/sci/ - Science & Math

What's the worst mathematical Wikipedia page?

https://en.wikipedia.org/wiki/Analytization_trick

sup piggots

why exactly is a linear transformation defined with these two properties in mind (over real vector spaces):

[math]T(\mathbf{u}+\mathbf{v}) = T(\mathbf{u})+T(\mathbf{v})[/math]
[math]T(c\mathbf{u}) = cT(\mathbf{u})[/math]

>>9401930
How else would you define it?

>>9401930
>why exactly is a linear transformation defined with these two properties in mind (over real vector spaces):
That's how it's defined for all vector spaces, not just real ones.

The only things you can do in a vector space are add vectors or scale vectors, so linear transformations are defined to "preserve" that structure.

This structure-preserving property is very useful when considering functions between two algebraic structures of the same type:
https://en.wikipedia.org/wiki/Homomorphism

>>9401944
thanks

>>9401930
like >>9401944 said, whenever you have some kind of structure, you should investigate maps which preserve this structure in some sense

in this case, it follows that every linear map can be represented by a matrix and composition of maps becomes matrix multiplication. this makes linear maps extremely easy to analyze. linear maps also have very clear geometric meaning: they are rotations, reflections, shears and scalings (i.e. the things you can do at photoshop).

>>9402008
>every linear map can be represented by a matrix
Wrong.

>>9402008
>thinks you can describe translations by a linear map.

>>9402031
Where was that implied?

>>9402044
>(i.e. the things you can do at photoshop).

'bout to blow my FUCKING brains out . FUCK LINEAR ALGEBRA. FUCK IT. FUCK IT. FUUUUCCCCK IT ALLLLLLLLLLLLLLL

>>9402075
>LINEAR ALGEBRA
literally the easiest maths

>>9402075
Linear algebra is one of the simplest mathematical topics. Just because you can't handle taking an ordered basis doesn't mean you should insult La-chan.

>>9402078
i have no idea how to change coordinates.

>>9402075
Why are you even at college if you can't even understand linear algebra? Why waste your money like that?

>>9401530
To the anon here >>9401135 I answered your question here >>9402063.

>>9402080
if you have a basis [math]\{v_i\}_{i\leq n}[/math] and want to change it to a basis [math]\{w_k\}_{k\leq n}[/math], then in particular you can write any vector [math]w_k[/math] in terms of sums of the [math]v_i[/math]. But then you have a system [math]w_k = a_{1,k}v_1 + a_{2,k}v_2+...+a_{n,k}v_n[/math] for every[math]k[/math]. So you can write the matrix of the transformation in terms of the [math]a_{i,j}[/math], and this is your transformation matrix to change any coordinate in terms of the [math]v_i[/math] to coordinates in the basis [math]w_k[/math].

>>9402075
linear algebra depends on the teacher

you can have either an autist that wants all proofs or someone that plugs and chugs into systems of linear equations for u engi majors

pic is a problem on the final that fucked pretty much everyone so he dropped it rofl

Why is/are there no linear transform(s) to get the transpose?

>>9402139
>Why is/are there no linear transform(s) to get the transpose?
What do you mean? The transpose is a linear transformation.

>>9402143
Aporogees for poor Engrish..


Why is there no way to find B such that

[math]A*B = A^T[math]

>>9402143
Aporogees for poor Engrish..


Why is there no way to find B such that

[math]A*B = A^T[/math]

>>9402111
Really? Did he not cover basic matrix factorizations in class, then?

(Though he should have put some quantifier on "n". As, "for any positive integer n".)

>>9402111
Is diagonalizing a matrix considered hard on Amerifatland?

Given square matrices [math]A,B[/math] and an inverse [math]M[/math] for [math]1-AB[/math], show that there is an inverse for [math]1-BA[/math] expressed in terms of [math]A,B,M[/math].

I want to start learning more about discrete stuff / combinatorics. I have a strong background in differential geometry and functional analysis / PDE. Is it possible to use this knowledge to my advantage? Are there scenarios in combinatorics where methods / intuition from the aforementioned fields can be applied?

>>9402158
What have you tried?

>>9402161
I know how to solve it. This is a nice exercise for you guys. You can try convincing yourself that 1-AB is invertible if and only if 1-BA is invertible; that can be done abstractly without expressing the inverse for one in terms of A,B and the inverse for the other.

>>9402162
I'm not a "guy".

>>9402152

its more of wtf is the question asking

>>9402111
That's a very simple problem, once you know that every self-adjoint matrix is diagonalizable.

>>9402163
faggot mentally ill nigger bitch ass

>>9402158
>Given square matrices A,B and an inverse M for 1−AB, show that there is an inverse for 1−BA expressed in terms of A,B,M.
Please no homework in this thread

>>9402111
>pic is a problem on the final that fucked pretty much everyone so he dropped it rofl
Which school for brainlets do you go to?

>>9402158
1-BA trivially has inverse BA-1, no need for M.

>>9402170
This is apparently an interview question from Microsoft.

>>9402173
Multiplicative inverse is what's asked for, not additive.

>>9402169
>faggot mentally ill nigger bitch ass
Are you okay?

>>9402171
saint louis university in stl

>>9402160
The closest interactions with combinatorics those fields have is ergodic theory. Mainly the representation theory of discrete amenable groups has some nice connections with geometric group theory and ergodic theory both of which are connected to combinatorics. There is some more abstract work that is connected with combinatorics under the guise of operads, specifically through Stasheff polytopes.

