous No.11307526]

>>11307507
> it's the hardest degree
drop the arrogance. will do you wonders.
>By that point I was expecting myself to already be working as a professor
do you know what it takes to attain such a position?

[Anonymous No.11307532]

>>11307385
Honestly? I'd go live with my fiance.

[Anonymous No.11307534]

>>11307526
>drop the arrogance. will do you wonders.
Is it arrogance when it's true?
>do you know what it takes to attain such a position?
A math degree then a masters then a doctorate then the job, quite simple to me, I'm pretty sure my parents will still be alive by that time.

[Anonymous No.11307544]

>>11307534
>Is it arrogance when it's true?
even if it is true, which really depends on what you mean by 'hard', there's no value in inflating one's ego with something as petty as a degree path.
>quite simple to me
uh huh...just hope you're smarter with math than you are with money.

[Anonymous No.11307585]

>e^st is the eigenfunction of LTI
what do they mean by that? i know eigen vectors from LA. what part of math studies eigenfunctions? func anal?

[Anonymous No.11307588]

>>11305390
>By Theorem 3.2, the Collatz Conjecture is false. End of story.

[Anonymous No.11307591]

>>11307585
yes

[Anonymous No.11307596]

>>11307544
Thanks for not helping at all :D

[Anonymous No.11307599]

>>11307585
>what part of math studies eigenfunctions
Quantum mechanics.

[Anonymous No.11307601]

>>11307596
I've helped plenty by being real with you. You just don't like what I've said, fair enough. Best of luck.

[Anonymous No.11307616]

>>11307601
All you said is 'slave yourself unhappily in a worthless job'.

Yeah, I ain't doing it, unless my aunt forces me to, let's just hope she doesn't.

[Anonymous No.11307625]

>>11307616
No, that's what you've seemed to interpreted from it which is unfortunate. I said exactly
>maybe try putting your damn math skills to work and TA/tutor people for money. get an on-campus job or w/e
i.e. use the skills you're spending so much time to learn in order to get some money OR get a campus job, which usually are bullshit low effort easy money, not this 'slave' picture of work you have conjured up. wage slaving is slaving at mcdons or some retail shit for 30 years. sometimes part time work isn't all that bad, since you know, it's PART time. just for a bit of change nothing more.
from what I can tell, you aren't being very realistic. you've got this fantasy bubbled in your head which sounds just like the ' study pure math btw 300k/yr any job i want' meme. hell, maybe it'll all work out, but man you gota think about if it doesn't.

[Anonymous No.11307630]

>>11307625
I have no experience in tutoring people, nor do I have any contacts that would recommend me or anything, but I'll try that, see how much I can make a month and how time consuming it is.

Also, you think it's easy getting a part-time job? I would be in one if they would hire me, but there's a job crisis around here.

[Anonymous No.11307639]

>>11307630
tutoring is easy, but you need to be able to convey what may seem like simple things to you in a way that doesn't come off as rude or condescending to people genuinely struggling with it. look for help center positions, set up your own ad, facebook, etc, etc.

depends on where you are, evidently and where you want to apply. the standard for employees for part-time work isn't very high, since they're easily replaceable/aren't usually invested in working there for over many years.

[Anonymous No.11307767]

>>11307630
Helping people on /sqt/ gives good tutoring experience. I used to tutor and the way I interact on sqt is pretty similar. It's all about helping the person to understand the fundamentals and then getting them to explain how to do specific problems to you (maybe not super directly, but it's never about giving them the answers). Also about explaining techniques to mathematical thinking.
But I'm going to be brutally clear with you. If you cannot handle a part time job and an undergraduate math degree at the same time, you absolutely will never be able to be a math professor. Please do yourself a favor and find an on campus part time job because this is a diagnostic to tell you if you can think about your math while doing menial things (say, working in the library) and if you can manage your time between multiple disjointed responsibilities. People who just know how to do math very very rarely make good academics.

