/sci/ - Science & Math

What is the absolute trace polynomial for GF(9) over GF(3)?

Isn't it P(x)=x^9+x^3+x?

Doesn't that mean that P(1) = 3 which isn't in GF(3)?

Threadly reminder to work with physicists.

Third for the real numbers are a meme

Put some work into this survey:

https://youtu.be/HrN7orXvu9k

>>10930547
I get the impression that's a really obvious lemma, but I can't make sense of it unless I have some restriction on the possible elements that a set can have.
But when I try to come up with some such restriction, it introduces the variable of the number of possible set elements and the formula goes through the window.

Lads, my uni's library has that nonlinear functional analysis set, but I think volume one is taken.
Do I go for it?

>>10930558
based soulless autist

How to prove this basic property of periodic real valued real functions?

>>10930558
I'm hiring.

>>10930944
The left hand side looks like the definition of periodicity, so what definition are you working with?

>>10930778
Probably not.

>>10931033
Physicist or mathematicans? What are you working on anyway? Measure theory?

what are some of your favorite lesser-known proofs for well known theorems, /mg/?

I spent all morning following and reconstructing cauchy's polygonal curve based proof for unique existence of solutions to ODEs. much better than the usual contraction mapping approach in my opinion. simultaneously constructive and theoretical.

>>10930558
I don't get it

>>10931123
Pic related is a nice argument (inb4 not a formal proof) for why
n choose 2 = 1 + 2 + 3 + ... (n-1)

There's a thread with some of those on SE.

Also, the first thing coming to mind when you ask the question that way, there's this good old meme proof
https://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes

Also in the spirit of >>10930709, I had uploaded this variant of the general diagonal argument which holds everything constructive as long as possible and generalizes the involved spaces:
https://youtu.be/rHsuesTdFLM
Take
>n : V -> V
>f : X -> V^X
be any two function
and let f_D : V^X be given by
>f_D(x) := n(f(x)(x))
Then
>∃(d:X). f(d) = f_D => ∃(d:X). n(v_d) = v_d, where v_d := f_D(d)
Special cases:
If n : V -> V has no fixed point, then f : X -> V^X is not surjective.
So in particular, with V={true, false} n = ¬, there's no injection into its power set
(which also holds in a Cartesian closed category)

Also, when you sit in the right Banach spaces, then the solution of
[math] x'(t) - x(t) = - f(x) [/math]
is
[math] x(t) = - \dfrac{ 1 }{ 1 - d/dt } f(t) = \sum_{k=0}^\infty f(t) [/math]

>>10931123
Pic related is a nice argument (inb4 not a formal proof) for why
(n choose 2) = 1 + 2 + 3 + ... + (n-1)
There's a thread with some of those on SE.

Also, the first thing coming to mind when you ask the question that way, there's this good old meme proof
https://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes

Also in the spirit of >>10930709, I had uploaded this variant of the general diagonal argument which holds everything constructive as long as possible and generalizes the involved spaces:
https://youtu.be/rHsuesTdFLM
Take
>n : V -> V
>f : X -> V^X
be any two function
and let f_D : V^X be given by
>f_D(x) := n(f(x)(x))
Then
>∃(d:X). f(d) = f_D => ∃(d:X). n(v_d) = v_d, where v_d := f_D(d)
Special cases:
If n : V -> V has no fixed point, then f : X -> V^X is not surjective.
So in particular, with V={true, false} n = ¬, there's no injection into its power set
(which also holds in a Cartesian closed category)

Also, when you sit in the right Banach spaces, then the solution of
[math] x'(t) - x(t) = - f(x) [/math]
is just
[math] x(t) = - \dfrac{ 1 }{ 1 - \frac {d} {dt} } f(t) = \sum_{k=0}^\infty f^{ (k) }(t) [/math]

E.g.

[math] x'(t) - x(t) = -t^2 [/math]

has

[math] x(t) = - \dfrac{ 1 }{ 1 - \frac {d} {dt} } t^2 = 2 + 2 t + t^2 [/math]

as solution:

[math] x'(t) = \frac{d}{dt}( 2 + 2 t + t^2 ) = 2 + 2t = x(t) - t^2 [/math]

Works for all polynomials.

----

I got interested in Markovs machine model today, is anybody into that?

>>10931194
that sure is one disgusting use of geometric series

>>10931209
Wouldn't actually think so, but I see how it's debatable. There's also variants where the same is done with integral operators then then
[math] \left( \int^t \right)^n t^m = \frac{(m+1)\cdot (m+2)\cdots (m+n)} t^{m+n} [/math]
Btw. I think I got a minus sign too much above, but whatever.

>>10931209
Wouldn't actually think so, but I see how it's debatable. There's also variants where the same is done with integral operators then then

[math] \left( \int^t \right)^n \, t^m = \frac{1}{\prod_{k=1}^n (m+k)} \, t^{m+n} [/math]

Btw. I think I got a minus sign too much above, but whatever.

>>10930944
nT = T+T+...+T
Induction and associativenes are your friends

>>10931252
Yeah I just typed it out when I saw I wrote left in >>10931102 but I meant right hand side.

Anyway, so
>P(0) and (P(m) => P(m+1)) => P(n)
with
>P(m) := f(x) = f(x+m*T)

Base case k=0 is true by reflexivity and induction step
f(x) = f(x+m*T) => f(x) = f(x+(m+1)*T)
is true via f(x+(m+1)*T) = f((x+m*T)+T) = f(x+m*T)

Reading Artin and having severe mental friction in understanding clearly every concepts.

>>10931425
Good you're reading it. I hope people do go on with it.

Anyway, are you literally copying the text from the book word for word in your notes?

>>10931425
Like what anon

>>10931437
Yes. This is my final form of reading level. Is it retareded?

>>10931469
Well I can't imagine you'd copy-write a 600 page book, that's the core issue I have with it.

>>10931445
When you read a lemma that a square matrix with a row of zeros or a column of zeroes does not have an Inverse, then it means it does not have a left inverse or a right inverse. Ok but can you mentally see this in split of a seconds and be satisfied? I can not. I will have to write down the formula and then look at it for at least 30 secs and then feel that yes it is true.

>>10931490
I copy wrote one book in Jan to Feb. It was 1000 pages. I am confident about the content.

>>10931496
lmfao which book?

Next I will copy write this big guy with solving all problems

>>10931425
You need know proof base Linear algebra for reading Artin.

>>10931496
And you felt like that helped you?

>>10931500
>>10931496
Game programming with Directx 12 by Frank D Luna

>Game programming with Directx 12 by Frank D Luna

>>10931504
Yes. Now I don’t need to go to index to find a solution to problem. I know where in book the exact problem is mentioned.

Btw i am copy writing Peter Norvig’s AI book too.

>>10931512
It was worth it. I understood the whole pipeline and remembered most apis to bootstrap a game engine with bullet physics support. My engine is lacking blender and unreal engine type modeling and transformation. Because of transformation i am going through all these Abstract algebra and linear algebra stuffs.

>>10931519
Take this
https://www.pbrt.org/

>>10931493
It's more evident once you have the geometrical volume interpretation of the determinant in mind. A zero row/column means that the image space of the matrix has a defect and it's clear that it can't be invertible on the image on the vectors mapping to zero. It comes soon and also that's that 3blue1brown video on determinants that may help.

>>10931514
Also, to come back to your previous question, yes now I think it's quite autistic.

>>10931524
Thanks.
Hmm I have Real time rendering by Tomas though, have not started.

>>10931123
The categorical proof that any continuous bijection between compact Hausdorff spaces is a homeomorphism. It's more roundabout than the usual proof, but it's still an interesting corollary of Beck's theorem.

>>10931534
Yes I know this(a fuzzy idea) concept of modifying the space by transformation, specially of vector spaces. Thanks to 3brown1blue video on Linear Transformation.
But why include a row or column of zero when it does not result in non zero entry? I mean if you are giving an example of a matrix why would anyone pick one with defect(From application point of view). Or you say that rows zeroes can be hiding in any row and reduction would reveal it and we cannot find the solution anymore?

>>10931524
Btw are you the author of this book? I wish I ever meet some author of good books.

>>10931524
Holy fuck Donald Knuth wrote that for this book? My god. Ordering now. Thank you very much. One day I shall finish this series.

>>10931552
No, No way.
>>10931554
Yes, book is amazing full modern Path Tracing in a single book, Autor wrote about New update for end year.

https://pharr.org/matt/blog/2019/01/23/over-one-tb-of-pbr.html

this nigga actually has autism not the fake attention seeking beta variety but high functioning obsessive machine consciousness lmao

>>10931567
Who me?
And yes I copy wrote Directx 12 book.