>>9402169
How old are you?

>>9402146
pls

>>9402146
If * is matrix product then it definitely is possible sometimes, like when A is invertible. Can you be more specific? Maybe you mean *when* is there no way?

>>9402075
This is why brainlets should stay away from universities.

>>9401535

Thanks for the laugh

I've seen the light. After journeying to hell and back, and mustering every last IQ point I have, I now understand where I went wrong. Forgive me, linear algebra. Take me back into your embrace.

>>9402397

If you're that brainlet who was struggling with LA, check this playlist out for a nice intuition based introduction to LA.It's pretty good and not too long either.

https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

>>9401530
Are these the most challenging maths textbooks of all time?

>>9402433
Are those TAOCPs even worth the time?

>>9402158
Cayley-Hamilton.

>>9401535
Why are physishits so retarded?

>>9402083
>solving other people's homework for free
Fag.

>>9402433
lel gb2 >>>/g/

>>9402674
>Fag.
Why the homophobia?

>>9402433
No, IUT is

>>9401535
Good lord it's actually real

>>9402686
>homophobia
Fag.

>>9402158
For small real numbers a and b
1/(1-ba)=1+b(1+ab+(ab)^2+...)a
=1+bma
Be inspired therefrom.

>>9402163
I'm not your guy, buddy.

>>9401930
Everyone else who answered this is educated stupid; I'll tell you the real reason because this is a good question not left to undergrads who just memorized this 2 years ago or whatever. Linear transformations are defined this way because it is just the precise way of saying "Knowing how the basis vectors transform tells you how ALL vectors in your space transform."

>>9402031
>>thinks you can describe translations by a linear map.
you can tho

>>9402031
[eqn]\begin{bmatrix}a & b & r \\ c & d & s \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} ax+by+r \\ cx+dy+s \\ 1\end{bmatrix} [/eqn]

not that other guy but you can represent translations with matrix multiplication, this is called homogeneous coordinates where you look at translations as being rotations restricted to a surface.

>>9402925
No, you need an affine map. Brainlet.

>>9402158
[math](1-BA)B = B(1-AB)[/math], hence [math](1-BA)B(1-AB)^{-1} = B[/math] and [math](1-BA)B(1-AB)^{-1}A = BA[/math].
Finally, [math]1-BA + (1-BA)B(1-AB)^{-1}A = (1-AB)(1+B(1-AB)^{-1}A) = 1[/math].
It's easy to check that [math]1+B(1-AB)^{-1}A[/math] is also a left inverse.

>>9401930
So that it is a homomorphism for the operations of vector spaces.
Generally a homorphism is a map between algebraic structures A and B (f:A-->B) with which:
computing in A and then sending in B
is the same as
sending in B and computing in B

Why study homomorphisms?
Because with them you can study part of B from the point of view of A, and vice versa.
How?
Partition A in cells where in each cell you have elements that are sent to the same element of B. i.e. a1,a2 are in the same cell whenever f(a1)=f(a2).
It is possible define "new" operations on these cells: (Cell where a1 is) * (Cell where a2 is) = (Cell where a1*a2 is) and this operation is independent of which elements of the cell you picked.
The cells along with those operations form a structure which is the same (except in names) as the image of A under f ( f(A) ).
This is called "First Isomorphism theorem".

Does anyone have the wolframalpha android app?
Is there a point buying it or is it the same as using the browser version?

>>9402928
Still not technically a linear map, since you always need the last component to be 1. They're elements of the projective general linear group on R^n, which is the quotient group of GL(R^n+1) by scalar multiplication (isomorphic to R*).

>>9402998
Of course it is a linear map, but it is not a translation in all of R^3. Still, it restricts to a translation on the plane {z=1}

>>9402158
In the same vein, but easier: Let A and B be square matrices such that [math]A+B=AB[/math]. Prove that A and B commute

>>9402146
Oh...you mean a B that works for all A.

If you set A = I then you get B = I so obviously that won't work.

>>9402164
it's asking you to prove that n-th root of a matrix exists if the matrix is hermitian and positive semidefinite.

>>9402111
This is a REALLY simple problem. It's there to test if you know the real spectral theorem.
Your class is retarded / you professor did a terrible job at teaching.

>>9402990
no point, he has enough shekels already

>>9402998
Linear maps are defined on vector spaces, which {z=1} is not. It's the restriction of a linear map on an affine subset.

>>9402855
>No, IUT is
IUT is a series of papers, not a textbook.

>>9402920
>Linear transformations are defined this way because it is just the precise way of saying "Knowing how the basis vectors transform tells you how ALL vectors in your space transform."
But that's not true at all, linear transformations are still defined that way even for vector spaces that don't have a basis.

>>9403203
meant to reply to >>9403012

>>9403217
>vector spaces that don't have a basis
I sense a rain of pro-AC posts incoming.

>>9403217
>vector spaces that don't have a basis
In non-retarded circles "vector space" means "free module over a field". Perhaps you meant to say that they were defined the same way for all homomorphisms of modules?

>>9403217
>vector spaces that don't have a basis
autism or ignorance

>>9403347
>>9403323
he means that defining linear transformations in infinite dimension is independent of AC, clearly

>>9403354
>he
I'm not a "he".

>>9403350
>autism or ignorance
Speak for yourself.

>>9403356
shut the fuck up, faggot

>>9403347
>In non-retarded circles "vector space" means "free module over a field".
You meant "module over a field".