[Anonymous No.11307768]

Is addiction to downloading books considered hoarding? I have this strange condition where even though I've learned enough on the subject I just can't stop and move on. it is some sort of inertia, I need to download another dozen or so books and look through all of them.

[Anonymous No.11307821]

>>11300167
i've been told the statement
∀P((0∈P∧∀i(i∈P-->i+1∈P))-->∀n(n∈P))
captures the principle of mathematical induction in second order logic.
why wouldn't the antecedent i∈P-->i+1∈P be sufficient, since we're using the universal quantifier on i? wouldn't that apply the condition to every element in P, including the base case?

[Anonymous No.11307837]

>>11307768
Yes. But, its productive hoarding if you’re reading even a small portion what you’ve downloaded.

[Anonymous No.11308014]

>>11307385
i don't know where you are from but in some places there are scholarships for students that have troubles at home

[Anonymous No.11308151]

>>11301918
>The Coomer Tao

[Anonymous No.11308168]

>>11300553
>assume G is finite
As opposed to assuming what?

[Anonymous No.11308183]

>>11307423
>stop posting with anime pictures
"No".

[Anonymous No.11308238]

>>11307821

Because "p => q" being true is not the same as "(p) AND (p => q)" being true.

IF Santa Claus exists THEN all children get presents during Xmas. (p => q is true)

But Santa Claus does not exist, and "p" is false. This does not in any way affect the fact that "p => q" itself is true.

Induction is Initiatiazation (p_0 is true, or at least some p_i is true, your starting point) + Heredity (p_i => p_{i+1}).

[Anonymous No.11308273]

>>11300167
> NSA discovers major vulnerability in Microsoft crypto system
>Exploit is literally

x to the power of 1 = x

[Anonymous No.11308298]

What is semigroup?

[Anonymous No.11308347]

>>11308298
a group with an identity problem.

[Anonymous No.11308348]

>>11308298
Please leave

[Anonymous No.11308362]

boys, could anyone give me a hand in suggesting some universities to apply to for undergrad summer research opportunities in applied mathematics? i'm finding it very difficult to actually find opportunities to apply to to get some experience in this field before applying for a PhD. would be much appreciated!!

[Anonymous No.11308375]

wtf are functionals?! Are they seriously just functions that map to the reals? If so, why should I give these functions a special name? I'm deriving the euler-lagrange equations and I feel like I'm missing some intuition about the nature of these things!!!

[Anonymous No.11308390]

>>11308375
yes. read a book on functional analysis

[Anonymous No.11308426]

>>11308375
it's not a precise term, but usually when F maps functions to numbers you would say that F is a functional
kind of like "a family of sets" means simply "a set of sets" but to avoid confusion you say the former

[Anonymous No.11308447]

Fuck I hate how most mathematicians use the matrix of a linear application and the application itself in the same context. Yeah I know they represent the same thing but it's not the same thing. A matrix is just a fucking array of objects and a linear application is a fucking function. A matrix is NOT a function, you CAN NOT apply a matrix to a vector, you can just multiply them. This trigger my autism.

[Anonymous No.11308461]

>>11308447
it resembles applying an operator

[Anonymous No.11308470]

>>11308347
heh

[Anonymous No.11308493]

>>11307532
Pros: she can warp you anywhere through interdimensional vaginas
Cons: too good to be real

[Anonymous No.11308499]

>>11308426
>but to avoid confusion you say the former
what confusion? I don't quite understand...

[Anonymous No.11308502]

>>11307385
Stop posting /adv/ shit on /mg/.
>>11307585
*Fun anal.
You'll also find those in stuff like operator theory, spectral theory, spectral geometry and even some numerical stuff.
>>11308168
>As opposed to assuming what?
What are you, some third removed shitter?
/mg/ is a strictly intuitionist finitist thread.

[Anonymous No.11308503]

>>11307437
Anime yields elegant and effective proof of several previously unsolved problems in mathematics. You can no longer have math without anime. Unless you want to live in the past, along with using an axe to shave yourself and chicken dung as shampoo.

