/sci/ - Science & Math

>>11482035
https://arxiv.org/abs/2001.08339

First for Vieta's Theorem. I'm currently going over it in Gelfands Algebra.

>>11482194
Damn, so close. I'll read your link in defeat then.

Threadly reminder to work with physicists.

>>11482035
I'm taking my first real upper division math course this semester and I'm really liking it. Now that lectures are cancelled, can anyone recommend any resources for a proofs based linear algebra class?

bump

>>11482035
when will mathfags learn
someone post uncle Ted letter

>>11482615

Norman Wildberger

Corona, please don't take Gromov.

>be me
>few years ago
>young and naive first year math student
>excited for my first linear algebra lecture
>be first one to wait infront of the lecture hall
>other students arrive
>too shy to start a conversation
>pretend to do something on my phone
>finally a mid 40s guy with short gray hair, clean shaven face wearing a knitted sweater arrives
>says good morning and opens the door
>fast forward a few minutes
>lecture begins
>prof is really charismatic and smiles a lot during the lecture
>sit in the front row and write everything down
next day
>be again the first one to arrive
>this time the lecture hall is open
>sit down in the front row
>the prof comes in
>he smiles at me and says good morning
>he sits down at his desk and attempts to make smalltalk with me about the weather
>he constantly smiles at me during that
>I feel very uncomfortable
>suddenly he talks about how he is lonely since his wife left him recently
>I just sit there and nod feeling very uneasy
>another student arrives and he stops talking to me and works on his laptop
>fast forward to the lecture
>he talks about surjective and injective functions
then something strange happens
>he claims that every surjective function has a right inverse
>he proves it
>didn't even ask for our consent to use the axiom of choice in this course
I was basically raped

The Weierstrass approximation theorem states that any continuous function on a compact real interval may be uniformly approximated by polynomials. The only constructive proof I'm aware of is via Bernstein polynomials - but I have another idea in mind.

Suppose [math]f:[a,b] \to \mathbb R[/math] is continuous. Let [math]P_n[/math] be a partition of [math][a,b][/math] with mesh [math]\lambda(P_n) = \frac{1}{n}[/math]. Let [math]p_n[/math] be the Lagrange polynomial which interpolates [math]f[/math] at the points of partition. Is it reasonable to expect that [math]p_n \xrightarrow[n \to \infty]{} f[/math] pointwise or even uniformly on [math][a,b][/math]? If not, could you suggest a counterexample?

>>11483136
You've never actually interpolated a continuous function with polynomials, have you?

>>11483152
I have not, but I take it that your answer is a big fat 'nope' then

Hello /mg/, I hope you all have a wonderful day

what are some cool mathematical objects related to dynamic of epidemics?

>>11482035
>socially distant edition

i only wanted to be a math major and have math friends

but i've been thrown back into a sea of chinese. they've killed me.

why am i always on the butt end of these horrid social experiments?

>>11483224

Because you live in a liberal shithole. Meanwhile a russian chad like me had no impact in my way of life since based Putin closed the borders since the beggining.

All right /mg/, let's settle this once and for all, what's the best algebra book for undergrads? Vote

https://www.strawpoll.me/19569179

>>11483159
So the issue would seem to be controlling the size of the oscillations between interpolation points. You could do things to make it work but since polynomial degree is increasing with interpolation precision, it seems like guaranteeing uniform convergence is a hard ask.

>>11483159
>>11483346
By the way, you should look into Runge's phenomenon and Chebyshev polynomials as a way to resolve your problems here. But now your proof is getting a little complicated!

>>11483235
Russia's borders are huge, you can't fully close them, it's impossible, sooner or later there will be people getting into the country ilegally, especially if the world gets fucked and Russia remains with low cases.

Also, this is not even getting into the huge problems with Putin's administration, he constantly spits in democracy and human rights.

>>11483136
If I recall Lagrange interpolation can get pretty wild if the continuous function oscillates. Though the compact interval can probably help uniformly bound the size of this oscillation.

hi what button do i need on a calulator for the fraction 851 divided by 999 to equal 23/27 instead when i press equal it gives me a decimal

>>11483061
>mid 40s guy with short gray hair, clean shaven face wearing a knitted sweater
>lonely because his wife left him

>>11482035
>tfw bombed manifolds final
Guess I can always be an algebrist.

hi what button do i need on a calulator for the fraction 851 divided by 999 to equal 23/27 instead when i press equal it gives me a decimal

anyone?

>>11483286
For #CoronavirusChallenge I am trying to make a complete solution manual for Dummit & Foote. Got through Chapters 0-1 in about 2 weeks. Boring as fuck but hopefully the later chapters get more interesting. I tried doing a solution manual for Ahlfors Complex Analysis last year but got distracted by actual work in the middle of Chapter 4

>>11484078
>Boring as fuck but hopefully the later chapters get more interesting.
Most of the exercises in Dummit & Foote are pretty bland, even in the sections with interesting content.
Dummit & Foote was engineered to be a standard course textbook, so they wanted a bajillion exercises per section to make it easier for lazy profs to use them as problem sets. The result of this is that most of the problems are pretty low-quality and often get repetitive.