>>9403354
>independent of AC
Yes, "every free module over a field is free" is independent of AC. Your point?
>>9403367
No, I meant "free module over a field".

>>9403369
>No, I meant "free module over a field".
Then your statement is not true.

Find one (1) source that defines a vector spaces as such.

>>9403364
>faggot
Why the homophobia?

>>9403381
>Find one (1) source that defines a vector spaces as such.
The book I'm currently writing.

>>9403386
I'm not a "homophobe".

>>9403386
Because faggots are not human.

>>9403354
But with AC comes a basis, and so that wording is there to tell xį does not require choice to be axiomatically true. Your interpretation of žůr post is incorrect.

>>9403481
What are your preferred axioms?

>>9403490
Axiom of Equality: every human is to be treated the same way.
Axiom of Infinity: there is an infinite amount of genders.
Axiom of Racial Purity: only white people (at least 57%) are to be considered human.

>>9403490
Ps. I'm not a "you". Please refer to me as "thou" from now on.

>>9403501
The second two can be derived from the axiom of faggotry

>>9403529
Why the homophobia? You should try homotopia instead. Just imagine you were sitting on the lap of some nice guy explaining him Quillen's model categories work, and he would then reward you with an intense kiss. So much more fun!

i know i'm retarded but i can't for the life of me get this same answer for the lcm

>>9403573
>i know i'm retarded but i can't for the life of me get this same answer for the lcm
What do you get?

>>9403583
408/25pi

>>9403586
>>9403583
oh fuck me i'm so stupid i realized what i've done.
i blame the calculator interface it confused me. nevermind i'll delete my posts now

>>9403590
>>9403583
I don't understand. What's the point of studying LCD's in the real numbers? The real numbers have no non-trivial divisibility structure. Everything divides everything. What the fuck?

>>9403591
it's a trick question. The answer is [math]\forall\epsilon>0:\epsilon[/math]

>>9403597
Is it really? Because it looked like a high school problem. I just thought it was shitty math education teaching retarded bullshit as usual.

>log in
>no new yous
>log out

I'm not sure what this question wants exactly. Is it something like, e.g. for (1, 1)-tensors [eqn]
\mathbf{T} = \left\{\begin{align}
& \mathbf{x} \in V_n \mapsto (\mathbf{\omega} \in V_n^* \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V^*_n, \mathbb{R}) = V^{**}_n = V_n) \in \text{Hom}_\text{Vect}(V_n, V_n) \\
& \mathbf{\omega} \in V^*_n \mapsto (\mathbf{x} \in V_n \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V_n, \mathbb{R}) = V^{*}_n) \in \text{Hom}_\text{Vect}(V^*_n, V^*_n) \\
\end{align}
\right. \\
\therefore \mathbf{T} \in \text{Hom}_\text{Vect}(V_n, V_n) \cup \text{Hom}_\text{Vect}(V^*_n, V^*_n)
[/eqn] and then I do something like that for all the other (r, s)?

>>9403657
yes

>Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms.

What the fuck. How is this possible?

>>9403776
>What the fuck.
Do you need to swear?

>>9403787
yes. how can two lines that are parallel "meet", especially "at" """"""infinity"""""""""? that makes no sense.

is there an extension of linear algebra to "non" linear transformations?

>>9403801
>is there an extension of linear algebra to "non" linear transformations?
https://en.wikipedia.org/wiki/Linear_algebra#Generalizations_and_related_topics

>>9403801
Pretty much algebraic geometry.

What do I need to approach Langlands program?

>>9403838
algebraic number theory, algebraic geometry, functional analysis, representation theory

What is calculus and why do we need it?

>>9403852
see pic

>>9403846
>functional analysis
why?

>>9403941
>why?
Infinite dimensional representations of Lie groups come into play

I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it,
[math]\Bigg(\mathbf{R},+,\times, \leq, |\cdot|,\tau = \{ A\subset \mathbf{R}\hspace{0.1cm} | \hspace{0.1cm}\forall x \in A, \exists \epsilon > 0 ,\hspace{0.1cm} ]x-\epsilon,x+\epsilon[\hspace{0.1cm} \subset A \},\hspace{0.1cm} \displaystyle \bigcap_{\substack{\text{A} \hspace{0.1cm}\sigma-\text{algebra of}\hspace{0.1cm}\mathbf{R}\\
\tau \subset A}}A , \hspace{0.1cm}\mathscr{L}\Bigg) [/math] -analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital [math]\mathbf{R}[/math]-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There is really a calculus, and these people are using it, but it is just a part of the filed they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field.
Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis.
All the so called calculus problems are really problems of real analysis.

>>9403984
This is overcomplicating things.

Those are a finite amount of finite tools trying to separate the inseparable in various ways.

>>9404009
it's a meme you dip

>>9403789
If you stand on a railroad and look straight ahead, you will see that the rails "meet" a long distance away in the horizon. Projective geometry deals with this notion

>>9403801
Yes, simply consider all functions instead of just linear transformations.

>>9404397
i see.

>>9404405
Please refrain from posting black people here.

>>9404424

>>9403984
IMO calculus is much more about linear transformations than real analysis.

Equations like:

[eqn]\int^b_a f(x) g(x) dx + \int^b_a k(x, y) h(x) dx = 0[/eqn]

are basically similar to tensor equations like:

[eqn]A_{\alpha} B_{\alpha} + C_{\alpha\beta} D_{\alpha}[/eqn]

>>9402021
But it's always true for finitely dimensional spaces

Are functions of the form:

[eqn] \sum^{\infty}_{k = 0} c_k \frac{\Gamma(x + k - a)}{\Gamma(x - a)} [/eqn]

real analytic?