>>10931582
Yes you, but you have my full support though I think maybe learning to systematize better and possibly develop a more efficient method for taking notes would help you.

I'm trying to understand deformations of complex structures of CY-manifolds. One thing that's making my head hurt is that I do not understand how to prove the stability of the dimensions of the cohomology groups under (infenitesimal) such deformations, or find a counterexample. Any ideas for a """""toy""""""" model to get a feeling for that?

>tfw don't understand, the equation works
>tfw brainlet

idk where I am going wrong on this one, the equation checks out when I plug that bitch into the variable.

>>10931763
15 * sqrt(5) is not 15

>>10931763
Mathematica tells me the neatest integer solution is
y(x) = 2 * 6^x + 3

Do you think in the future they are gonna look back at us and think that we must have been morons for using axioms that lead to things like Tarskis’s paradox yet we accepted them?

>>10931971
nobody is accepting banach tarski paradox, learn what a measurable set is

why is the use of the capital pi notation so limited
can I use it for anything other than 1x2x3x4...?
if not, why not just use !

>>10931975
>nobody is accepting banach tarski
like everyone who accepts axiom of choice, also accepts banach tarski

>>10931987
and all it says it that we can't expect to reasonably measure volumes of extremely weird sets

>>10931998
that's not what banach tarski says

>>10931982
if you make the index of the product symbol have an order corresponding to the order of multiplication, then you can use it to denote any multiplication in a noncommutative group. I've seen it used for things like construction of an orthonormal basis (Householder orthogonal Decomposition) or a series of similarly transforms (eigenvalue algorithms for a matrix's Schur form).

>>10931987
That's a bit a of strong claim. I wonder if there's "natural" additional consistent axioms which, when adopted, disable you from proving the Banach-Tarski decomposition.
I doubt that's true because Banach-Tarski already works in R^3, but I don't know for sure.

Any tips on learning watered-down, practical ("engineering") probability and stats? As in what would allow me to reason about randomness, write a simulation or two, etc.

How do I write notes when reading math texts? I don't write notes and that negatively impacts my learning capabilities

Quick anon! Aliens are to destroy the world, but won't if you find a sequence of functions [math] f_n [/math] and [math] g_n [/math] which are both uniformly converging to [math] f [/math] and [math] g [/math] respectively but the product [math] f_n g_n [/math] doesn't!

The normed Gaussians parameterized by their variance make for an integral kernel that lets one represent the direct distribution. Is there a rational kernel which probably does the same job?

>>10932194
Learn measure theory. Calc-based probability is incomprehensible nonce.
>>10932396
Write down theorems and definitions, sketch geometric intuitions.
You could try to write out non-geometric intuition, but in my personal experience it does nothing.
>>10932404
On an arbitrary domain?
Set g_n=f_n=x+b_n, with the b_n's converging to some b, and the domain is the entire real line.

>>10932516
Oh no, now the aliens are wanting the result of the Ramsey number [math] R(6,6) [/math]. QUICK ANON!

>>10932542
137!

>>10932546
Just a sec, this could take a while.

Can someone recommend me a good modern topology book?

>>10932788
What's modern topology?

>>10932788
>>10932899
Higher Topos Theory

>>10933115
Absolute Mad Man

>>10933115
>Lurie
>readable

Why is the single text on derived algebraic geometry in libgen written by representation theorists? Is the subject
>that
bad?

>Lang is a meme
why do people say that?

>>10931987
But not as a paradox.
Because there really is no paradox, yes, you can cut a ball into non measurable sets and rearrange the pieces to get a measurable set of twice the size, but this really isn't so surprising, because you are talking about non-measurable sets.

>>10932019
Not exactly, but it literally only works because the decomposition is made into non-measurable sets, which is the resolution to the "paradox".

Just calculated that being an academic costs about $3 million dollars in nominal opportunity costs over a lifetime, vs industry.

>>10933180
Higher-Things abstract something until somebody found useful.

>>10931551
>From application point of view
In applications it usually means something went wrong, other considerations are of numerical nature, you might have a row which is very close to zero, numerically which can lead to very bad numerical results.

>Or you say that rows zeroes can be hiding in any row and reduction would reveal it and we cannot find the solution anymore?
Yes.

>>10933159
Is there any other book that covers modern topology?

>>10933180
I think you're thinking about it backwards.
Derived geometry wasn't developed by algebraic geometers who thought "I bet I can come up with an even more abstract version of what I already do---and then convince people to care".
It was developed by rep theorists and homotopy theorists who had lots of hand-wavy geometric arguments---but when pressed would have to admit that they weren't formal.
From this perspective you have a god-given "geometric object" to which you wish to apply geometric intuition and developing the machinery necessary to do so is an obvious step.

Why do some dislike the terse textbook style?
I personally love the lemma-theorem-proof-corollary style.

>>10933159
Lurie is actually very understandable, given the topics he writes about.

>>10933636
As an introduction it may not be the best. Especially if you tend to need some kind of motivation to retain what you're reading.

>>10933636
Lots of people do not experience autism the way that you do.

>>10933251
because his writing style is like a graduate student who doesn't understand when detail is not required. he always comes at things from an abstract perspective, even when it hurts his ability to convey a message.

>>10930547
are there any notable difference in the properties between a square and rectangular matrix? I guess in the context of solving a system of linear equation a square matrix would be n equations with n unknowns? And a rectangular one has more equations than unknown variables, or more unknowns than equations?
Is that correct?

pls no bully

>>10931554
Is this worth reading as a mathematician with an interest in programming? I've already started SICP.

>>10933999
checked

>>10934050
A square matrix *can* be invertible, in which case a system has an unique solution.
Otherwise it's basically the same thing.
Yes, a square matrix is n equations and n unknowns. Some of those might be linear combinations, tho.
Yes, if you count linear combinations of equations as equations.

Why can't the simplex method handle negatively value variables?

>>10933383
If you only care about the stable homotopy category, Higher Topos Theory is both too much and too little. Reading something like the Kervaire invariant 1 paper would be a better use of your time.

There is no mention of stable infinity categories in HTT. You have to go to Higher Algebra for that.

Guys what could I do with a math degree? I have no idea what I want to do but I was only ever good at math in school

>>10934242

tech industry

in terms of creativity(not a job etc), what can i do with lots of knowledge in graph theory? just software stuff?

>>10934284

uhhhhhhh

>finished undergraduate degree a few months ago
>taking two years off before graduate school
>working full time
>trying to study in my spare time
>barely motivated
>been reading novels/learning Chinese instead of doing math
>wondering if I should even bother with grad school at this rate
Any suggestions? Should I try a different book? An open problem?

>>10934599
>taking two years off before grad school

>>10934050
>I guess in the context of solving a system of linear equation a square matrix would be n equations with n unknowns? And a rectangular one has more equations than unknown variables, or more unknowns than equations?
Well, yes. But it is a lot more helpful to think about a matrix as a linear transformation, a non square matrix is this a mapping from a vector space of one dimension to a vector space of a different dimension, which among other things means that eigenvalues do not make sense.
Always keep in mind that the span of the columns of a matrix is the image of that matrix, which makes the behavior clear.

Also the notion of an inverse obviously doesn't make sense, but there are alternative approaches through pseudo inverses, just like Eigenvalues can be substituted with the somehow similar concept of singular values.

>>10933269
Well if you study a bit on the side at uni, you can get a new grad job as a quant and make $500k your first year
>>10934067
It's worth it if you're already sufficiently advanced in both CS and maths that you can benefit the most out of it
https://functionalcs.github.io/curriculum/

>>10930547
Could we have progressed this far if we never found out numbers?
The concept of 1+1=2 is so fundamental that it is applicable to remotest of universe.

>>10934599
U need to ignite the need of studying. Right now I am on Lol binge.

>>10934608
Literally see nothing wrong with this. You can always go back and get a masters to springboard yourself into a Ph.D. Not every ones life is a perfect linear path.

>>10934132
The simplex algorithm doesn't distinguish between the initial problem variables and slack variables. Each constraint is transformed from a.x>b to an equation a.x-y=b and a constraint y>0 where y is a slack variable. Thereafter, all variables are assumed non-negative.

You can get around the issue by solving for the initial variables in terms of slack variables, eliminating them from the problem and thus eliminating the implicit x>0 constraints. So long as the variables don't appear, they can't be chosen as entering variables which would result in a solution being incorrectly deemed optimal.