>>11484078
Heh, several people have tried that.

>>11482249
WHERE THE FUCK WERE YOU IN THE LAST THREAD YOU IMBECILE

>>11483992
on my cheap-ass Casio it's one that looks roughly like a . Check your model's manual, I'm sure you can find it online

>>11484123
fuck, the board filters certain ASCII symbols. It's one that looks like a bottom-right corner piece, such as U+259F here: https://en.wikipedia.org/wiki/Box-drawing_character#Unicode

>>11482194
Did you coauthor this, yukariposter?

>>11484108
>WHERE THE FUCK WERE YOU IN THE LAST THREAD YOU IMBECILE
Under quarantine.

>>11484078
Holy shit, you're fucking awesome if you manage to finish it. Can you please skip to the chapter on Rings for now? That's what I'm currently studying and it would be tremendously helpful if I had a solution manual for it. Thanks for you hard work man.

>>11484099
Several people tried to prove fermat's last theorem, did it stop andrew wiles from succeeding?

Several people tried to prove a millenium problem, did it stop grigori perelman from succeeding?

Get the fuck outta here with your defeatist attitude.

>>11482249
KYS.
Physics is for niggers, fuck off

>>11484276
>Racism outside of /pol/
You realize you're sticking out like a sore thumb here, right? That shit is just pure cringe on the science and math board, my man, don't mistake us for troglodytes.

>>11484276
>niggers
Why the racism?

>>11484276
>for niggers
Kikes and niggers, actually.

>>11484275

Cringe

>>11483211
here are two quite relevant objects: the number of dead people and the number of infected people

>>11484222
sweet trips, but no unfortunately I'm not an algebra god so I really want to go through sequentially and get this material down

>>11483136
the answer is no, read this:
https://en.wikipedia.org/wiki/Runge%27s_phenomenon#Problem

>>11484501
Well, then go fuck yourself.

>>11484222
>>11484544
Tried my hand at it.
Dude this shit sucks so much, Jesus Christ. Stopped at 14.
https://anonfile.com/79l2w8icod/dude_pdf

>>11484748
Yay, thanks, you're awesome.

Anyone here interested in the philosophy of mathematics? I've been reading pic related, but the first chapters are quite boring to be honest. What do you guys think of PoM? Cool subject IMO to study as a amateur

>inb4 Reddit label
learn to enjoy a good meme

>>11484854
What sort of father expects their child to take an interest in formalisms of quantum mechanics that still don't have a rigorous footing in mathematics? Shame on you.

>>11484854
I saw that one on Facebook, so it's even below Reddit-level, be ashamed fucking newfag.

>>11484863
>still don't have a rigorous footing
Well, there are many techniques such as Atiyah-Bott localization or Feynman push-pull correspondence that allows you to define the path integral rigorously. If your fields are holomorphic sections into a finite-rank vector bundle you can even count the number of integrals you have to do with Riemann-Roch. The main problem is if your theory is dynamical/non-topological and the gauge group/OPE algebra is infinte, really.

Now THIS is a good funny image.

>>11484885
Do you find pleasure in stating meaningless and wrong posts? Idiots will believe you said something relevant at all, congrats...

Hey guys I just want to check if my solution to an easy Hatcher exercise is correct (it's just i'm confused buy all the gluing stuff). P.91 he wants us to show that when you quotient EG (built as a delta complex) by G you get a covering. I'm just showing that it is free (that's obvious) and the action is proper (with the discrete topology on G) because a compact subspace K crosses at most a finite number of cells (idk if they are called like that as for CW-complexes, i just mean the open image of one of the maps delta^n->EG), and then it's a matter of easy combinatorics to show that gK intersects K for at most a finite number of g. Is that it? Thanks

>>11484992
No, my post is fine. Don't worry about it little boy, it's not for you.
>>11485009
Yeah that seems ok.

>>11485019
OK now that i've got someone, where does the g_1.f(g_2,...,g_n) come from in group cohomology? specifically the action of g_1. I get that you obtain H*(G,Z) (with trivial action of G on Z) from the singular cohomology of K(G,1) but how in hell can you get H*(G,A) with nontrivial action of G on A from a functor from Top since there's no way that K(G,1) or any space remembers about the group structure of G.