Is there a formula for converting between this form and a polynomial series?

>>9404437
The Schwartz kernel theorem also shows the same thing for integral transforms.

>>9404405
watch n j wildberger math history projective geometry very helpful and intuitive

>>9401930
invariance under action
>>9401943
stoopid
>>9401944
handwavy
>>9402008
finite-dimensional
>>9402920
estupido, this is one of many consequences
>>9403217
low iq

>>9404526
all me

>tfw really struggling with the theory of L^p spaces.

Should I just end it all? I get physically sick when doing relevant exercises and it's anything more advanced than some super simple Hölder application.

>>9404591
samefag

>>9404591
wew

I'm in highschool, which math topics should i focus on other than Calculus?
Geometry?
Discrete math?

>>9404805
Introduction to mathematical proofs and reasoning

>>9404805
What's discrete math?
I would say trigonometry or algebra in general. Trying linear algebra will put you ahead, it's really easy.

>>9404805
http://4chan-science.wikia.com/wiki/Mathematics#Proofs_and_Mathematical_Reasoning
http://4chan-science.wikia.com/wiki/Mathematics#Matrix_Algebra

>>9404806
>>9404814
You don't need proofs unless you're gonna major in math.

redpill me on topos

>>9404424

>>9404620
Just keep practicing, that stuff gets easy. Just remember Holder, Egorov, Jensen, convolutions, things like [math] |g| = |g|^{1/p} |g|^{1/q}[/math], and that if you want to show that a sequence goes to zero a.e., you can show that the integral of its infinite sum is finite. Among other things.

What are the simplest examples of [math]\infty[/math]-groupoids?

>>9404805
underage b&
Start with some linear algebra and basic proofs

>>9404814
>>9404806
Thanks!
>>9404809
Discrete math is topics like introductory Logic, Set Theory, Combinatorics and so on
>>9404818
I plan on majoring in math, but I'm not 100% sure, might go for compsci or physics
>>9404908
Isn't linear algebra very hard?

>>9404429
That's some nice point of view - actually, linear ODEs behave like systems of linear equations..
But dealing with integral and differential equations also relies heavily on topological (resp. "analytical") arguments for many finite-dimensional arguments don't work any more...

>>9404918
>Discrete "math"
It's not a field of math strictly speaking, much like "combinatorics". It's best to ignore it entirely and study logic and set theory in their own right.
>Isn't linear algebra very hard?
No. You'll have to learn it anyway since it's used everywhere even outside of mathematics.

>>9402978
>cell
What?

>>9404805
Linear Algebra for sure.

>>9405008
Is blocks the correct term? I don't remember. I am not anglo.

>>9405007
>study logic and set theory
Why?

>>9405206
'Coset' is the word you're looking for.

>>9405317
But, isn't coset a term for groups only?

>>9404866
A topos is the natural model for intuitionist higher-order logic.

>>9404898
The fundamental infinity-groupoid.

>>9405394
Vector spaces are groups under addition.

If you just have a set, then the word should be "partition".

>>9405426
Not exactly. The partition is the set of all equivalence classes
You can call them equivalence classes/classes/blocks. But cosets are a special case.

>>9402433
Nop, whole books just use linear algebra,few calculus and some easy combinatorics.

As textbook algebraic geometry hartshorne will be more hard book for students.

>>9403657
>[math]V_n^{**} = V_n[/math]
Only if [math]V_n[/math] is finite dimensional. Also replace [math]n[/math] by [math]x[/math] to denote tangent vector spaces [math]T_xM[/math] at the point [math]{\bf x}[/math], on which this [math](r,s)[/math]-tensor business makes sense.
>[math]{\bf e}_i[/math]
What space is this basis for? [math]T[/math] should act on the basis for [math]V_n[/math] and the basis for [math]V_n^*[/math] at the same time, not separately. You need to put what you've written down together.
>[math]{\bf T} \in \operatorname{End}(V_n)
\cup \operatorname{End}(V_n^*)[/math]
No. You need the exterior algebra.
>>9403776
>How is this possible?
Have you never looked at a photograph in your life before?
>>9404429
General Volterra equations of the first kind can't be cast into an ODE tho. One way to construct solutions is via the resolvent map, which exists depending on the topology of your function space.
>>9404439
Expand [math]\frac{1}{\Gamma}[/math] in its Laurent series and you'll see that at specific points [math]x \in \mathbb{R}[/math] it acquires infinitely many negative powers.
>Is there a formula for converting between this form and a polynomial series?
You mean formal Laurent series? Use the Cauchy integral formula.

>>9405599
>>[math]{\bf T} \in \operatorname{End}(V_n)
>\cup \operatorname{End}(V_n^*)[/math]
Meant [math]{\bf T} \in \operatorname{End}(V_n)\cup \operatorname{End}(V_n^*)[/math].

>>9405420
Every infinity groupoid is equivalent to a fundamental infinity groupoid.

>>9405426
True, but I was talking about a general algebraic structure.
Here:
https://en.wikipedia.org/wiki/Isomorphism_theorems#General

I'm studying quantum information theory using pic related. Anyone ever study this before? Not sure where to go. Would like to perhaps study more general C*Algebras as I have covered a lot of that topic.

I've also heard of quantum information having ties to geometry.

>>9405604
this picture is funny because it's true

>>9405604
You are comparing notation for abstract theory and notation for computation.