[Anonymous No.11308504]

>>11308503
no it doesn't

[Anonymous No.11308506]

> tfw you do anal

[Anonymous No.11308510]

https://www.youtube.com/watch?v=MKluX5zMXz0

[Anonymous No.11308512]

>>11308506
>tfw you do complex anal

[Anonymous No.11308517]

>>11308512
>tfw you do non-standard anal
>far-too-gone-coomer.webp

[Anonymous No.11308518]

>>11308512
>tfw you do Clifford anal

[Anonymous No.11308527]

>>11308502
>Stop posting /adv/ shit on /mg/.
What're you gonna do about it if I don't?

[Anonymous No.11308541]

>>11308527
I'm gonna stop helping you with your homework in /sqt/.

[Anonymous No.11308560]

>>11308447
It's isomorphic in every way. "Apply" is almost always the same as "multiply" in the context of group actions, modules, etc. It's perfectly reasonable.
The real issue is that a linear map only becomes a matrix after a choice of basis, but most of the time that's also completely uninteresting information.

[Anonymous No.11308563]

>>11308362
Check out the mathprograms.org website where you'll apply to most REUs, it should have links to the undergraduate math ones if you find the right page. Plenty of them are applied math, but you'll have to look and see what you like. Also many projects straddle the line between pure and applied and can use someone with numerical / other applied experience.

[Anonymous No.11308565]

>>11308502
>fun anal
>mentions a bunch of "other" fields
>they're all just subsets of fun anal

[Anonymous No.11308577]

>>11308565
>spectral geometry is a subfield of functional analysis
I mean, it's not wrong.
But it's sort of besides the point, you know?

[Anonymous No.11308637]

>>11302175
>What do the non-free parts of the homology groups correspond to?
Torsion holes.

>What the hell does cohomology correspond to?
Coholes.

[Anonymous No.11308651]

>>11308563
thank you very much man

[Anonymous No.11308671]

>>11308238
Thanks anon!

[Anonymous No.11308681]

>>11308447

>>11308560
>this

For a code monkey, here's how to see the logic:

function matrix_multiplication_apply(A:Matrix, v:Vector)
return Av

through currying/partial application of a given matrix A, this creates a single possible function:

function applyA(v:Vector)
return Av;

There are precisely as many of the linear applications "applyA" as there are matrices A to build them with; additionally they'll function the exact same way as original multiplication of their A with any v (for all v).

Your complaint is the same as saying that "linear application in R, and real numbers themselves, are not the same thing", well, quite the contrary: they are, so long as you're in a context where the two are isomorphic. That's precisely what an isomorphism teaches us: elucidating which concept can be considered "mathematical synonyms" based on context.

To be precise with this example, taking the algebras (R, +, *) and (L(R), +, °), we have:

a + b = c <=> (x -> ax) + (x -> bx) = (x -> cx)
a * b = c <=> (x -> ax) ° (x -> bx) = (x -> cx)

Here, you have a perfect correspondence of structures, elements and operations on all levels. That's an isomorphism. It's "having two different allowed ways of seeing the same single abstract concept".

The only issue with matrices is that there's only an isomorphism with linear applications once a basis for your vector space/algebra is defined. But that's just because most math works without the need to introduce coordinates in the first place.

[Anonymous No.11308689]

>>11308447
The matrix defines the transformation. Consider each column of a matrix from left to right. Each column states the position of the canonical basis vectors after the transformation. As such, a new basis is created, which spans a new space. The matrix defines the transition from your regular orthogonal [math]\mathbb{R}^2[/math] to the new space.

[Anonymous No.11308690]

>>11308689
I should clarify, it's [math]\mathbb{R}^n[/math] but it's simplest to visualize for [math]\mathbb{R}^2[/math]

[Anonymous No.11308696]

Where can I find a definition of the product of two discrete probability spaces, without getting too far into the subject of probability theory?

For example, if I have a probability space D where I flip a coin, [math]D\times D[/math] should be the probability space "Flip two coins."