>>11485032
You can treat [math]f[/math] as a cocycle representing a class in [math]H^*(BG,\mathbb{Z})[/math]. Then you force equivariance under the [math]G[/math]-action and the cocycle condition [math][g\cdot,\delta] = 0[/math] gets you an alternating sum.
>how in hell can you get H*(G,A) with nontrivial action of G on A from a functor from Top
We have [math]H^*(BG) \cong H^*_G(\bullet,\mathbb{Z})[/math] for which you can then apply UTC to send [math]H^*_G(\bullet,\mathbb{Z})[/math] into [math]H^*_G(\bullet,A)[/math] for Abelian [math]A[/math]. These maps are classified by [math]G[/math]-fixed points of [math]\operatorname{Hom}^{s,t}(\mathbb{Z},A)[/math] and the derived objects thereof.
>no way that K(G,1) or any space remembers about the group structure of G
But [math]K(G,1)[/math] remembers [math]G[/math] through its loops? Or are you asking for something else?

>>11485095
>representing a class in [math]H^*(BG,\mathbb{Z})[/math]
Sorry, I meant [math]H^\ast(EG,\mathbb{Z})[/math]. The [math]G[/math]-equivariance sends you down to [math]H^\ast(BG,\mathbb{Z})[/math].

>>11485019
>The main problem is if your theory is dynamical/non-topological and the gauge group/OPE algebra is infinte, really.
This is plain obvious and what was implicit in the post you quoted. You just posted that shit to show-off to brainlets that keep raising your undeserved fame.

>>11485181
It really isn't. Quantum theories such as in-equilibrium many-body QM, [math]G[/math]-QFTs for finite [math]G[/math], YM, TQFTs and CFTs don't fit that description and indeed have well-defined/rigorous notions of "average over histories". You need to stop pretending you know what you're talking about anon, because you don't.

>>11482640
Ted is a tranny

>>11485095
OK thanks I need to learn the utc sorry

>>11485095
can i ask you a question though? How do you manage to not get lost in the myriads of details and technicalities and unimportant theorems? not intellectually really but I mean how do you not lose your appreciation of and dedication to math and your desire to get into it as a career. Algebraic topology particularly is so full of technical shit (delta and CW complexes, thre excision theorem and clutching functions in K-theory kind of disgust me or at least they really did at firsg).

>>11485772
and also the results of algebraic topology are really boring for some, like what do I care about division algebras. I know I don't care about diophantine equations either but somehow it doesnt bother me so much in NT.

>>11485778
and (sorry for multiposting), is there a result as pretty as the Hasse-Weil bound on the number of points on elliptic curves in AT? (with a proof as pretty too) and what is the easiest one to understand?

>>11485778
>what do I care about division algebras
if you asked me, this is a textbook example of an interesting theorem. it's comprehensible by everyone, it's kind of surprising, and using AT to prove it is pretty creative. I mean you could have said something like "what do I care if a space has (7n+4)th trivial homology with coefficients in some weird ring if (3n+2)th homology has zero torsion" and I would totally get your sentiment, but this is kind of strange.

>>11484768

Yeah, probably interesting but they won't pay you for doing this and you may go completely mad.

>>11485944
You mean by every first year math students, laymen don't understand division algebras. Maybe my appreciation of them is at fault but i really couldn't care less about them (I know there's some stuff about them and the Brauer group or something but I'm not knowledgeable enough about that). Although to be frank the problem I have with AT is not that it is useful in proving theorems about divison algebras, I just don't really know why i don't like it, i feel it's too 'smooth' and 'flexible'. What I like about AG and NT are the 'rigidity' of it and the pretty cristalline structure, but i may be i'm just being weird. I don't know if people have had similar feelings.

Also I think you're making a false dichotomy with your 7n+4 3n+3 example, bc it's normal for the technicalities to feel unmotivated from the outside (I hope there aren't big theorems that read like that).

t. reddit spacer

Do fields of arbitrarily large cardinality exist?

>>11486156
>choose a set [math]A[/math] of your preferred cardinality
>take [math]\mathbb{Q} [A][/math]
>it's a domain, so you can add in all the inverses
>just_works.jpg

>>11486156
no, for example there is no field with 10^1000000000000000000 elements

>>11486241
but there's one with 11^1000000000000000000

How many algebraic structures are there? Groupoid, semigroup, monoid, group, ring, field, vector space, algebra... What else?

>>11486389
you have listed all of them, in fact grothendieck proved in his seminal 1965 paper that these are the only algebraic structures

>>11486389
You can create any you want you fucking retard

>>11486389
Exactly as much as you need for your research.
Homework: read any article about quasifields.

>>11486389
category, magma, semi-magma, algebroid, cogroup, cocommutative coassociative coalgebra, etc.

>in quarantine, not on campus
>can't get terms textbooks from University library

How can people cope with the pain of studying topology?
Don't get me wrong, it's beautiful, but it's also slowly consuming my mortal soul

>>11486491
>Not having Principles Of Mathematical Analysis, Walter Rudin downloaded

>>11486491
>not knowing libgen

>>11486528
I just learnt libgen
rules of nature

hi what button do i need on a calulator for the fraction 851 divided by 999 to equal 23/27 instead when i press equal it gives me a decimal

>>11486502
general or algebraic ?