Topologists are as messy when it comes to computation (i.e. spectral sequences)

>>9405620
Proof?

>>9402928
but that's affine and not linear transformation

>>9405924
[math]\[\left| - \right|:\operatorname{Kan} \rightleftarrows \operatorname{Top} :\operatorname{Sing} \][/math] is a Quillen equivalence in the standard model structures of the two categories.

>>9405933
*
[math]\left| - \right|:\operatorname{Kan} \rightleftarrows \operatorname{Top} :\operatorname{Sing} [/math]

>>9403217
norman pls go

>>9403381
It's not untrue, it's just redundant since module over a field is always free

>>9405952
>it's just redundant
No, it's simply incorrect.

>>9405952
It's actually not a bad definition. It allows you to do linear algebra without globally assuming the axiom of choice. He's still being an ass about it though.

>>9405961
>It's actually not a bad definition. It allows you to do linear algebra without globally assuming the axiom of choice.
But he/she did assume the axiom of choice.

>>9405961
>It's actually not a bad definition.
It's bad because now it becomes extremely difficult to tell whether certain modules over a field are vector spaces or not.

There's countably many finite cardinalities (0,1,...).

Are there countable many infinite cardinalities? (ordered by power sets?)

>>9405982
There are countable many countable infinite cardinalities but uncountably many infinite cardinalities in general.
Uhh I mean
>infinite cardinality
No such thing exists xDdxXDd

>>9405988
>There are countable many countable infinite cardinalities
Isn't there only one countable infinity?

>>9405992
[math]\omega + 1,\omega + 2 ,\dots[/math]

>>9405996
>ω+1,ω+2
Do those not have the same cardinality?

>>9406004
Cardinals, ordinals, what's tha differenece???

>>9406004
they have

>>9405992
Finite sets are countable, little one.

>>9402075

>>9405977
Just don't use arbitrary modules over a field, that should be fine for you and your kind since you have already decided to lose generality by working over a field in the first place.

>>9405965
>But he/she did assume the axiom of choice.
I didn't though. You could even assume its negation and that would still be the correct definition of "vector space".

>>9405977
>it becomes extremely difficult to tell whether certain modules over a field are vector spaces or not

Yes, which is something you would need to prove explicitly in a constructive universe. This is the point of constructivism, it forces you to prove things in a constructive/more informative way.

>>9401530
So I just learned about [math] Li(x) [/math] from my number theory book. It's applications to number theory are fun, but what I want to know now is what new integrals can I find antiderivatives for using this beast?

How can I get into relation theory? Something better than Halmo's pls

>>9405599
H-how do you know so much of so many different fields of maths? Or am I just retarded?

>>9407594
Finishing college

>>9407594
Not him but that's pretty standard info for undergrads and first year grad students in math. Basic facts about manifolds and dual vector spaces are encountered at the undergrad and grad level, so is complex analysis, the only one that isn't standard is the ODE stuff, unless you're doing applied math

>>9407640
>him

How much maths should I know to stop hating myself for being retarded?

>>9407647
Jesus fucking Christ

>>9407729
>Jesus fucking Christ
Are you okay?

>>9407731
No. I want this meme to stop. We know you are just pretending to not be male. I bet you aren't even actually trans in real life.

>>9407739
>No. I want this meme to stop. We know you are just pretending to not be male. I bet you aren't even actually trans in real life.
But this isn't about me, it's about her.

>>9407741
>her

>>9407747
Yes?

>>9407739
>I bet you aren't even actually trans in real life.
He is pretty pathetic, so I wouldn't be surprised at him being "tr*ns".

>>9407753
>He

>>9407757
That's a bit of a bizarre question.

Here's a problem that I'm struggling with:
find absolutely integrable function f: R->R, such that f is zero outside the interval [-1, 1], and the convolution of f with itself is constant on interval [-1, 1].
Is there any such function except for zero?

>>9407715
No amount will suffice. It is not related to anything mathematical.

>>9407729
>Jesus fucking Christ
Why the blasphemy?

>>9407796
>No amount will suffice. It is not related to anything mathematical.
How about the amount of maths I need to know so that I don't fear people considering me a fucking retard?

>>9407827
That amount is obviously related to the population you surround yourself with. On the other hand, the fact that you are even asking a question like this makes me think you will most likely be permanently percieved as a retard.

>>9407827
>fucking
Why the vulgarity?

>>9407741
>her
When did you ask for my pronouns shitlord?

>>9407715
A finished course in Algebraic Geometry, then you're slightly less retarded.

IUT if you want the hate to stop.

>>9407849
>On the other hand, the fact that you are even asking a question like this makes me think you will most likely be permanently percieved as a retard.
How so?

>>9407904
Because the question was ill-formulated. The reason was given in my post. Btw look at this cat inside this sock. It is cute.

>>9407917
I don't understand why people will perceive me as a retard.

>>9407925
Teehehee~

>>9407768
Fourier transform

What would be a good math textbook for an of age highschooler taking AP Calc BC? The one I use for my class is very problem heavy- and while that's great and all, I feel like my understanding of the topic is pretty shit.

>>9407989
https://www.amazon.com/Methods-Mathematics-Calculus-Probability-Statistics/dp/0486439453
https://www.amazon.com/Elementary-Calculus-Infinitesimal-Approach-Mathematics/dp/0486484521/
https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mathematics/dp/0486428869/
https://www.amazon.com/Calculus-Lifesaver-Tools-Princeton-Guides/dp/0691130884/
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/

>>9408048
>amazon.com
>not libgen.io

>>9408058
But piracy is I L L E G A L!!!