Is 'factoring' a local to global property?
That is, if I have ring homomorphisms [math]A\to B[/math] and [math]A\to C[/math] that factors locally, that is, for every prime [math]\mathfrak p\subset A[/math], [math]A_\mathfrak p\to B_\mathfrak p[/math] factors as [math]A_\mathfrak p\to C_\mathfrak p \to B_\mathfrak p[/math], can we factor [math]A\to C\to B[/math]?

>>10931123
There is a very nice proof of the existence of a Jordan-Chevalley decomposition for matrices (ie. an expression T = D+N, where D is semisimple and N nilpotent, with the condition that DN = ND) using Newton’s method

What book does /sci/ recommend for a brainlet with a degree in humanities who wants to learn HS-level math?

>>10934960
Yes, there is a factorization iff [math]\ker(A \to C) \subset \ker(A \to B)[/math], ie. if and only if [math]\dfrac{\ker(A \to B) + \ker(A\to C)}{\ker(A \to B)} = 0[/math].
This vanishing is a local-to-global property.

>>10934960
>>10934983
I guess a more general question that would encapsulate this would be, for two [math]R[/math]-modules [math]M,N[/math] such that there exist homomorphisms [math]M_\mathfrak{p}\to N_\mathfrak p[/math] for every prime [math]\mathfrak p\subset R[/math], can we construct a (unique?) global homomorphism [math]M\to N[/math].

I could imagine if [math]R[/math] has a unique prime which in particular trivializes [math]N[/math] and/or [math]M[/math], this could go very wrong.

>>10935011
or two algebras, rather

havent logged on to /mg/ in three months, what have I missed?

>>10934770
>Could we have progressed this far if we never found out numbers?
Greeks got quite far without them.

>>10931194
>Also, when you sit in the right Banach spaces, then the solution of (latex)
I remember experimenting with odes and finding the bottom formula, shere you treat the differential as a number. Any book/area to find more information about this?

>>10935107
Where*

there goes a rapcuck

>>10935107
Well... Banach space theory?
https://en.wikipedia.org/wiki/Neumann_series

>>10933999
Is this a common sentiment?

>>10934050
https://en.wikipedia.org/wiki/QR_decomposition

>>10934067
How could the answer be no?

>>10935016
i dont know if this is the right timing but the >>>/lit/ guy came and went
also
>logged in

>>10935217
what

>>10935107
It's called the Neumann series
https://en.wikipedia.org/wiki/Neumann_series..
It's used and abused in QFT, in particular to derive the Dyson equation [math]G^{-1} = G_0^{-1} + \Sigma[/math]

>>10935338
you will end up a rape victim after i'm done with you

>>10933999

> maths
> not caring about the details

what happens when you omit the details?

>>10935338
Whats QFT?
I remember abusing this method to solve equations like
$$ y^{(n)} - y*e^{t} =0 $$

>>10931263
>>10931252
Thanks, this property may seem obvious by geometry of horizontal translations but i knew it have to be proven. Anyway, what i understand is induction over natural numbers and the same but for integer numbers is required in what i want to proof.

Maybe i just would proof f(x)=f(x-mT), but then i would need to define also f(x)=f(x-T). I only have f(x)=f(x+t). I need negative integers as well but i see we are proving for n=0,1,...,k,k+1,...

How to complete this proof then? Thank you again.

>>10931263
>>10931252
Thanks, this property may seem obvious by geometry of horizontal translations but i knew it has to be proven. Anyway, what i understand is induction over natural numbers and the same but for integer numbers is required in what i want to prove.

Maybe i just would prove f(x)=f(x-mT), but then i would need to define also f(x)=f(x-T). I only have f(x)=f(x+t). I need negative integers as well but i see we are proving for n=0,1,...,k,k+1,...

How to complete this proof then? Thank you again.

>>10935437
Quantum Field Theory

>>10935445
>For all n in Z. P(n)
can be rewritten as
>P(0) and for all n in N. P(n) and P(-n)

>>10934907
Don't you think two years off is a bit too much especially if you're working full time and not studying ? I'd say taking one year off to do some research and get prepared to go into grad school my be a better idea.

What do you guys think about pic related ? It seems more reasonable than the other "math guides" you often see on /sci/.

Is this guy correct?

I'm a CSfaggot and I want to learn category theory.
I forgot pretty much everything I learned about calc and lin alg. I remember the concepts (most of them), but I forgot the theorems and if you ask me to do even a babby tier u substitution problem I'll have to google how to solve it.
Should I be able to learn the basics of category theory it anyway, or do I forget about it? Not interested in reviewing calculus and algebra.

>>10935500
Category theory is algebra.
Anyway, I suppose Adwodys book will do.
But if you're afraid of reading math, then it won't work.

>>10935217
>log in to /mg/
>...
>log off

>>10935505
Oh shit he's got online lectures too. Thanks anon

>>10935587
People also like Bartozs, although that guys just a programmer and he's got a book on getting thin too

>>10934978
read the sticky and the wiki

they have generously outlined dozens of texts for brainlets like yourself to make use of based on just how brainlet you happen to be desu

>>10935478
isomorphic, not equal

>>10935505
>not suggesting based Emily's "please like my homo type theory" book

>>10933999
his multivariable calculus book reads like a charm

>>10935505
>Adwody
>not Boirceux

>>10935691
Not sure if I know it, what do you mean

>>10935699
http://www.math.jhu.edu/~eriehl/context.pdf
Category Theory in Context, Emily Riehl

>>10935468
Relax, we're working on a wonderful guide right now in >>10935689

>>10935708
Ah well, yeah I know of this guy but it's not an intro afaik

Can anyone tell me what I'm missing here? I have a hard time relating mod p to mod p^2.
Pic related

>>10935738
Use the frobenius automorphism.

>>10935500
You should probably ask yourself why you want to learn it in the first place.
Category theory is useful because it gives a top-down perspective on a lot of different math. Of course this is only useful if you actually know any of that math and you clearly don't, so even though you could in theory read a basic category theory book you'd gain almost nothing from it.

Category theory in programming is probably around 98% meme, if that's where you heard of it. It's mostly just overcomplication for overcomplication's sake.

>>10935726
it's very much a balls-deep approach, posting it here for the meme.
although since I'm going through it rn - I can tell you that it's quite an interesting experience

>>10935468
Beware of anyone trying to sell you some one size fits all guide to learning mathematics.

Having said that, this looks to have some good choices in it, though maybe the applied math route looks kind of cruddy.

Hey, brainlet trying to understand how this works. What does it mean that the variables should be suitably constrained?

>>10935771
Ah, okay I'm all for balls deep and memes, so props

>>10935780
Not all Exponentiation rules over the reals carry over to the complex numbers - Wikipedia the complex never entry and look for the arithmetic/Exponentiation rules

>>10935747
I did, and so far I proved that a^78 is not congruent to 1 mod 13, but this is still information I have in mod 13.
I have no clue how to go to mod 169 from this.

>>10935776
Yes I know that you should not trust everything about such guides. However I liked this one since I was looking for a book that covers pretty much all the high school subjects like Precalculus by Axler in order to get back in shape, while learning new stuff with How to Prove It. This guide gives some good pointers imo.

>>10935850
Take the quotient map from Z_169 to Z_13, and remember that 13^2=169.

>>10935478
He is abusing the notion of equivalence relation

>>10935780
exponential rules need that x>0 & y>0 and somentimes hold when x=0 or y=0 (or both, remember or is inclusive disjunction and xor is exclusive disjunction)

>>10935780
For any complex number k (which includes real k), there are two solutions to z^2=k. √k denotes the "principal" square root of k, which is the one whose argument is in (-π/2,π/2], i.e. the one closest to the positive real line. The other square root is -√k with an argument in either (-π,-π/2] or (π/2,π].

Multiplying complex numbers adds their arguments. If you multiply two complex numbers with arguments in the range [0,π/2] the argument of the product doesn't necessarily lie in that range; it may be in (π/2,π]. So while the product of the principal square roots of two complex numbers is *one of* the square roots of their product, it isn't necessarily the principal square root; it may be the negation.

The square roots of a negative real are imaginary, i.e. they have arguments of -π/2 and π/2, so the one with argument π/2 is the principal root. So if a and b are both negative reals, √a√b = -√(ab), i.e. the principal square root of their product is the negation of the product of their principal square roots.

For positive reals, the principal square root is always a positive real, and the product of two positive reals is a positive real, so √a√b is always equal to √(ab).