>>11486577
General. Just for you to know, I'm not a math major, but since I'm studying fuzzy inference, I had to take some basics of topology. I currently have a small assignment which I just can't seem to solve

>>11486571
you'll need a plastic button or maybe a metal one depending on the type of calculator you have

>>11486577
ALGEBRAIC N scientific

>>11486589
>General
Don't.

>>11486589
general topology is the math equivalent of filing your taxes
nobody likes it, and there's no way to make it enjoyable. It's just something you have to force yourself through so it doesn't bite you in the ass later.

>>11486868
>general topology
T0 space, T1, T2, T2½, completely T2, T3, T3½, T4, T5, T6... If I wanted to study such shit, I'd become biologist.

>>11486502
>>11486868
when you deal with the very basics like bases, compacity, connexity, it's really impressing how well it works. Things like the compact open topology are really clever (you take compacts because they're the only subsets whose continuous images are necessarily closed, hence f(K) is closed in U and hence has some wiggle room, hence it's reasonable that the set of f such that f(K) \in U be open). But the things like tychonoff, urysohn or tietze, almost nobody needs them and if you do you just open wikipedia. But the general notion of a topology is really one of the great triumphs of the axiomatic method.

>>11486882
again, nobody does that, nobody needs that, it's just mediocre nerds putting it on wikipedia. I'm French and I've literally never heard about T anything irl.

Hey all, so I have a quick probability theory question for something I'm working on (not homework). Let's say I have a sequence of iid random variables and some continuous function g(), could be bounded or unbounded. The RVs are nice, no weird Cauchy shit.

Now the sample mean of the iid random variables converges to the population mean by SLLN. If I want to say g(sample mean) converges in probability to g(population mean) that's just continuous mapping theorem. But is there anything equivalent that says that the sample mean of g(random variables) converges to g(population mean)? Surely there has to be a name for this.

>>11486890
>I'm French and I've literally never heard about T anything irl.
Funny, in France topological compact space is defined as T2 space (Hausdorff) and each of its open covers has a finite subcover.

>>11486889
except tychonoff in functional analysis i guess but it's so easy to remember

>>11486882
b-but anon... I am a biologist...

>>11486889
We didn't get that far though. We saw the Scott and the Sierpinsky topology for example...

>>11486589
>being unable to solve problems in General Topology
General Topology is one of those subjects where you can genuinely make up for the lack of talent with memory. Just memorize all the theorems.
>inb4 talking from experience?
Pretty much.

>>11487096
The issue is, how do you solve stuff if you're lacking talent and a lot of background knowledge?

>>11487107
You don't, you're fucked.

>>11484078
Where will you be posting it wen it’s done?

>>11487094
yeah i don't even know the scott topology and sierpinsky is the meme fractal shit i think? sorry but your class is probably shit. The only counter examples worth remembering in topology are the cantor set, the hawaiian earing, maybe the line with two origins. The others you see once and then you can forget them or you should be able to make them yourself (like the stuff with sin(1/x)).

>>11487140
the sierpinski topology is [math]\{\emptyset, \{a\}, \{a,b\}\}[/math] on two points
you're thinking of the sierpinski triangle

>>11487090
From my experiences people who are into science, but can't mathematics become biologists.

>>11487165
oh ok, i hate the people who give these kinds
of things name

>>11483061
what is a right inverse? as opposed to a left inverse? do you mean invertible? where is AOC used in this, i saw the proof but it was about diensionality and bases, i dont remember choice

>>11487296
waaahhhh I can't be bothered to actually learn why things are called the way they are waaahhh

>>11487598
b is a right inverse of x => xb = identity

>>11482035
What the hell is the G on this picture?

>>11487817
oh i see, but isnt the right inverse just the left inverse inverse? because T*T = I and TT* = I so T = T**

>>11487913
The [math]M[/math]'s are tensor fields and [math]G[/math] is a gauge transform.

>>11487770
based Putnam Poster might've killed himself

>>11487936
Wait a second.
Random shirtless video dude was the Putnam Touhou poster?

>>11487947
I think so

>>11487986
Based on?

Do you guys read biographies about mathematicians?

>>11487996
I read a couple erdos biographies, cool guy
>>11487936
rip

>>11487996
Sometimes. I also enjoy "slice of life" sort of stuff like the book in pic related, or this short text by Steve Smale: http://www.cityu.edu.hk/ma/doc/people/smales/pap107.pdf

What's the difference between an antiderivative and a primitive?

>>11484992
It's what avatarfagging does to your brain.
They try to inflate their ego making hundreds of psud posts.
KYS yukaritard

>>11482632
I'm going to follow this. What are the last two books called?

>>11488325
Keep in mind that misha's list is heavily biased toward symplectic geometry/complex geometry

How the fuck do I remember theorems and proofs? I find after like a year or so I've forgotten most stuff I studied the year before.

>>11488331
tyvm

>>11488339
Anki

>>11488325
If you try to follow this you have brain damage

>>11488325
>What are the last two books called?

>>11488552
i do have brain damage

>>11488325
>What are the last two books called?