>>9408062
>But piracy is I L L E G A L!!!
Which memecountry do you live in?

>>9408064
Oh, I live in the memiest of them all. In which there are no restrictions placed on libgen either. I was just helping that person justify linking books on sale insteadd of ones free for everyone to take.

>>9407989
>>9408048
And if you are a weeb

https://www.amazon.com/Manga-Guide-Calculus-Hiroyuki-Kojima/dp/1593271948

confidence is low lads

long story short: i'm 30 years old and going to college in a few weeks. my placement test is in about a week and depending on how i score i'll be placed in either pre-college(non-credit) courses or college level courses. there's a lot of pressure

the goal is to graduate with a STEM degree so knowing maths is a must. it's been 12 years since i've been in an academic setting and my math skills are as bad as you can imagine. i've been studying with khan academy for three weeks and i'm up to algebra(factoring quadratics, polynomials, nested fractions and the like).

to get to my point, i'm struggling, and i'm beginning to doubt i'm smart enough for STEM. getting every problem is really wearing me out. i'm learning but it's going very slowly. word problems are what i'm struggling with the most(write an equation to solve x word problem). i fucking hate word problems. i feel a lot dumber than i used to be, and the hopes of placing into college level courses are non-existent. i've accepted that i'll have to spend a semester learning the basics. i just hope that i'm smart enough to get into a proper physics/mathematics course by the fall semester.

i do enjoy doing math which is surprising. i thought i'd hate it. solving a hard problem is very satisfying. i'd like to make a career out of it; astrophysics or something along that vein, but i realize that's a long ways down the road. i'll probably be close to 40 by the time i graduate but i honestly don't care anymore.

anyway, i guess i'm posting here to see if anyone else is in a similar situation, and if being in a classroom setting with a proper teacher will make it easier to learn, and if my intellect will ever return

>>9408106
>tfw no manga guide to IUT

>>9407594
Read.

>>9407917
>Btw look at this cat inside this sock. It is cute.
Does me finding the cat cute require the axiom of choice, categories can get a little hairy like that.
>>9408123
Depending on the degree the workload can get pretty heavy, I'd say have a back up to a stem degree that you have confidence in getting.

>>9408244
No. It only requires a little spark of animal love.

>>9408244
>Does me finding the cat cute require the axiom of choice
No, the category of cats having enough projectives suffices.

The wiki suggests A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre in the computer science for intro to proofs, but it doesn't include number theory which is something that I'm interested in.

Is there another comparable book that touches on number theory?

>>9408255
So, what do we call the category of cats...[math]CAT[/math]?
>>9408403
>Is there another comparable book that touches on number theory?
Are you specifically interested in number theory as related to compsci? Knuth's art of programming has the relevant number theory

>>9408123
Wtf? What took you so long to go to college?

Sorry, I know that's not helpful. I don't know what to tell you, because you're not really being specific. Word problems are typically pretty easy to translate into equations, but I don't think I've had to do such problems since I was in elementary school or something.

Do you have a decent grasp of algebra? Trigonometry?

>>9408181
"no"

>>9408123
Unless you have a good plan, try my experimental study guide :3
1: Finish Khan Academy math subjects up to Algebra 2 and have done the tests
2. Pick up Serge Lang's "Basic Mathematics" and do everything except Trigonometry chapter. Now you will have clear and rigorous grasp on College Algebra and able to prove few things and learn bonus things like Binomial Theorem
3. Go back to Khan Academy and only do Trigonometry chapter. Because trig is fairly easy and at college level it's nothing but memorizing identities and formulas, you can learn more about trig later in calculus. ALSO, Khan Academy exercises are very intuitive so I recommend going back to Lang's book to do the trigonometry exercises there.
4. Now you are well prepaired to do Calculus. Depending on which letter you choose to major in STEM, you can choose a Calculus book on appropriet difficulty. So you can pick up a soft Calculus book for E like Stewart's Early Transcendentals or if you choose to go for a better major like M you can pick up Spivak's Calculus which is rigorous.
4.5. At this point you're doing well so you should do another book to help you through problem-solving and proving things to ease your mind during difficult exercises and heavy texts like Spivak's. I recommend "How to prove it" by Velleman and/or "How to Solve it" by Polya. Polya's is important if you need help solving things in general, while Velleman's is great foundation for later studies in university like Analysis and Linear Algebra.

>>9408841
5. At this point you should be very familiar with Calculus and easily get into university if you do entrance exam applications. Now you can go find your major, and do some headstart studies by checking the curriculum of your class. Or alternatively you could do things like:
Re-visit and skim through some other calculus books to get a better and clear grasp on things and be polished.
Try more rigorous books and topics like Linear Algebra, Number Theory and Multivariable Calculus.
Master problem-solving by doing things like archived Olympiad problems and some difficult ones on Brilliant.org
Whatever you choose to do is up to you and about how much you like mathematics or if you need to get better understanding of things. The point is that it's important to keep doing math and not take too long breaks.
6. Make your own study guide and plan ahead :3

>>9407981
Please elaborate, i thought about fourier transform but it didnt help.

>>9408923
Yeah actually after thinking about it, I realized that it didn't actually solve the problem. You could have deduced what you want under the stronger assumption that f*f is constant everywhere (because then the fourier transform of f is continuous, and its square is a multiple of the dirac mass at 0, hence zero)
I'm not sure what to do about this

>>9408403
The only "number theory" in a discrete math courses is Fermat's Little Theorem / Euler's Theorem and Euclid's Algorithm.