What conditions should a periodic function on the n dimensional Torus satisfy for both its Fourier series and the series of its partial derivatives to converge to the function and its derivative? The paper I'm reading puts a weird conditions that I can't seem to justify by asking that the integral of the function be 0. I suppose that the function must also be at least C^2

>>10936418
Carleson's theorem for Fourier.
Being analytic for Taylor (which is circular, but life is like that sometimes).

>>10936460
>Carleson's theorem
Does it apply for more than 1 variable? If so, I only need my functions to have integrable weak derivatives?

>>10935396
you don't have to build up all of mathematics by digging it out of dirt. it is ok to explain at a high level first. If you give every detail and always start at first principals, you will prevent your mathematics from being meaningful to the greater mathematical community. It's fine to do this in advanced books, but if your text has "fundamentals" or "intro" in the title, you should explain and derive from a simpler perspective. In the context of Lang, I would say his "fundamentals of differential Geometry" is particularly bad at this.

>>10935327
there was some dude who would just post the following in response to almost every post
>For discussion of x, please refer to >>>/lit/

Was I supposed to take lin alg before calc 3? If not, I'm still unsure of what path of classes I should take.

>>10936713
>there was some dude who would just post the following in response to almost every post
I'm not a "dude".

>>10936767
My apologies.
>>10936742
One can usually take them in either order. Afterwards, one might take a differential equations course, or a course in abstract algebra, or in real analysis.

>>10930547
wrong when s=1

>>10930547
I'm at less than high school levels in math due to mental problems causing me to miss much of high school.
I think I'm stuck at an 8th grade level but not entirely sure
my attention span is pretty much broken but I can work through things if its more interactive or if I go really, really fucking slow

is there any hope for me and where should I start?
looking to get good enough to program video games or something, I dont have a solid reason besides I just like math

>>10937003
Khan academy? Videos tend to be relatively short and focused and I think they have interactive exercises now.

This is such god-tier textbook.
http://neuralnetworksanddeeplearning.com/

It is providing very deep insights on how neural networks work.

Does anyone know how to automatically number theorems, definitions, lemmas, etc. in org-mode when exporting to html?

>>10937399
Doesn't seem to be about science or math.

I'm reading a textbook about baby Field Theory, they introduce field extensions as vector spaces over the original fields.
Is my intuition right when I think that R is an infinite vector space over Q? If so, would an irrational real number be represented as the Cauchy series of rationals that converges towards it, or would it be represented by something else entirely?

>>10937654
Why would neural network be "not science', dude

>>10937658
It is.
No, not at all. Abstract away the ordering and topology on the reals and stricly consider the field algebraically.

>>10937658
I would say that it is not always a good idea to think about the form definition of the reals.

>Is my intuition right when I think that R is an infinite vector space over Q?
You can identify each rational number with an equivalence class of elements of "Q^infinity", which probably form a vector space.
But that seems like a bad way to think about the real number and really leads to no additional insight, except that a series is just something like an "infinite vector" which leads to the concepts of l^p spaces.

> If so, would an irrational real number be represented as the Cauchy series of rationals that converges towards it, or would it be represented by something else entirely?
If anything it is represented by an equivalency class of cauchy series whether it is rational or not is entirely irrelevant.
But this is not the only definition of the real numbers.

>>10937658
You're right about the infinite dimensional vector space thing, but not about the sequence thing. With axiom of choice, every vector space has an algebraic basis, but the one for R over Q is very very difficult to imagine. It is also uncountable (and algebraic, so one can take just finite sums).
This means you're saying that any real is a finite linear combination of elements from this basis with coefficients from Q, and also that no element in the basis may be represented as a finite linear combination of the others.

>why would neural networks not be science
Why would they be? Science involves a specific method which is used to generate and support models. Neural networks are merely a tool. Scientists may use neural networks for their work, just as they might use a pen. But the pen is not science.
The study of neural networks fits better under the name software engineering (because you are making the tool, not discovering it.)

>>10937674
>>10937702
>>10937704
So there's no explicit representation? Alright, thanks for clarifying.

Should I put "Bachelor of Mathematics" on my resume? At college 90% of my classes are maths related. But I'm not sure how the official title translates.

>>10937735
>so there's no explicit representation? alright.
I don't know that this is getting through to your head. It's not "alright." It's enormous, and it's horrific. This "object," the basis for the reals over the rationals, is literally unconstructable. There is no way to pin it down or fully describe it. Provably so.
Imagine that I just put a few lines of mathematical argument in front of you and the conclusion was that unicorns exist. But when you ask me, well, show me a unicorn, I tell you that it's literally impossible for me to ever show one to you. Even if we can spend all the time we want traveling to it, no matter what. I can prove to you that it exists but also that you will never see it, touch it, or experience it beyond my loose description.
These are the implications of what I said. Is everything still "alright?" Is this "mathematics?"

>>10937738
What is your degree called?

>>10937738
Look at your degree and put whatever the exact official degree title is on your resume. Degrees that are actually formally titled a "Bachelor of Mathematics" are very rare. The only school I know of that gives them out is Waterloo. Almost all places will give you either a BSc or a BA in Mathematics, which is what you should write.

>>10937744
It's a bit of a letdown, yeah. However the limitations of the field of mathematics is also makes it that much interesting, in my opinion.
This particular problem stresses the need of a broader vision of math, which is appealing to me. In this context, the pure algebraic approach is not sufficient to get a satisfying construction of the real numbers, you need some topology and some analysis.

>>10937746
It is not so clear what it's called in the English language. It translates literally to "graduated mathematician". I don't see why I wouldn't put "Bachelor of Mathematics" if maths is 90% of my coursework. Wouldn't it be worse to put "Bachelor of Science" when I don't know the first thing about chemistry, physics, biology, etc?

>>10937773
This must be something other people have had to figure out. Have you looked it up on other websites? Sounds like a pretty common concern.

>>10937767
Yes, of course. The reals are not an algebraic object. Nor are the complex numbers; indeed, the fundamental theorem of algebra is elegantly proved in dozens of ways through analytic techniques.
This is why algebra is for hacks, and is the children's playpen of mathematics.

>>10937783
Well I wouldn't say Algebra is for children as I'm not even a real mathematician, I study physics
I just like math

>>10937654
>>10937704
Neural networks are based on math.
It's an algorithm that minimizes a certain function.

>>10938006
Just like linear regression, or logistic regression is math.
Specifically, statistics.
Although, the theorerical side on neural networks isn't as expanded yet.

>inb4 stats aren't math

>>10937786
I don't really give a single fucking shit what you'd say, nasty physishit. I am saying that the study of algebra is the mathematical equivalent of being confined to the children's playpen, and I'm fucking right. Algebra is a worthless venture which seeks to bastardize perfectly intuitive geometric and analytic structures, and turn them into discretized trash.

>>10938006
Algorithms aren't math. And it does so stochastically, statistics isn't math.
>>10938020
Please, do your best to explain to me how statistics is math.
Statistics is a tool used by scientists. There are mathematically rigorous frameworks on which statistics is built. None of this implies that statistics is a part of mathematics.

Redpill me on model theory

>>10936418
plz help

>>10938120
What is mathematics to you?

>>10937399
Hey, I know I've been posting links in /mg/ a lot in the last months and start to feel bad about it, but in this case let me point implemented the example (i.e. the first half of the book) in one go here

https://youtu.be/z2aq21lMw40

>>10938120
>Algorithms aren't math. And it does so stochastically, statistics isn't math.
What's your goto definition of math, man? What do complexity theorist say about it?
There's huge load of theoretical work in the field as well. In fact, a lot of people complain that too many of the conferences are too theoretical.
Theorems like
https://en.wikipedia.org/wiki/Universal_approximation_theorem
are fairly straight measure theory'esque results.
There's also stochastic differential equation approaches for the motions through parameter spaces (with dimensions in the thousands) which try to set theoretical bounds on how easy it is to relearn a network from one task to another - it's not Engineers doing that, let's say.

>>10938165
>>10936418
>asking that the integral of the function be 0
Is that a condition on the spectrum, maybe? What does it mean if you express it in terms of the fourier coefficients?

How do you guys get rid of brain fog? I sometimes (e.g. today) have these days when I just can't seem to get anything inside my head, but yesterday I was literally unstoppable.

Mathlet here, is there a classification for topological spaces/rings/fields/vector spaces as there is one for finites simple groups?

>>10930547
Test

>>10938267
There is stuff like "solid rings" etc, and these have been classified. Look up for example Bousfield-Kan.