[Anonymous No.11308726]

>>11308375
Have you studied linear algebra? Just asking. It's easier to explain if you have.
It's a map that takes a vector and outputs a scalar.
With geometric vectors, you could think about the dot product of a vector with itself. It takes a vector and returns its length squared.
When seeing functions as vectors (you can add them and scale them and the output is still a function), if you take the definite integral of a function, you're effectively mapping a function to a scalar. Which scalar? Depends on the limits of integration.
An example is the potential energy of a wire hanging between posts. You can have a function that describes the height of the wire with respect to its position, and taking the integral of the height times the linear density will give you the total potential energy of the wire. So which position will the wire take when dangling from the post? The one that minimizes the functional (potential energy).

[Anonymous No.11308816]

>>11308726
>Have you studied linear algebra?
I have. Thank you for the helpful response.

[Anonymous No.11308836]

>>11308696
There's a lot of finicky measure theory to it, but essentially, the set of possible occurrences is [math]A \times B[/math], and for an event [math]C \subset A[/math] and an event [math]D \subset B[/math], [math]P(C, D)=P_A (C) \times P_B (D)[/math], where [math](C, D)[/math] reads as "C and D" the way you'd imagine.
>what if the event in [math]A \times B[/math] cannot be given as (C, D), [math]C \subset A[/math] and an event [math]D \subset B[/math]
Then you need to give it as an union of disjoint events of the form [math](C, D)[/math] and sum.

[Anonymous No.11308866]

>tfw microlocal anal

[Anonymous No.11308886]

>>11308447
is 42 a function? it can be in the context of currying. same thing

[Anonymous No.11308934]

Prove that [math]\mathbb{C}\bigoplus_{\mathbb{R}} \mathbb{C} \not\cong \mathbb{C}\bigoplus_{\mathbb{C}} \mathbb{C} [/math]

[Anonymous No.11308973]

>>11308725
>>11308735
>>11308751
>>11308754
>>11308761
>>11308762
>>11308767
>>11308770
>>11308773
>>11308779
>>11308784
>>11308788
>>11308789
Thoughts on this?

[Anonymous No.11309008]

>>11308934
the proof is trivial and left as an exercise to the original poster

[Anonymous No.11309017]

>>11308934
what does the plus in circle means

[Anonymous No.11309083]

>>11309017
tensor product

[Anonymous No.11309107]

>>11309083
i havent used those in 3 years, but i think as a vector spaces over R these objects have respectively dimensions 4 and 2 so they are different

[Anonymous No.11309114]

>>11308866
Should have attached a 1x1 pixel image.
>>11309083
> [math]\oplus[/math] stands for the tensor product
Does it, tho?

[Anonymous No.11309206]

>>11300167
[eqn]
\log_{0.02}\sqrt[3]{25} = x \\
\sqrt[3]{25} = 0.02^x \\
\sqrt[3]{5^2} = \frac{1}{200}^x \\
5^\frac{2}{3} = 200^{-1 \cdot x} \\
5^\frac{2}{3} = (5^2 \cdot 2^3)^{-x}
[/eqn]
No idea how to solve this, textbook says answer is [math]\frac{-2}{3}[/math].

[Anonymous No.11309225]

>>11309206
what book?

[Anonymous No.11309260]

>>11309225
It is a highschool textbook from the third world and there is no english version of it, so I doubt that info is going to be useful. One of the authors is Gelson Iezzi.

[Anonymous No.11309266]

>>11309206
my friend it's simply not -2/3, put it into wolfram to check
it's fucking 2020, learn to use your computer

[Anonymous No.11309297]

>>11309206
>>11309260
Either you screwed up earlier in the question (or copied it down wrong) or your third-world textbook is shit, because the answer is wrong.

[Anonymous No.11309303]

>>11309206
1. You've got an extra zero in there. Replace log_{0.02} with log_{0.2} and you'll get the right answer.
2. Your working is wrong. 0.02 is 2/100, not 1/200.