Just read one of the three intro books here:
http://4chan-science.wikia.com/wiki/Mathematics#Number_Theory

>>9408923
What is the Fourier transform of a function constant on [-1,1] and zero everywhere else?

>>9408923
It was actually more complicated than I thought, I'll have to keep thinking about it

>>9408979
It's not assumed to be zero outside of [-1,1]

>>9409013
Assume f(x) is zero inside [-1,1]. Then rect(x)*f(x) = 0 everywhere, so sinc(w) convolve F(w) = 0 everywhere, i.e. the integral of sinc(w) F(a-w) is 0 for all a. Perhaps differentiate the integral wrt a?

>>9405604
>pic
Nice OC. Saved.

What is a binary operation? Is it a function from the set AxB to C? Any other properties/definitions?

>>9409278
like when youdo operations on digits in base 2
0*0=0
0*1=0
1*0=0
1*1=1
AND

>>9409278
A binary operation [math] \rho [/math] on a set S is a subset of [math] S \cross S \cross S [/math] such that [math] (x,y, *) \in \rho \ \forall x,y \in S [/math]. Generalisations to binary operations on multiple (distinct) sets are left as an exercise to the reader.

>>9409292
Errata: [math] \rho \subseteq S \times S \times S [/math].

>>9409292
Oh, and * is uniquely associated to the pair (x,y).

>>9409309
Errata: [math] \forall x,y \in S \exists ! * \in S, \ (x,y,*) \in \rho [/math].

>>9401530

Where can I learn about topics related to p-adic 'density'? What I mean is, imagine you have a p-adic rational with infinitely many non-zero digits. You now multiply it by another p-adic rational. How many non-zero digits will the resulting number have?

E.g. in binary, 11111111111..... = -1 in the 2-adics. If you multiply this by 1111111111... you get 1. So you got a very 'sparse' number (1) from multiplying two very dense numbers. What I'm interested in is a theory for when the numbers aren't necessarily algebraic or rational. Will two 'dense' numbers generally speaking multiply to form a sparse number? Will a sparse number and a dense number tend to form another sparse number?

>>9401530
Why do topologists write [math]\mathbb{C}{\mathbb{P}^n}[/math] but geometers write [math]\mathbb{P}_\mathbb{C}^n[/math] ?

Why do topologists write [math]\mathbb{C}{\mathbb{P}^n}[/math] but geometers write [math]\mathbb{P}_\mathbb{C}^n[/math] ?

Man why math teacher is a literal retard.

>>9409809
Can you re-post it but not in Arabic this time?

>>9409413
Because it's called the [math]\mathbb{C}[/math]omplex [math]\mathbb{P}[/math]rojective space and not the other way around.

>>9409817
>xyeet doesn't know german

>>9409841
Most of the people currently living in Germany don't either.

>>9409845
So we must help our Germanic brethren by preserving their language. Thank you for providing a good reason to know it.

>>9409817
Oh sorry, it's just a find the maximum/minimum exercise with [math] { x }_{ i } \ge 0 [/math] and the formula to minimize being [math] 5{ x }_{ 1 }-{ x }_{ 2 }+{ x }_{ 3 }+2{ x }_{ 4 } [/math]. I have no idea why he is fucking around with the stuff in the brackets when he already has 1s everywhere. Seems like a dumbass thing to do but what do I know.

>>9409840
or l'espace [math]\mathbb P[/math]rojectif [math]\mathbb C[/math]omplexe

>>9409840
Or, you know, projective space over C

>>9409880
> l'espace [math]\mathbb P[/math]rojectif [math]\mathbb C[/math]omplexe
How about no.

>>9409413
Because [math] \mathbb{P}_\mathbb{C}^n = \operatorname{Proj} \mathbb{C}\left[ {{x_0},...,{x_n}} \right][/math] and [math] \mathbb{C}{\mathbb{P}^n} = \left( {{\mathbb{C}^{n + 1}}\backslash \left\{ 0 \right\}} \right)/ \sim [/math] .

Both spaces are interesting to a geometer, but only the latter to a topologist (because the Zariski topology is boring from a topological perspective).

>>9408685
Then enjoy being treated like a brainlet for the rest of your life.

>>9409331
Your usage of "dense" and "sparse" is non-standard. As to your particular question, I have no idea.

>>9401535
https://en.m.wikipedia.org/wiki/Analytization_trick#/issues
>This article does not cite any sources
Hmmm...

>>9408123
Most likely you're going to get placed at the level you need to be at in order to have a fair shot at progressing it's not really worth stressing too much about just keep chugging through problems. I wouldn't isolate myself to khan academy either sometimes it is bit unhelpful. Studying math on your own is hard compared to with a class with an experienced professor to guide you I wouldn't get too down about the frustration you might be feeling now.

People like to be all up their own ass on this board about math but at least for where you are skill is a byproduct of work. The more time you spend with a pencil to a page working problems or taking notes the more your skills will improve.

>>9411261
Samuel has a nice amerimutt face.

>>9408124
be the change you want to see in the world, anon

i think /sci/ is a hellhole but langlands is my guy
what are you all up to? i left for software last year and i have this silly hope that in my spare time i can toy with the conjectures i was too cowardly to chase while in academia
i'm a few years out of date on bsd and that's the project for january

>>9411659
>i have this silly hope that in my spare time i can toy with the conjectures i was too cowardly to chase while in academia

Go for it!