>>11487811
i hope you're bait

>>11482632
it's been a long time since i haven't been to /sci/, i saw this list but i took it as satire, is it? does mg stand for gromov?

>>11488570
>i saw this list but i took it as satire, is it?
No.

>does mg stand for gromov?
Look at the name of this thread.

>>11488573
ah ok, sorry. but nobody ever studied that hard and fast, you have to be 35 y/o at least to know about both hodge structures, seiberg-witten, deligne proof (you could get there earlier i guess bout you have to do research). And I think nobody knows those + gromovian geometry and really advanced AT. And no you can't learn about the index formula in sophomore year. Plus there's a lot of things lacking like idk arakelov theory

>>11488576
inferior w*stoid cuck can't handle the BRC (big Russian curriculum)

>>11488589
My country has 150% of their number of fields medalists with a third of the population.

>>11486899
But we don't call them T2 or Hausdorff spaces. We just call them "séparé".

Just started getting into mathematics recently, but the sheer amount of high-school level math I have to relearn since I forgot all of it takes away my motivation. To fix this, I watch channels like Mathologer, Numberophile, 3b1b to still be able to encounter more abstract maths. Do you know any similar YouTube channels, with almost only math output?

See if you could solve this, /sci/, as an exercise. It's not homework, don't worry. I already have my solution I will post afterwards.

Let (M, J) be any topological space. For generalized sequences (nets) (Fsubi) [for all i in I] of subsets of M one defines the limes inferior of Fsubi and limes superior of Fsubi and convergence analogously as for sequences. Show that every generalized sequence of subsets of M contains a converging generalized subsequence. (Hint: use Tychonoff's Theorem)

I didn't use Latex.

I said...

i is composed of nets, so the family I of M has the finite intersection property, and due to

A : Fsub1 ⋂ Fsubi [for all values of 1 ≠ i ∈ I] = Ø

we have the topological space (M,J) being compact. Consider

Set of all subsequences where all A exists = B

If A is not true, then for a sufficiently high ī, it is true for all i ≤ ī, defining the Li(Fsubi) as the difference (Fsubī) - B = Li(Fsubi). In the sequence [Closure of]Fsubī U infinity [invoking Alexandroff compactification], clearly there are infinitely many values of i for which A is false, which defines Ls([Closure of]Fsubī).

Due to the compactness of (M,J), all subsequences are also compact, and due to Tychonoff's theorem, for ī sufficiently high,

∏ Fsubī = Fsubī

∏ Fsubi = Fsubi

Therefore, due to the properties of the Alexandroff compactification,

∏ [Closure of]Fsubī = ∏ Fsubi = Fsubi, defining a converging subsequence for ī sufficiently large, in relation to an Alexandroff compactification of Fsubī, converging on Fsubi. Therefore because ī has been chosen sufficiently large,

Ls([Closure of]Fsubī) = Li(Fsubi) = Ls(Fsubi) = Li (Fsubi). Because Li (Fsubi) ⊆ Ls (Fsubi), Fsubi has been defined as the Cauchy sequence, or closed convergence, of M.

>>11488666
>Let (M, J) be any topological space.
No question that starts with "any topological space" is worth doing.

See if you could solve this, /sci/, as an exercise. It's not homework, don't worry. I already have my solution I will post afterwards.

Let (M, J) be any topological space. For generalized sequences (nets) (Fsubi) [for all i in I] of subsets of M one defines the limes inferior of Fsubi and limes superior of Fsubi and convergence analogously as for sequences. Show that every generalized sequence of subsets of M contains a converging generalized subsequence. (Hint: use Tychonoff's Theorem)

I didn't use Latex.

I said...

i is composed of nets, so the family I of M has the finite intersection property, and due to

A : Fsub1 ⋂ Fsubi [for all values of 1 ≠ i ∈ I] = Ø

we have the topological space (M,J) being compact. Consider

Set of all subsequences where all A exists = B

If A is not true, then for a sufficiently high ī, it is true for all i ≤ ī, defining the Li(Fsubi) as the difference (Fsubī) - B = Li(Fsubi). In the sequence [Closure of]Fsubī U infinity [invoking Alexandroff compactification], clearly there are infinitely many values of i for which A is false, which defines Ls([Closure of]Fsubī).

Due to the compactness of (M,J), all subsequences are also compact, and due to Tychonoff's theorem, for ī sufficiently high,

∏ Fsubī = Fsubī

∏ Fsubi = Fsubi

Therefore, due to the properties of the Alexandroff compactification,

∏ [Closure of]Fsubī = ∏ Fsubi = Fsubi, defining a converging subsequence for ī sufficiently large, in relation to an Alexandroff compactification of Fsubī, converging on Fsubi. Therefore because ī has been chosen sufficiently large,

Ls(∏ [Closure of]Fsubī) = Li(∏ Fsubi) = Ls(Fsubi) = Li (Fsubi). Because Li (Fsubi) ⊆ Ls (Fsubi), Fsubi has been defined as the Cauchy sequence, or closed convergence, of M.