>>10938267
bre, did you even try to google "classification of finite simple rings" or something along those lines, before asking? (rhetorical question)

>>10938267
>topological spaces
For one dimension and two dimensions, yes. Not sure about the rest, since I'm not a topologist. Maybe 3 and 5, definitely not for 4.
>rings
There aren't that many finite rings.
>fields
There aren't that many in general.
>vector spaces
There are extremely few of those.

This problem doesn't make sense to me.

let [math]\psi(L)\rightarrow K[/math] be an onto homomorphism, let I be an ideal of L, and let J be an ideal of K.
Show that [math]I_\psi[/math] is an ideal in K, and [math]J_\psi^{-1}[/math] = {a | a [math]\in [/math] L, [math]\psi (a) \in J [/math] is an ideal of L.
(the inverse is on psi in J-subpsi inverse, I don't jow how to apply a superscript onto the subscript).

I don't understand why this MUST be an ideal. Why couldn't I send everything in I to the upper half of K, where it would be an ideal filter /dual ideal instead of an ideal? or split the homomorphism in half so it sends half of I to the upper part of K and half of I to the lower part. so long as the rest of L is mapped to every element in K? Unless by "ideal" he is including filters/dual ideals as well? but even then, it doesn't necessarily HAVE to be an ideal in K, does it?

>>10938299
This is about lattices btw

Anyone has a good math chart? Like a complete guide from 0 to advanced maths?

>>10938328
Yeah.
Here's the true champion's route.
http://libgen.is/search.php?&req=math&phrase=1&view=simple&column=def&sort=year&sortmode=ASC&page=7
Start with Sur l'integration des equations aux derivees part. du 1er ordre

>>10938261
Then why did you stop?

>>10932155
Axiom of Determinancy

>>10938395
I didn't. Yesterday turned into today.

>>10938267
Classification of finite fields is trivial and any undergrad algebra course should mention it. Same for vector spaces.

Artin-Wedderburn theorem implies that all finite simple rings are matrix rings over some finite field.

Classification of topological spaces is pretty much impossible unless you restrict your attention to very specific kinds of spaces.

>>10938187
Mathematics is the study of formal axiomatic structures for the sake of understanding the structures themselves. Algorithms involve invention, which is not mathematics. Statistics is a scientific tool. Tools are not mathematics.
>>10938206
See above. Complexity theory is the study of algorithms, not the creation of them, so complexity theory is math.
There are plenty of areas in math which use tools from statistics.

>>10938509
Mathematics is a subclass of computer science/computation.

>>10938516

>>10938521
It literally is. A mathematical proof is a construction of a program, where the type in the program is a logical connective in the math. They're the same thing.
All of mathematics is forming tautologies off some branch of the Heyting lattice, and a mathematical proof is a computation of the tautology.
Read this basically: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
Everything in mathematics boils down to a computation, thus mathematics is a subclass of computation. The study of computation is the study of the foundations of mathematics, it's limits, etc.

>>10938430
Thanks, but I meant consistent with choice, replying to that guys claim the claim that assuming choice would always imply Tarski.

>>10938509
But wouldn't you say the range of possible algorithms given some model of computation is fixed and thus also up for your notion of discovery. Is finding prove not math - it's very closely tied to creating algorithms? And relatedly, that those tools mentioned have mathematical properties to be studied (and, with them, e.g. optimized).

Not replying really to you (as you seem to be comfortable with your definition), but for general interest, there's a cute Wikipedia page with various mathematicans definition of mathematics
https://en.wikipedia.org/wiki/Definitions_of_mathematics

>>10938532
I don't think that extreme is capturing the wealth of mathematics either.
While chains of argument amount to program execution, the descriptive buildup - the mathematical practice e.g. when classifying structures - isn't in itself the algorithm that confirms the claims validity.

>>10938438
>it just happened to me
not going to make it
>>10938532
this is lunatic

>>10937744
If a proof is not constructive it is not a proof at all, and isn't mathematics.
The so called uncomputable reals don't exist, and the reals are isomorphic to the natural numbers.

>>10938579
It's infinite so it in no way dilutes the beauty and wealth of math. In my eyes, it actually proves that mathematics is discovered and not invented, as it rigorously and empirically sets mathematics on a real foundation.
Mathematicians, then, are "travelers" who go into this lattice and discover the tautologies within it.
>>10938586
>this is lunatic
How?

>>10938587
I don't think you can do much better than embedding the nats as a semiring into the reals, so I guess you mean just in bijection? I think even in the most restrictive schools, the reals and the naturals are just in subcountable relation to each other.
Don't fight boys.

Coming back on topic - what's a nice book on order or lattice theory. Let's start with something not all too dry.

>>10938600
>>10938600
Gratzer.
There's an anon who's been reading it and asking questions about lattice theory every now and then.

>>10938608
I am that anon, and I still haven't figured out the solution to >>10938299

>>10938611
It's really easy, I'd feel bad just giving you the answer.
If you absolutely need a tip: since the homomorphism preserves orders, the preimage map also does.

>>10938608
Cool thanks. Seems even to be google-able here
http://www.xituju.com/download/file/20170704/20170704153281618161.pdf

I think it's even at my local library.*
That's something I ever feel I should need better when getting across filters in stochastic processes or maybe some forcing tricks (or the adjunctions Galois theory?), but I never feel like it. I even like combinatorics more. Does somebody who really like it give a pitch?

*Speaking of which, is the guy who proposed Artin for reading it on /sci/ still alive. It died faster than I predicted.

>I don't jow how to apply a superscript onto the subscript).
a_{ b_c^d }
[math] a_{ b_c^d } [/math]

I can't answer the question, looks like there might be a pretty butterfly-like picture associated with that thing. Can you take the inverse image and mutlply stuff to it to see that it behaves as an ideal back on the pre-image? What's the theorems mentioned close to where this exercise is from?

>>10938619
Oh shit thanks, wow that's embarrassing

>>10938637
Actually, short mistake.
It doesn't preserve order, it just "doesn't violate" order.
So if f(a)>=f(b), then it can't be that b>a.

>>10938608
>>10938629
Damn, that book's goal a shitton of exercises.
I also observe that he freely globally wrote Boolean in lower case letters, even in paper references that don't do so.

I've picked up a book called Quantum Field Theory for Economics and Finance, but it seems to assume I'm a physicist who knows tensor fuckery with [math]\partial _{\mu}[/math] and [math]\partial ^{\mu}[/math] and I genuinely don't have the absolute foggiest what the second one stands for and why they keep moving indexes up and down.
Does anyone have some sort of summary for physics notation?

>>10939052
https://en.wikipedia.org/wiki/Musical_isomorphism

>>10938509
>for the sake of understanding the structures themselves
lol

>Algorithms involve invention, which is not mathematics.
>invention
invention, discovery, meaningless words

>Statistics is a scientific tool
so is math

>Tools are not mathematics.
meaning mathematics can't be useful?

>>10938509
>Algorithms involve invention
Algorithms exist in the Platonic realm waiting to be discovered.

>>10939052
https://en.wikipedia.org/wiki/Raising_and_lowering_indices

>>10938509
>Algorithms involve invention
Not more then the rest of mathematics, the study of algorithms generally happens in the context of a formally defined axiomatic system.
"Inventing" an algorithm doesn't seem less mathematical then "inventing" ways to solve DEs and algorithms are usually considered in the way you demand.
This even goes as far as that algorithms are the result of the study of algorithms when searching for an "optimum".

>Statistics is a scientific tool.
So is linear algebra, or analysis or stochastics as any engineer, chemist or physicist will tell you.
I do not see the relevancy, you can consider statistics purely axiomatically, or you can use it as a tool.
And while one of them might not be considered mathematical, the other one definitely is.

>>10939084
The other responses to my post merit some level of respect and quietude on my part. They are valid philosophical worldviews, whether or not I agree with them.
Your post, on the other hand, is merely linguistic babble. I don't get what you find so funny about my assertions? And you seem to make all these baseless claims.
I'll answer one of them. No, mathematics cannot be useful. Mathematics may be used, but anything which is produced specifically to be used is not mathematics. For this reason, pure probability theory is mathematics while statistics is not. Similarly, numerical analysis is on a borderline - those who study the theoretical error bounds of certain approximation processes do mathematics while those who implement approximation processes do not.
Sure, it is possible to do mathematics with neural networks if you're solving problems related to the theoretical efficacy of neural networks (i.e. convex optimization, etc) but you are no longer doing mathematics the instant you begin to want to make a neural network to do something yourself.