[Anonymous No.11309304]

>>11309206
1/200 is not 0.02

[Anonymous No.11309368]

>>11309083
if it's the tensor product then it's easy because they have different dimensions

[Anonymous No.11309370]

>>11308973
My thoughts are that it's not a qusstion

[Anonymous No.11309462]

>>11309297
>>11309303
These anons are right, I screwed when I copied it to notebook, I feel stupid now.

[Anonymous No.11309487]

Mental how the most amazing patterns and revolutionary discoveries were made long before computers/numeric methods. The euler formula, the Taylor series, various interesting polynomials, curves, the Gamma function, strange attractors, the mandelbrot set, the list goes on and on. And now that we have super fast computers, nobody has come up with anything even remotely as significant. Well we have found some new primes. Thats it? No bright minds to come up with new ideas? new patterns? new beautiful curves? Crickets.

[Anonymous No.11309498]

>>11309487
>nobody has come up with anything even remotely as significant
What about the vast progress made in basically every facet of machine learning and AI?

>inb4 that's not maths

[Anonymous No.11309504]

>>11308816
I'm glad I could help.

[Anonymous No.11309506]

>>11309008
kek

[Anonymous No.11309511]

>>11309017
I think it means direct sum.

[Anonymous No.11309522]

>>11309498
>i said inb4 so you can't argue it is not math
but it is not. it is just number crunching based on existing ideas and theories.

[Anonymous No.11309547]

>>11309522
>existing ideas and theories
Be honest anon, do you know anything about machine learning more than the threeblueonebrown video you watched explaining neural nets? Can you explain how things have progressed so rapidly since the invention of backprop in the 60s? Hint: the answer isn't "computers got faster"

As an example, LSTMs are (imo) a far greater discovery than "various interesting polynomials".

Convergence and optimality proofs as well as statistical interpretations and justifications are now vast components of ML that dind't exist 50 years ago.

[Anonymous No.11309566]

>>11309547
LSTMs are a far greater discovery than, say, the orthogonal polynomials? Or the Taylor polynomials? Or elliptic polynomials? Be honest.
(I don;t know what LSTM is).

[Anonymous No.11309572]

>>11309566
>Taylor polynomials
Nah
>elliptic polynomials
Nah
>orthogonal polynomials
Yeah

[Anonymous No.11309575]

>>11309572
>orthogonal polynomials
Those are the most useful ones from those four, tho.

[Anonymous No.11309578]

>>11309575
What are they used for?

[Anonymous No.11309580]

>>11309578
Finite element methods.

[Anonymous No.11309582]

>>11309575
>>11309578
I ask this because Taylor polynomials are literally everywhere and the https we are talking with right now probably uses elliptic curve crypto

[Anonymous No.11309590]

>>11309582
Listen, I'm trying to google what the fuck are elliptic polynomials, and I'm having next to no success.
But I don't think they're elliptic curves.

[Anonymous No.11309612]

>>11309590
I just assumed anon meant elliptic curves since they are polynomials.

[Anonymous No.11309642]

>>11309582
>elliptic curve crypto
I also hate it how algebra fags take all the credits because they retrofitted and pigeonholed someone else's discovery into their beloved fields and groups and what not. like cryptography wouldn't be possible if we didn't know about " muh F21" or whatever. weierstrass is rolling in his grave.

>>11309590
>>11309612
yes i meant elliptic curves and yes they are polynomials.

[Anonymous No.11309649]

>>11309642
Children know that elliptic curves are polynomials.
But I'm not sure if elliptic polynomials are elliptic curves.
https://www.crcpress.com/Elliptic-Polynomials/Lomont-Brillhart/p/book/9780367398200
"The term elliptic polynomials refers to the polynomials generated by odd elliptic integrals and elliptic functions."

[Anonymous No.11309655]

>>11309649
Such a beautiful, well written page, holding a wealth of information.
http://mathworld.wolfram.com/EllipticPolynomial.html

[Anonymous No.11309668]

>>11309649
every polynomial of a degree > 1 is a curve, i,e, gradually deviates from a straight line.
proof: take the derivative.