Are there any infinite groups that have finite automorphism groups? (apart from (Z,+) of course)

>>11487915
no. there are functions that have right inverses but no left inverses. for example the function
[math]f: R2 \to R[/math] with [math]f(x,y)=x[/math] has a right inverse, namely [math]g(x)=(x,0)[/math] but it has no left inverse.
We can show that every surjective function [math]f:X \to Y [/math] has a right inverse(here X, Y are arbitrary sets). Here is the proof:
Choose (using AOC) for every [math]y \in Y [/math] an element [math]g(y)\in f^{-1}(\{y\}).[/math] Here [math] f^{-1}(\{y\})[/math] is always nonempty since f is surjective. Now [math]g:Y \to X [/math] is a left inverse to f. This is actually even equivalent to AOC

>>11487598
>i saw the proof but it was about dimensionality and bases
it is about arbitrary functions not linear maps. anyway, every surjective linear map has a right inverse that is also linear. This needs AOC too but it is a little bit more subtle. for the proof you need that your vector space has a basis. And "Every vector space has a basis" is equivalent to AOC

>>11488762
The group of dyadic rationals under addition modulo 1 has automorphism group C2.
My question for /mg/:
Does there exist an uncountable group with automorphism group C2?

>>11488762
free group on n generators?

** TL;DR: Is there any enlightening connection between formal language theory and topology? **

I'm taking a course on finite automata and formal languages. We've just introduced the concept of regular languages. The very essence of regularity seems to revolve around finiteness: a language [math]L[/math] is regular iff it has finitely many Myhill-Nerode equivalence classes, iff there exists a DFA which generates it, etc.

Often in math, it's enlightening to generalize such characterizations by replacing "finite" with "compact". So my question is: Is there a topology that may be defined on [math]\Sigma^*[/math] (space of all finite strings over a finite alphabet) such that it would "interact nicely" with the known notions of regular languages, context-free languages, etc?

>>11482632
>>11488573
Why is this list so biased towards Algebraic Geometry?

>>11488842
free group on 2 generators a, b has infinitely many automorphisms:
for any nonzero integer k, define the map [math]f_k:\mathbb{F}_2 \rightarrow \mathbb{F}_2[/math] by [math]f_k(a)=ab^k[/math], [math]f_k(b)=b[/math], extended naturally to all other words.
Then it is easy to check that [math]f_k[/math] and [math]f_{-k}[/math] are inverse to each other.

>>11488858
yea you're right

>>11488851
it's just a meme, bro

>>11488762
Consider, for example, [math]G = \mathbb{Z} \cross C_2[/math].
Suppose [math]f:G \rightarrow G[/math] is an automorphism.
Since (0, 1) is a unique element of order 2, we get [math]f((0, 1)) = (0, 1)[/math].
Since (1, 0) and (0, 1) generate G, we also have that f((1, 0)) and f((0, 1)) = (0, 1) generate G.
We conclude that f((1, 0)) is one of (1,0), (-1,0), (1,1), (-1,1). So there are finitely many automorphisms of G.

>>11488880
that was supposed to be [math]\mathbb{Z} \times C_2 [/math].

>>11488762
fuck i never realised aut(Z) was trivial. it's funny i've never studied automorphisms of a groups now that I think of it

from the preface of Foundations of algebraic topology by Eilenberg, Steenrod
>A directed path in the network represents the homomorphism which
is the composition of the homomorphisms assigned to its edges. Two
paths connecting the same pair of vertices usually give the same homo-
morphism. This is called a commutativily relation. The combinatorially
minded individual can regard it as a homology relation due to the pre-
sence of 2-dimensional cells adjoined to the graph.
what exactly did they mean by this? can you detect commutativity of diagrams by some kind of homology relation?

>>11488926
shit I fucked up the greentext

Difference between Subspace and Span in Linear Algebra?

>>11488914
it's not trivial. it's the cyclic group of order 2

>>11488931
it doesn't make sense to say that a set is just "span". it's always "span of something".

>>11488836
>Does there exist an uncountable group with automorphism group C2?
I don't think so. Either the group is non-abelian or the group is abelian.
The obstruction to the former is that if [math]ab \neq ba [/math], conjugation by a is an automorphism which fixes [math]a[/math] and moves [math]b[/math], while conjugation by [math]b[/math] fixes [math]b[/math] and moves [math]a[/math], so you by default already have two automorphisms.
I think you can do some structure of abelian groups arguments for the latter.

>>11488931
subspace = nonempty subset of the vector space that is closed under addition and scalar multiplication

span(S) = smallest subspace containing S (where S is some subset of your vector space)

>>11488943
>structur of abelian groups argument
but the structure theorem is only for finitely generated abelian groups

>>11482035
Based pic

>>11488953
Weren't there some analogues for non-finitely generated abelian groups?