>>10939109
ALGORITHMS AREN'T FUCKING PLATONIC YOU FUCKING MORON!!!! DO NOT FUCKING IMPLY THAT SHITTY LISTS OF LE EPIC INSTRUCTIONS ARE ANYWHERE NEAR THE ETERNALLY ELEGANT TENETS OF MATHEMATICAL TRUTH! PROOFS ARE NOT IN THE PLATONIC REALM, AND SO NEITHER ARE ALGORITHMS. PROOFS ARE OUR DESPICABLE, HORRIFIC COPING MECHANISM WITH THE SHINING LIGHT OF MATHEMATICAL THEOREMS.
Look at any accomplished mathematician - all have experienced a number of instances of "divine intervention" during which a result merely appears before them and the path to it is clear. There is no "process of proof" here, this is the proper way to do mathematics, to stumble into a perfect result unwittingly.
We are limited by our blindness to the platonic realm, and thus must explore it with our hands and with our feet. In the same way that Wiles stumbled about a pitch black house for years and years, turning on a light in each room as until he proved Fermat.
An algorithm CANNOT be discovered in that way.

>>10938113
Autism
Algebra is fun to learn and useful

>>10939610
What about Eratosthene's shieve

>>10939679
why in the fuck would eratosthene's sieve be platonic in any way
it's so inelegant it isn't even funny

>>10939692
It just works

>>10939674
>Autism
Incorrect, and irrelevant.
>Algebra is fun to learn
In a similar way to how building blocks are fun to stack, yes. Fun for an infant on the spectrum of mathematical maturity.
>Useful
Mathematics is not "useful." Mathematics should not exist to be "useful." If the best you can do is describe Algebra as "useful" then you clearly do not know very much about mathematics.
Even I would not deign to condemn algebra as "useful."

>>10939702
The real beauty of the art of mathematics, to me at least, is how the abstract relates to the concrete.

>>10930547
I swore I'm going to read this and solve every exercise
>over 900 pages

how long do you think it will take me?

>>10931506
Wait, was that book good or not?

>>10939889
The "economic" edition (softcover, lower characters, and shipped internationally rather than just for America) has a little over 500 pages.

I don't think it's a good textbook, though. There is an obsession with formality for formality's sake ; in many cases, the author states theorems in a context where they aren't really useful or enlightening, and does not explain their concrete applications.
Reading is a challenge (much like Rudin and advanced textbooks in general, a good chunk of the proofs are omitted and you are to complete them, which is a good exercise), the exercises are hard, and you need some familiarity with other stuff.
I'd say 6 months to 1 year, depending on your motivation.

>>10939065
>>10939177
Thanks, I think I've got it.

Do points D and B have a general name? Like how a vector has an initial and terminal point.

>>10939889
Lang is a meme.

>>10939610
>PROOFS ARE NOT IN THE PLATONIC REALM, AND SO NEITHER ARE ALGORITHMS.
This is a non-sequitur.

>>10939973
>Do points D and B have a general name?
Corners.

>>10939702
>Even I would not deign to condemn algebra as "useful."
Mathematicians use "we", not "I".

>>10938299
is this about the big disjunct V property and.or the inverse function is well behaved for ideals?

>>10939702

>>10938113
You are either trolling (hopefully), or a dumbfuck. Saying that algebra or analysis is worthless is like deciding that
certain tenses, or words from the second half of the dictionary are worthless in English.
Sure, you can probably write a lot of stuff without them, but there may come a time when it is decidedly the best way to phrase something and you won’t get around it unless you resort to writing very clunky things.
Only someone with an extremely limited understanding of math would write something like this.

Is there anything like the minimal superpermutation problem but with the change that the included permutations don't need to have their elements be contiguous, just in order?
As an example, the minimal superpermutation of 1,2,3 is 1,2,3,1,2,1,3,2,1 but you can also extract all the permutations from the shorter 1,2,3,2,1,2,3 if you skip over elements.

>>10939702
The reals have the same cardinality as the integers.

>>10939610
But an algorithm is nothing but a Theorem?

Take Banach's fixed point Theorem, it is the claim that under a given set of circumstances a certain series will converge. That is all an algorithm is, a theorem which allows explicit calculation.

>>10940224
>The reals
Defined how?

>>10939924
thanks for detailed answer. I'll do it anyway. do you recommend some other book?

>>10940291
The definable reals (the only computable ones) are the same in cardinality to N. The so called "uncomputable reals" don't exist - an existence proof that is not constructive isn't real and isn't valid.
If a mathematical object is not computable in finite description, then it is not real and doesn't exist in the platonic realm nor in the material universe (a nonterminating output of finite description is valid, so the algebraic reals are valid).
Cantors diagonalization is wrong because it involves doing a computation on an infinite input which is invalid a priori. The reals as equivalence classes of rationals are countable, and the reals as a union of equivalence classes of countable sets is countable. So the reals are countable.
No matter how much it pisses you off, you will NEVER GET beyond the fact that mathematics is computation, and if something is not computable than it is not mathematical. The limits of computational inscribe the limits of mathematics. Computation is a superclass over mathematics.

>>10940306
What do you think of Weihrauchs approach to computable analysis. There's also a nice looking book.

Also, I don't think you can decide upfront if a defined real is computable, so it doesn't look like you can have a total map from N into your reals. That seems like an issue for saying they are the same cardinality.

Btw., since you formulate your answer in terms of cardinalities yourself - do you take any issue with the classical set theory in which other set relations than finite, countable/denumerable/subcountable are formulated in?

>>10940119
Oh, I'll gladly use algebra when I have to. I mean good lord how else do you expect me to prove Brouwer's fixed point, or to manipulate solutions to quasilinear parabolic PDE? That's all algebra is, a tool. Have fun digging up my toolkit.
>>10940059
Mega cope.

>>10940281
An algorithm is the set of instructions satisfying a theorem, saying, "this is how you do this"
A theorem would be "this can be calculated in this time or with this precision"
An algorithm exhibits the truth of the theorem, in much the same way a proof does.

>>10940306
I can pretty easily define the reals by taking the set of functions from N to {0,1,2,...,9}, and applying some very simple restrictions and a very simple equivalence relation. I don't see how the set is in any way not constructable, even if individual elements may not be.

>>10939990
It is not. An algorithm is an exhibition of the truth of a mathematical statement (x can be computed in n time with e precision). It is the same as a proof. One could translate any proof into an algorithm or vice versa merely by changing the language.
Proofs and algorithms are the same in that they are the manner in which we verify mathematically true statements without being able to see them face to face. They are the basis for a model of mathematical truth (I mean model in the scientific way, not the logician's one).

>>10939788
And many would disagree with you about that. I would say that mathematics is beautiful in that it allows us an escape from the concrete, and better yet, to study the concrete as though it were abstract, doing away with its materiality.

I'm having severe imposter syndrome, especially when it comes to 'understanding' topics, which is when I'd say I'm able to experiment with my head and connect them with other areas. I've only just finished my second year of undergraduate study in mathematics, but I feel as if I constantly need to go back to the fundamentals, to see how everything is derived when it isn't immediately obvious. It becomes detrimental because I end up spending a lot of time revisiting said elementary concepts. What do I do? I want to be confident in my abilities and I want to be able to think about the topics I've studied in a recreational manner but this lack of confidence and understanding hinders me from doing so. Any advice would be appreciated.

>>10940441
You're still in undergrad. What sorts of things are you doing that are so far off from fundamentals?
If it helps, at different points during the semester look back through your notes for a course and select some of the most important facts you've proved to just take on faith. I probably couldn't tell you how to prove Arzela Ascoli now, but I sure as hell think it's a neat way to prove certain sets of functions are compact. Going back and reframing the "big things" in your head as though they're axioms is how you build several layers of mathematical abstraction in your learning, which is not really something anyone is expecting you to do yet, but it could be helpful.
When I say layers of abstraction I mean like, you have a basis of knowledge in some basic logic and set theory, and then you have a layer built upon this of very basic structures like say vector spaces and groups and some of their most basic results, and then there's a layer above that with more complicated objects and some of their basic results. It's not too often when you'll have to pull a result from one layer and use it in another, so you can just stash it away for a bit and if you think "hmm, why was this true again" in the top layer you can go back a layer down to prove it. But you won't have to go to the start.
Of course all of this takes a long time to hone and build and the majority of people after their 2nd year studying math have very very little knowledge. If you have enough to where it's an ordeal to go back to the basics ever, then you definitely shouldn't worry about your abililies. It happens to everyone.

>>10940306
>The definable reals (the only computable ones) are the same in cardinality to N.
What analytic/algebraic properties do they have?