[Anonymous No.11309679]

>>11309642
>I also hate it how algebra fags take all the credits because they retrofitted and pigeonholed someone else's discovery into their beloved fields and groups and what not. like cryptography wouldn't be possible if we didn't know about " muh F21" or whatever. weierstrass is rolling in his grave.

[Anonymous No.11309783]

>pushforward

[Anonymous No.11309936]

>>11308577
>you know?
Yes, I do.

[Anonymous No.11309980]

>>11300673
wh-what is this?

[Anonymous No.11309987]

>>11306580
good question
>>11306590
interesting

[Anonymous No.11310009]

>>11308493
Con: she eats you.
Or it might be a pro if you're into that.

[Anonymous No.11310028]

if i were to extrapolate the excellent /mgay/ advice to work with physicists to instead work with mathematicians (am a physicist), what maths should i know (and what would be the appropriate textbooks to use)?

[Anonymous No.11310045]

>>11310028
Fun anal (OA and *-rep theory, PDE/optimization), topology (alg and diff), geo (anal and diff), alg (Lie and affine Lie and rep theory).
You can pick up a mathematician at the shelter and he'll teach you all that stuff.

[Anonymous No.11310074]

In R^n given the regular topology, if we give S^(n-1) (the unit sphere) the subspace topology, will this be discrete? Because we can find balls in R^n that only intersect single points (singletons) of the sphere.

I am pretty sure, just checking.

[Anonymous No.11310080]

>>11310074
>will this be discrete?
No
>we can find balls in R^n that only intersect single points (singletons) of the sphere.
Give an example

[Anonymous No.11310081]

>>11310074
The balls have to be open anon...

[K@ !wLYVBQU9kU No.11310098]

Hi /sci/
I'm mostly from the arts, so I'm not sure how different this is from pre-existing concepts. The project stemmed from my desire to enter into STEM more, and is based off of my firsthand observations of nature (biology, geology, etc.)

I develop alternate numerals (characters) that highlight number relationships. There are a few sets, but I collectively call them "Reactive Element Neo-Numerals",
or "REN-Numerals" for short.

They do not have set values in the way that Hindu-Arabic numerals do. Rather, they aim to highlight states, phases and number relationships in various bases.

K@ - 1/4

[K@ !wLYVBQU9kU No.11310102]

>>11310098

If you find any errors, please let me know. I'll try my best to correct them, and answer questions.
Also, if there are any pre-existing related theories you think I should study more, I'm always looking for new material to read.

Most of this has been done on paper so I will be making graphics to share information as it is requested, or becomes more relevant.

K@ - 2/4

[K@ !wLYVBQU9kU No.11310104]

>>11310102

I hope to see a much more metaphysically attuned world in our future, and will contribute what I can.
Developing/learning these various number systems "upgraded" my perception in a very tangible way. It feels like like taking a "red pill" with a very, slow onset. It has made my vision and hearing are clearer, my spacial and bodily awareness increased...

K@ - 3/4

[K@ !wLYVBQU9kU No.11310108]

>>11310104

K@ - 4/4

[Anonymous No.11310112]

[Anonymous No.11310139]

>>11310081
>>11310080
Ok yeah, I'm retarded, sorry. The open sets will be CIRCLES on the surface of the sphere.

[Anonymous No.11310144]

>>11310098
>>11310102
>>11310104
>>11310108
Anon I suggest you read an introductory maths textbook to learn how to write maths. Instead of dumping a bunch of information at once, it is much clearer if you build your theory up incrementally, defining each object (and how they interact) one at a time.

What you have written is completely incoherent to an outsider. Maybe you're onto a great idea but no one will understand you unless you slow down and elucidate your ideas.

[Anonymous No.11310146]

>>11310139
Balls are solid anon...

[K@ !wLYVBQU9kU No.11310209]

>>11310144
Thankyou for the feedback, and I apologize for the incoherence. I'll do just that.

Trying to translate the ideas into base-10 feels like trying to write out tonal Chinese in the alphabet, there's a lot that's lost in translation.