>>11488836
Let G be a group. Let's recall some stuff from group theory:
For any [math]g \in G[/math], the function [math]f(x) = g^{-1}xg[/math] is an automorphism of G, these are called inner automorphisms, the set of all those maps is denoted by [math]Inn(G)[/math]. This is clearly a subgroup of [math]Aut(G)[/math]. Furthermore we have that [math]Inn(G) \cong G/Z(G)[/math]. If [math]Inn(G)[/math] is cyclic then it is trivial.

So if [math]Aut(G) = C_2[/math], then [math]Inn(G)[/math] is either trivial group or [math]C_2[/math], so either way [math]Inn(G)[/math] is trivial so G is abelian.
now I don't know what to do next

>>11488339
PLEASE I need serious answers ONLY

>>11488965
Why do you always go and post the exact same argument I do?

>>11488969
Start teaching anon. That'll help

>>11488651
Professor Leonard is my favourite.

>>11488953
Right, better argument: we consider the group [math]G[/math] as acting on a set [math]S[/math]. Choose an element [math]a \in G[/math]. [math]S = Fix_a \cup Mov_a[/math], the set fixed by [math]a[/math] and the set where [math]a[/math] acts non-trivially.
Consider the full subgroup that fixes [math]Fix_a[/math] and the one that fixes [math]Mov_a[/math]. Naturally, the entire group decomposes as the direct sum of these two groups, and then we can take the [math](-1, 1)[/math] (invert on one coordinate) automorphism.
The trick is now showing that, because the group is uncountable, [math]Fix_a[/math] is nontrivial.

>>11488973
he finished his argument, as opposed to you

>>11488997
Your argument fails spectacularly if we consider the trivial group action.

>>11488999
>noooooo you need to hold my hand and explicitly write the conclusions you can't just outline the argument and expect me to figure myself out like an adult

>>11488926
no algebraic topology chads here that can answer this?

>>11489010
>the trivial group action
The argument doesn't need to work for any group action, tho.
As long as it works for one of them, it's aight.

>>11489018
I know that. What guarantee do you have that it works for even one group action?
My example shows that it doesn't hold in general, so you would need to construct an explicit action with the desired properties. You didn't do that, nor do you know that this is even possible in the first place.

>>11488836
>Does there exist an uncountable group with automorphism group C2?
Apparently the answer is yes (for arbitrary cardinals, actually). There are some citations to old papers (looks like this problem was addressed in the 60s) here at the bottom of the first page
>https://core.ac.uk/download/pdf/82394413.pdf
but the autism going on in the cited constructions is really next-level disgusting. There doesn't appear to be a human example of such a group, at least not that I can find.

>>11482035
how do i start math if i never did any math ?

>>11489028
Right, whatever.
Anyhow, I found this result in an abelian group theory book. If it admits any decomposition where [math]\mathbb{Z}_2[/math] isn't a summand, we can take the invert one coordinate automorphism, so we might as well consider the group as indecomposible.
If it's cocyclic, it's necessarily countable.
Then, we can assume that it's a torsion free abelian group.
>>11489082
FUCK.

>>11488933
eheh

What are some music you listen to when you're studying math?

>>11489178

Once you start getting bored and wanting to listen to music, it's a sign you should stop studying and come back later. If you can't even do a single hour without music, visit a doctor cause you're gonna need some meds!

>>11489178
Cumbias, NMH, Banda, Molotov, USSR folk songs and meme songs.

>>11489178
Tohou ost
Gamma wave sounds (they help with concentration)

>>11489178
https://youtu.be/2GjPQfdQfMY

>>11489082
I'm the one who asked the question. Thanks for the answer.
This was a bonus question on a problem set for a 2nd year algebra course I took at uni last year. Needless to say, I didn't solve it. And yeah, the construction really is very autistic.

>>11488847
Have you been asking this question recently with CS in general?

>>11484768
Great topic.
Haven't heard of you book, tho.

What's a problem with philosophy of mathematics is that the more you understand about foundations/formal logig/structuralism/etc., the more you find that people talk about the philosophy of mathematics without having a good understanding of that stuff and it seems to invalidate a lot of ideas. Basically, it's quite inaccessible and lonely subjects. This might be because any sort of progress is hard to quantify. So like literature, people just dig themselves into a framework they more or less agree with and then won't get out of it anymore. At 40++, you stop changing your mind (effectively for pragmatic purposes) in general and then some day you die.
Due to this bottomlessness, you could come to the conclusion that one should just keep on sticking to the formalities of foundations before being another dogmatic.

>>11483286
I found various German algebra books to be more readable than any English one.

>>11489392
Most people can't read ~leangrylanguage though

>>11489407
Not sure what that means, but if you're saying that most people can't read all the languages, then sure.

I was inclined to say Lang, except Lang is far too think. I like his general approach but don't know how improve on it either. My own texts also always seem to get too long.
I like Aluffi, but not as an introductory book. Not for students who want to get into academia, anyway.