>>10940306
>The definable reals (the only computable ones) are the same in cardinality to N.
Not necessarily true - the statement that all reals are definable is consistent with ZFC, so there are countable models of ZFC in which there are uncountably many definables. See https://arxiv.org/pdf/1105.4597v2.pdf
>The so called "uncomputable reals" don't exist - an existence proof that is not constructive isn't real and isn't valid.
I guess proof by contradiction is invalid then. Might wanna rework basic logic.
>The reals as equivalence classes of rationals are countable, and the reals as a union of equivalence classes of countable sets is countable.
This is just false.

>>10940417
Again, look at Banach's fixed point theorem.
On one hand it is a theorem, undeniably so, on the other hand it says that a certain series will converge against something.

An algorithm is just a theorem which states that given a certain set of operations applied in a certain way gives a certain result.

>it is not a low level language anymore.
No, it simply doesn't.
Should I go on with examples?
The Euler method for a first order initial value problem is the mathematical theorem that for a step size nearing zero the Euler method will approximate the unique solution to the ODE arbitrarily well.
In no way is this a proof or "like" a proof, in fact the proof is entirely absent.

>>10940436
But an algorithm needs to be proven?
I could claim that you get the greatest common denominator if two numbers by taking the square root of the product and rounding to the nearest integer.
This obviously is an algorithm, but the only interesting thing about an algorithm is the theorem which says what it computes.

An algorithm is fundamentally different then a proof and I really fail to see any analogy.

>>10935217
>logged on

test $1+1 = 2$
\[ \int fdx \]

>>10940550
how does one gets to write with LaTeX on /sci/ ?

>>10940552
[,math,] \frac{ a } { b_c } [,/math,]

without the commas

[math] \frac { a } { b_c } [/math]

>>10940480
If you move the goalposts a bit there's quite a bit of interesting stuff between [math]\mathbb{Q}[/math] and [math]\mathbb{C}[/math].
The main highlights are
[math]\mathbb{Q} \subset \bar{\mathbb{Q}} \subset \mathbb{P} \subset \mathbb{C} [/math]
Although it's very much not "all definable numbers" the period algebra [math]\mathbb{P}[/math] is essentially all the numbers you can obtain through integrals. For example, integrating 1/z around the pole gives [math]2\pi i[/math]. The main approach to the Hodge conjecture I know proceeds via trying to understand the period algebra. This is definitely an area of study that has been under-utilized and is going to heat up in the coming decades.

>>10934172
>If you only care about the stable homotopy category
???
HTT is about topology, HA is about algebra (and in order to do algebra it starts by setting up linear algebra). I agree that right now there's a need for a modern linear algebra textbook, because if you want to learn linear algebra the best method seems to be reading Adams' blue book and HA simultaneously.

I wouldn't recommend reading Kervaire Invariant 1 to someone who wants a first introduction to linear algebra.

So I know this is asked like a gazillion times, but can anyone post that Teach yourself Math textbook flowchart, please? I don't want to be a brainlet anymore, time to get serious.

>>10940552
>>10940556
Looks even better if you use the [,eqn,] [,/eqn,] tags.

(without the commas)

>>10940589
or y'know just tell 'em to read the wiki article

>>10940561
>Adams' blue
Stable Homotopy and Generalised Homology from the 70's is what I found.
Why is it good?

Also, those integrals are probably not all computable. Surprises me you'd respond to that - I might be confused tho

>>10940592
I should have

>>10940487
>I guess proof by contradiction is invalid then. Might wanna rework basic logic.
Proof by contradiction works in the form of P implies contradiction, therefore not P. However, not-P implies contradiction, therefor P is false under constructive logic. This has already been "reworked" by Heyting and intuitionistic logic exists over classical logic, so it's done.
Not-not-P is not the same as P, because not-P is not "the inverse of P is true" but rather "there does not exist a proof for P". Thus not-not-P is actually "there does not exist a proof for not-P" but that says nothing about whether there exists a proof for P.
Read up on constructive logic.

>I guess proof by contradiction is invalid then. Might wanna rework basic logic.
Proof by contradiction works in the form of "P implies contradiction, therefore not P". However, "not-P implies contradiction, therefor P" is false under constructive logic. This has already been "reworked" by Heyting and intuitionist logic exists over classical logic, so it's done. The statements of P and not-P correspond to "this is a proof of P" and "there does not exist a proof of P", its not about "P is true" or "p is false", because if P is true then there will be a proof of it, and if P is false then there is no proof proving that P is true.
Thus not-not-P is not the same as P, in the same way "not not true" is the same as "true", because not-P is not "the inverse of P is true" but rather "there does not exist a proof for P". Thus not-not-P is actually "there does not exist a proof for the nonexistence of a proof for P" but that says nothing about whether there exists a proof for P.
Read up on constructive logic. It's an objectively more thorough and superior form of doing mathematics. This is a good intro to it:

https://www.youtube.com/watch?v=zmhd8clDd_Y

I deleted the original post and posted this one for clarity.

>>10930558
Instead of being half retarded, be both a mathematician and a physicist

>>10940565
pls help...

>>10940413
>Have fun digging up my toolkit.
Well sure, that’s what research is. Digging up things so that others don’t have to. If you think your work will be viewed any differently by people that are not literally working on your exact subject, then I don’t think you have a very realistic idea of how math research works.
Sure, what you do feels concrete and useful to you, but it sure as hell does not to anyone else, even within your general area. Even an "analyst" that is working in an area slightly removed from your own probably will not understand the motivation for what you are studying.
There is no need for self-aggrandizing. Everyone is digging up their own narrow path in the mathematical landscape for reasons that only they (and hopefully a handful of other people) understand, and that is all there is to it.

>>10940496
>>10940506
Yeah, that makes sense. The euler example turns out to actually be a very simple and elegant counterargument. I was wrong. Apologies.
Seems like algorithms can be math. In this sense, yes, a statement of the form "this structure, which we call a neural network, does this optimization process when we run it" could be a mathematical theorem.

>>10940866
Of course you're right. But from my perspective, the stuff I'm doing is important and beautiful and the stuff algebraists are doing is pithy trash that I use on occassion. And my perspective is the important one to me. So that's what I'm going to listen to.
>no need for self-aggrandizement
No need to not self-aggrandize either, is there? Not if I'm going to sit snugly in my field. Why would I bother having respect or admiration for others with different interests? Sounds like a lot of wasted energy I could spend on stuff I give a shit about instead.

>>10940940
Well if you are going to go out of your way to shit on other people’s work, I think you could just as well spend that time and energy learning the first thing about it.
It does not seem like you have a lot of stuff going on

>>10940637
>why is it good?
I suppose I should clarify, I'm specifically referring to part 3 (each of the parts can be read independently) minus section 4 (which should be avoided---just take it on faith that there is a tensor product). Ultimately the issue is that historically there was a trade-off between how many good properties your category of spectra has and how easy it is to set up. Adam's blue book is simple and gives a nice introduction to the basic properties, but it's not robust enough for many later applications. Other expositions tend to introduce far more technical constructions in order to avoid hell much further down the road. For an introductory account this is terrible because the student will simply have to take it on faith that "if you do it the easy way you'll pay later".

If you want to think about the category of spectra and its properties, you should think about the stable infinity category of spectra as set out in HA. If you want to think about a particular object of this category, then you should think about it terms of Adams' blue book. The two complement each other very nicely.

>those integral are probably not all computable
They are uniquely defined and you can always cook up a finite-length formula which approximates them arbitrarily well. I'd call that "computable".

>Surprises me...
It seemed like an honest question. Had the person he asked responded he would have gotten an answer that made him go "oh it's just a bunch of crankpot wank". Most mathematicians, having had that experience before would answer the question with "it's a bunch of crankpot wank".

I figured I'd fight back against the widely held misconception that there's nothing but crackpots between [math]\bar{\mathbb{Q}}[/math] and [math]\mathbb{C}[/math].
The real answer to that question is: here be dragons.

>>10940983
I shit on people's work while I'm pooping or eating. It's very low energy and therapeutic. And that's not the only thing I do on 4chan, I answer lots of sqt questions and generally engage in normal discussion.
Socializing in a number of forms is important to maintaining high energy levels for mathematical work, in my opinion. Gets the communicative brain moving.

>>10940487
Quickly put together a clarification on
>I guess proof by contradiction is invalid then
and also proving classically equivalent LEM's constructively here:

https://youtu.be/Fyfvwu1dejA


>>10941042
You you mean it's good for the linear algebra needed to discuss spectra, or it has a more broader insight for the field of linear algebra?