This was my first numeral set. "The Corregan Cube", can be read as either binary, quaternary or hexidecimal. Its fairly self explanatory.

K@ - 1/3

[K@ !wLYVBQU9kU No.11310212]

>>11310209

K@ - 2/3

[K@ !wLYVBQU9kU No.11310217]

>>11310212

[Anonymous No.11310219]

>>11300167
>maths

[Anonymous No.11310223]

>>11308298
An associative magma.

[Anonymous No.11310227]

>>11308347
You mean monoid.

[Anonymous No.11310379]

>>11307187
2. 24

Prime sums r 31+7+2

31 - 7 = 24


I’m happy with this one achievement desk XD

[Anonymous No.11310438]

>>11308447
>>11308447
You are right, matrix is not a function, formaly is the representation of the image of a finite function like this:
[eqn]M:[/eqn]
[eqn] \qquad I \times J \rightarrow \mathbb{F}[/eqn]
[eqn] \qquad (i,j) \mapsto M(i,j)=x[/eqn]
Where: [math] i \in I=\{1,2, \dots ,m\} [/math], [math]j\in J=\{1,2, \dots ,n\}[/math] and [math]x \in \mathbb{F}[/math], [math]\mathbb{F}[/math] is any field.

For example, abusing notation, let:

[eqn]A = \pmatrix{2 & 4 & 55 \\ 1 & 6 & 12 \\ 5 & 5 & 31 \\ 0 & 0 & 4} [/eqn]

Let see what this "function" A yields with different inputs:

[math]A(1,3)=55[/math]
[math]A(3,3)=31[/math]
[math]A(4,1)=0[/math]

So [math] A [/math] is just a function which inputs are ordered pairs. But then, we get rid of the function notation and simply use index notation:

[math]A_{13}=55[/math]
[math]A_{33}=31[/math]
[math]A_{41}=0[/math]

So whenever you think of matrix indexes, think of this "matrix", for the same example:

[eqn]A = \pmatrix{(1,1) & (1,2) & (1,3) \\ (2,1) & (2,2) & (2,3) \\ (3,1) & (3,2) & (3,3) \\ (4,1) & (4,2) & (4,3)} [/eqn]
This last "matrix" is really an array of ordered pairs, visual presentation of the domain of A. And what we really call matrix is that array of images of each element of the domain.

Fun fact: the matrix of ordered pairs can really be a matrix if we choose to represent complex numbers as ordered pairs, and their entries are all integers, they are called gaussian integers.

Fun fact 2: m and n are integer numbers, but we can have infinite entries for a matrix, in fact it were used by Werner Heinsenberg in his famous matrix mechanics, a formulation for quantum mechanics that have been surpassed by the fun anal formulation (although you can talk about a certain isomorphism here too).

Exercise for the reader: find how this "matrix as a function" notion is related with "matrix a linear transformation".

[Anonymous No.11310451]

>>11308447
You are right, matrix is not a function, formally is the representation of the image of a finite function like this:
[eqn]M:[/eqn][eqn] \qquad I \times J \rightarrow \mathbb{F}[/eqn][eqn] \qquad (i,j) \mapsto M(i,j)=x[/eqn]Where: [math] i \in I=\{1,2, \dots ,m\} [/math], [math]j\in J=\{1,2, \dots ,n\}[/math] and [math]x \in \mathbb{F}[/math], [math]\mathbb{F}[/math] is any field. For example, abusing notation, let:
[eqn]A = \pmatrix{2 & 4 & 55 \\ 1 & 6 & 12 \\ 5 & 5 & 31 \\ 0 & 0 & 4} [/eqn]Let see what this "function" A yields with different inputs:
[math]A(1,3)=55[/math]
[math]A(3,3)=31[/math]
[math]A(4,1)=0[/math]

So [math] A [/math] is just a function which inputs are ordered pairs, [math]m=4[/math] and [math]n=3[/math]. But then, we get rid of the function notation and simply use index notation:
[math]A_{13}=55[/math]
[math]A_{33}=31[/math]
[math]A_{41}=0[/math]
So whenever you think of matrix indexes, think of this "matrix", for the same example:[eqn]D(A) = \pmatrix{(1,1) & (1,2) & (1,3) \\ (2,1) & (2,2) & (2,3) \\ (3,1) & (3,2) & (3,3) \\ (4,1) & (4,2) & (4,3)} [/eqn]This last "matrix" is really an array of ordered pairs, visual representation of the domain of A. And what we unfortunately really call matrix is that array of images in the same spot as the corresponding element of the domain.