What kind of software are you doing?

>>9411667
"mobility" for an automaker. it's not as exciting as one might think but my only relevant experience was as a hobbyist, so i'm quite grateful

How do I visualize an Euclidean 4D space

why is this wrong?

>>9412193
Visualize an n-space and set n = 4.

>>9412228
Can you Latex it? Your disgusting handwriting is illegible.

>>9412228
ups basic mistake sorry for the brainletism

>>9412256
apart from the arctg (which is merely different) it's not bad

Is there a Stable Hurewicz Theorem?

i.e. Given an (n-1)-connected spectrum [math]E[/math] , is [math]{\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)[/math] an isomorphism for k<=n ?


I can't find a reference.

>>9405007
>look at how retarded I am

>>9412815
>>look at how retarded I am
Who are you quoting?

>>9412193
Are you assuming choice?

>>9412193
So you know how in microsoft paint or adobe photo shop you have the color scale, let's say from red to blue, well consider this, you know how to visualize 3d space, then at each point imagine being able to assign this red-blue color scale. In effect what you've created is a 4d space where the fourth dimension is represented by color (since you can associated a color to a numerical quantity based on it's placement on the scale, you could say the edges of the scale are assigned to plus and minus infinity so that you never actually reach the edges of the scale). From there it's easy to gain some intuition about the properties of objects in 4d space.

>>9413041
>you know how to visualize 3d space
I don't...

>>9405309

Pretty much all of modern mathematics uses sets as mathematical objects; its pretty useful to know their properties.

Logic is the formalism by which all proofs are written. Understanding logic makes prof writing/reading much easier.

>>9413130

Depends, formal logic can cease to be useful to most areas of mathematics. Once you go down the set theorist or logician path you will never come back.

>>9413130
>Pretty much all of modern mathematics uses sets as mathematical objects; its pretty useful to know their properties.
>Logic is the formalism by which all proofs are written. Understanding logic makes prof writing/reading much easier.
The post that was replying to said to study them "in their own right". Anything beyond the most elementary of properties of set theory and logic is irrelevant to most mathematics

>>9413041
Tried your method but still can't visualize a 3-sphere in my head

>>9412193
you don't, only through methods in R-3 can we approximate what manifolds might look like in higher dimensions.

>>9413318
>manifolds
Their existence is only a conjecture at this point.

>>9413252
That doof forgot to mention that a single point could have a line extending from red to blue, which still may give you some intuition about 4d, but is not so easy to visualize.

>>9413252
If you want, here's a playlist that tries to explain how to visualize higher dimensions through various methods, for the 3-sphere in particular they employ the hopf fibration. They form a nice little series (a couple of hours long but it's fun so who cares).
https://www.youtube.com/playlist?list=PLelIK3uylPMGj__g3PeO9yg1QTCiKhwgF

>>9413335
pls go away

>>9413579
What are you trying to say?

>>9413891
fuck you and fuck off

>>9413896
You are very rude.

>>9413896
>fuck
Why the vulgarity?

Calculus is shit desu

>>9408123
I'm doing an HNC in Electronics Engineering, hadn't done maths since GCSEs and was rubbish at it then, got a C and struggled and now my course is pretty much all maths.

It's a cunt but I'm getting through it with help. I found Khan Academy takes too long, I think once you get back into maths some of the stuff comes back to you.

You might be slow though, it takes me longer to memorise things and actually understand them. So far I've got the top grade on both my assignments so far though so I'm getting through it.

>>9413928
>>9413931
expect he's 100% right, so fuck off.

>>9414318
>expect he's 100% right, so fuck off.
Do you need to swear?

>>9414323
yes

>>9413986
agreed

>>9413986
why?

>tfw will NEVER be as smart as my proffs
AAAAAAAAAAAAAAAAAAAAAAAAAAAAA

>>9414407
new

Can someone give me a quick rundown on the current state of math ? I'm curious as to what are the important fields of mathematics today, and what problem they're trying to solve.

>>9415071
>Can someone give me a quick rundown on the current state of math ?
Too broad of a question, but if you look at https://arxiv.org/ a lot of research falls under the headings of:
Algebraic Geometry; Algebraic Topology; Analysis of PDEs; Category Theory; Classical Analysis and ODEs; Combinatorics; Commutative Algebra; Complex Variables; Differential Geometry; Dynamical Systems; Functional Analysis; General Mathematics; General Topology; Geometric Topology; Group Theory; History and Overview; Information Theory; K-Theory and Homology; Logic; Mathematical Physics; Metric Geometry; Number Theory; Numerical Analysis; Operator Algebras; Optimization and Control; Probability; Quantum Algebra; Representation Theory; Rings and Algebras; Spectral Theory; Statistics Theory; Symplectic Geometry

>>9415089
Thanks, do people just specialize themselves or is it possible to, for example, learn every branch of Algebra.

>>9415248
Depends on what you mean by learning. Whatever you do, you should finish your undergraduate with some knowledge in as many of the areas listed above as possible and, if you can, take a wide range of graduate classes.
The reason for this is that your interests might change and that you never know where a problem could take you.
That being said, people do specialize, but algebraists usually have (at least) some working knowledge of most other parts of algebra.

should proof of academic enrollment in as a math/physics/chemistry major be requirement for browsing /sci/

bump

why was hilbert a brainlet

killing this thread with no survivors

>>9414407
>>9414407
>>9414407
NEW
>>9414407
>>9414407
>>9414407