>>10940487
Quickly put together a clarification on
>I guess proof by contradiction is invalid then
and also proving classically equivalent LEM's constructively here.
You can also take it as a babby intro to the Curry-Howard correspondence between algorithm and proof

https://youtu.be/Fyfvwu1dejA


>>10941042
You you mean it's good for the linear algebra needed to discuss spectra, or it has a more broader insight for the field of linear algebra?

>>10940413
>Mega cope.
t. seething analyshit

>>10940822
read the fucking wiki it is quite thorough you probably don’t have remotely enough exposure to elementary math yet to begin “planning” self study. When you finish with fundamental subjects then come ask for specific recommendations.

>>10940565
>So I know this is asked like a gazillion times, but can anyone post that Teach yourself Math textbook flowchart, please?

How do I read a Math paper?

>>10942968
I'd think if you try to read this, then you should already know.

Anyway, my general approach is to read the abstract, then skim through the pages and look at the pictures, maybe read the figures. Read the conclusion, and then go back and read it in full, first with not necessarily the biggest attention to detail.
Take note of the authors and acknowledgements too.
Takes notes, find your style at that.

To the faggot saying neural networks aren't math:

http://colah.github.io/posts/2014-03-NN-Manifolds-Topology/

I want to prove that every plane through the origin in [math]\mathbb{R}^3[/math] can be identified with the kernel of a linear functional [math]f \in (\mathbb{R}^3)^* [/math].
I know this trivial using annihilators but here is my attempt at proving it without using them (Mainly I'm doing this because this was mentioned as a motivating example):
Let [math]P[/math] be a plane through the origin with normal vector [math](a,b,c)[/math]. Now let [math]f_{a,b,c} \in (\mathbb{R}^3)^*[/math] given by [math]f_{a,b,c}(x,y,z) = ax + by +cz [/math]. Then if [math](x,y,z) \in P [/math] it follows that [math](x,y,z) \in (\mathbb{R}^3)^* [/math].
My problem with this is that it seems tautological in a sense, if someone spots something wrong with this please tell me

>>10943342
I meant that [math](x,y,z)\in Ker(f_{a,b,c} [/math] in the last part

>>10940561
I consider topology the study of homotopy invariant notions on spaces (whether you use simplicial sets or compactly generated Hausdorff topological spaces). So this is the study of Kan complexes, not the Joyal model structure. My demarcation of the subject is just as arbitrary as yours but it's probably closer to whoever asked the question.

hey bros lets make Shafarevich 's Linear Algebra & Geometry the new sci meme book, its dope :)

>>10943342
A linear functional f:R^3--->R can be represented by [a b c] for some a,b,c in R.
(In general any linear map R^n-->R^m can be represented by a mxn matrix)

(x y z)' in R^3 is in the kernel of f
if and only if
(a b c) (x y z)' = 0
if and only if
ax + by + cz = 0

Given any plane passing through the origin, you can vary a, b, c however you want and you'll get its equation from the equation of the functional.

>>10943406
Amazing book and Advance calculus by shlomo steinberg too!

>>10943342
There is an isomorphism [math]V \to V^*[/math], the functional corresponding to [math]v[/math] is [math]\varphi_v = \langle v,- \rangle[/math]. Clearly [math]\ker \varphi_v[/math] is the orthogonal complement to [math]v[/math], in other words the hyperplane having [math]v[/math] for a normal vector.

>>10943415
>shlomo steinberg
lmao

I was just thinking about stability of the Newton method. In particular to what in the complex plane amounts to
[math] \frac { d} {dt } z(t) = f(z(t)) = i\, z(t) [/math]
(where in each Newton iteration step you'd actually step outside the circle a bit and thus drift away)
Now I came up with one approach to systems like this (whether or not this is a good idea or not) and since I'm sure somebody did something this before, I'd like to get a reference or a name:

The idea is first to introduce a grid of some small distance around the origin. Let's say from -2 to +2 in both x- and y- direction and grid length some negative power of 10. Then compute the gradient f(q) at all those point q, choose a step length [math] \gamma [/math] and find the point closest to [math] q_{\gamma} := q + \gamma \cdot f(q) [/math]. At this point you have a map [math] \Gamma(q) := q_{\gamma} [/math] that for each stating point represents a lookup table for how to traverse the whole trajectory for [math] \frac { d} {dt } z(t) = f(z(t)) [/math] (introducing offset due to [math] \gamma [/math]).
(One could, instead of "take nearest point", also do something more clever, i.e introduce a bias that tries to make the trajectory stay true to some invariance properties, e.g. that the trajectory doesn't change Euclidean norm too much in the above example.) I'm not sure if the pre-computation of the connections is computationally an issue (will of course depend on your gamma)
Does this method have a name?

>>10943342
I feel there's a way to obtain the result by using the exact fibre sequence [math]1\rightarrow O(2) \rightarrow \operatorname{St}_3^2 \rightarrow \operatorname{Gr}_3^2 \rightarrow 1[/math] and the [math]3[/math]-lemma.
However I have my own problems to deal with right now so I'll leave the details to the reader.

Is Linear algebra by Shilov a good book for a beginner? I had linear algebra in Uni 4 years ago, but I've forgotten most of it. I want to brush up and start applying lin alg in programming.

>>10930547
I listen to Mozart and Beethovan while doing maths sometimes when there is noise outside my house. Is this good?

>>10943682
Pretentious.

>>10943058
Why is topology is full of purists and crusty old fucks who refuse to believe what they do could be useful for anything other than geometry or physics?

>>10930547
can you high IQ anons help a brainlet out? Let’s say we have a system like pic related. We have 1 button per row with 4 slots per row. How many configurations are possible? I think 12*4 = 48, right?

I keep confusing myself and i cant into combinatorics because sub100 IQ

>>10943971
How is this not just 4^4? Where is the 12 coming from?

>>10943971
the general formula for r picks out of a s-long set is
[eqn]
s^r
[/eqn]
you could even do it the concrete way multiply the amount of possibilities for each row

>>10943682
you have bad taste

>>10943682
I listen to the music that the yt algorithm recommends to me, or to a playlist of such tracks I saved over the years

>>10943682
>>10944100
Agreed. You should listen to Bach only.

Question
Im trying to self learn linear algebra. Axler's LA done right was recommended so I bought it, but I'm finding it difficult to follow. Given that I am more interested in the applied side of things, should I instead get a book that is more application-based? Or should I continue this one?

>>10944269
It seems this is your first la book? If so, I’d suggest trying another book. If you are interested in applications, I recommend strangs lectures, and maybe a pdf of his book.

>>10944328
thanks, I'll look at the MIT OCW lectures

>>10930547
Does Jordan Peterson misrepresent Gödel here? I've seen people accuse him of it but I lack the knowledge to make heads or tails.

>>10944517
His caharcterization is correct but the arguments drawing from it are sketchy and philososhittery.

>>10941619
Is there any mathematical statistics book that would fit in here?

>>10943682
you literally have cringetaste

Is Art of computer programming a cap for brainlets? I can't get passed 1 chapter. Or the author is bad?

>>10944517
Anytime you cite the incompleteness theorems outside the context of a consistent first order axiomatizable r.e. theory it is a misrepresentation.

>>10943058
shut the fuck up you literal moron
and why the homophobia

>>10943682
>>10944100
>>10944212
lol fuck off idiots, listen to chopin

>>10944998
>the author is bad?
lmao can you imagine someone posting this unironically

>>10943348
The study of homotopy invariant properties of your favorite model for homotopy types is homotopy theory.

Topology is the study of space. Historically this meant either topological spaces or topoi. Today, "modern topology" almost certainly refers to infinity topoi. In the future it may refer to some further refined conception of space.

>>10945034
Fuck u cognitive biased person. Have you ever read the book? Knuth is closest faggot who doesn't know how to write book properly. There are better authors in other subjects who write properly. Take KandR C. It is an example of proper authoring .

What is the proof that gcd of two numbers divides their difference?

>>10945028

>>10945090
GCD(a,b) divides any linear combination of a and b trivially

>>10944517
Gross oversimplification and just flat out wrong.

>>10945090
I mean (a-b)/c = a/c - b/c, for c being gcd(a,b).
I don't know what more to say.

>>10945045
Why the homophobia?

>>10944517
No. He speaks of some Godel, so you don't need to worry about him misrepresenting Gödel.

>>10945429
Why the homophilia?

What does "form an algebra" and "family of integrable functions" mean?

>>10946238
Family is a funny name for set.
Form an algebra means that they're closed under sum, multiplication by scalar and convolution.