Fun fact: the matrix of ordered pairs can really be a matrix if we choose to represent complex numbers as ordered pairs, and their entries are all integers, they are called gaussian integers.

Fun fact 2: m and n are integer numbers, but we can have infinite entries for a matrix, in fact it were used by Werner Heinsenberg in his famous matrix mechanics, a formulation for quantum mechanics that have been surpassed by the fun anal formulation (although you can talk about a certain isomorphism here too).

Exercise for the reader: find how this "matrix as a function" notion is related with "matrix as a linear transformation" notion.

[Anonymous No.11310471]

>>11300167
Have a complementary anime girl

[Anonymous No.11310596]

>>11310451
3/10 formal matrix definition
what if I want to index my elements starting from 0 instead of 1? Would that not be a matrix anymore? You also want to have empty dimensions and those are not uniquely recoverable from the matrix (the function) itself so you'd want to save it together with matrix
So a matrix is an ordered triple (f,m,n) where m,n are natural numbers (could be equal to 0) and f is a function defined on I times J where |I| = m and |J| = n

[Anonymous No.11310617]

Not sure if this is the right thread for this but fuck it I'll try anyway.
I've got a few offers for PhDs in Math - Between a university/ city you would prefer to live in, and a project/ supervisor you would prefer to work with, which is more important?

[Anonymous No.11310722]

>>11310617
depends on the way you like to work and your talent level. If you think you can basically handle your phd by yourself with little input then I would go for the more prestigious uni/supervisor. Otherwise a supervisor who actually gives a shit about you is very nice, also for connections for postdoc positions etc.

[Anonymous No.11310727]

redpill me on homotopy theory

[Anonymous No.11310790]

>>11310139
Yes, you're right.
>>11310146
retard

[Anonymous No.11310852]

>>11310727
It's fun, but it's also very painful.

[Anonymous No.11311002]

>>11310727
Like. Yeah. You only care if they're so6rta similar. For example a circle and a fat circle. The same thing. Cos they similar.

Fuck Jews.

[Anonymous No.11311100]

why am i seeing a new general and no warning here?

[Anonymous No.11311108]

>>11311100
>why am i seeing a new general and no warning here?
Why would you need a warning?

[Anonymous No.11311111]

new thread?

[Anonymous No.11311112]

>>11311108
that's why: >>11311111
check the digits too. is a binary code basically.

[Anonymous No.11311117]

>two posts a minute apart
>next sequence number
god this board is slow
we need more IQ threads

[Anonymous No.11311144]

>>11311112
>that's why: >>11311111
We don't hand-hold brainlets in this neck of the woods.

[Anonymous No.11311184]

>>11311144
it is called courtesy. you are just a douche.
the catalog is scary and poisonous and best avoided. ideally you stay in your thread and then move on to the next one and never see the catalog. someone just should make sure to create new threads on time without bitching too much about that.

[Anonymous No.11311417]

>>11311184
>it is called courtesy. you are just a douche.
A courtesy for brainlets perhaps, but a non-necessity for anyone who's familiar with using an imageboard.

[Anonymous No.11311629]

>>11311417
no, retard, every heard of race conditions? another retard like yourself could create another thread and there would 3 or 4 duplicate threads. even the most degenerate generals post links for that reason.

[Anonymous No.11311864]

>>11311111

>>11310872
>>11310872
>>11310872

[Anonymous No.11311933]

>>11311864
ah! thanks anon. finally. i was waiting here hoping someone would post the link. i am coming