>>11489014
Sounds like he's talking about what we would today call the nerve of a category. You're not going to get any new commutativity information out of it though.

So, is category theory really as groundbreaking as some people make it sound? If the aswer is yes, then why?

>>11489461
no, its soulless

>>11488926
>>11489014
See the simplices in pic related and think really, really hard.
>>11489449
Don't think that's it, tbqh.

>>11489465
Oh right, just remembered it.
We call the red line [math][/math], and we orient everything so it goes down and or to the right.
Then [math]-g-f+r[/math] is triangle 1,[math]-r+h+j[/math] is triangle 2.
Then, [math]-g-f+r-r+h+j=-g-f+h+j[/math] is the full square, which vanishes because [math]g+f=j+h[/math]

>>11489461
Yes, but that was in the 40's, although I'm more interested in the foundations aspect of it, which came 2 decades later.

>If the answer is yes, then why?
Because you have a language that describes things structurally/relatively. This is in contrast to e.g. set theory, where typrically all objects are nested stacks above the empty set. The language ends up being ridden with artifacts.
Now whether you care for all of that in the first place is another question, but that's more of a question on what your purpose in life is in general.

>>11482035
just a reminder that math and numbers is a big load of bullshit with no practical or philosophical application just a bunch of made up squiggly lines for losers who aren't smart enough for the humanities

>>11489465
yea I first also thought about something like this.
>>11489478
how do you justify writing it as +?

>>11489556
>numbers have no application
nigger ever heard of prices?

>>11489366
Uh, no, why?

>>11489598
>how do you justify writing it as +?
Because the entire thing is abuse of notation and analogies, I suppose.

>>11488847
One can prove that homotopy groups of CW spaces are finitely generated groups with finite presentation, so we can write them as a(n invertible) language. This then allows you to construct interesting loops on spaces using formal language theory. For example on the torus [math]\mathbb{T}^2[/math], the Thue word lets you construct a formal loop [math]S^1\rightarrow\mathbb{T}^2[/math] that never winds around the same "hole" three times in a row while having infinite winding number.

>>11489609
Just because someone asked a similar question recently.

>>11489629
Makes me think of topics in this direction
https://en.wikipedia.org/wiki/Geometric_group_theory

Maybe though he had use of topology for language theory in mind, not so much the other way around. I imagine, since word problems and all are close to logic itself, you'll often be able to express some subtopics in all fields using enumerative and combinatoric questions.

>>11489603
>taking obvious bait

>>11486389
>>11486477
lattice

>>11489622
I mean aren't you losing important information if you just take composition of functions as abelian?

>>11489677
I'm not taking it as abelian, tho, I'm using plus because simplices.
Specifically notice how we have [math]g+f=j+h[/math] implies that [math]-g-f=-j-h[/math], and then [math]-g-f+h+j=-j-h+h+j[/math]

What's the best book on homotopy theory?

>>11489815
Which covers homotopy type theory as well. Thanks.

>>11489820
lel

>>11489815
>>11489820
There exist homotopy theory books which also cover homotopy type theory?
I do know that there are htt books that cover homotopy theory, tho.

>>11489963
>>11489976
?????????
Am I being trolled???

>>11489815
>>11489820
>>11489963
>>11489976
>>11490007
please discuss your gay garbage elsewhere
just throw those books away and study algebraic geometry

>>11482615
Linear Algebra and its Applications, Lax

>>11490015
Algebraic geometry and homotopy theory are deeply interconnected

>>11486505
>>11486528
go back you fucking faggots

>>11487913
omg I can read it. I can read it and I understand it.... have I finally made it??

>>11489755
shouldn't it be -f-g = -h-j ?

>>11489178
lofi hip hop beats to relax/study to

>>11489815
Fomenko Homotopical topology

https://www.youtube.com/watch?v=SYWaQe0qmNI&feature=youtu.be
based

>>11490423
beautiful song

>>11489388
what books on philosophy of mathematics do you recommend?

>>11487996
I enjoyed this
https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/pragacz.pdf

>>11488339
Learn theorems that rely on them. Keep learning new theorems continuously so nothing leaves your mind. If you forgot it, then you didn't really need it.

>>11488931
Span takes some vectors and gives a subspace. The smallest subspace that contains the vectors in particular.

>>11489461
It makes some things very convenient to express. Say for example a presheaf. Without category theory it's pretty awkward. I don't really see why some people have a problem with it. It comes out naturally when you try to look at certain objects in topology or geometry. I think some people take it too far though, look at nLab for what I mean.

>>11489815
Bertlmann. Anomalies in Quantum Field Theory.

>>11490649

>>11490649
What are some of those anomalies? Please explain in layman's language

>>11490649
He was actually my QM Prof during my undergrad years.
At one point he didn't seem much chance to make it in QFT and went into optics. Also related to the fact that we have some famous quantum optics experimentalists around here who push it.

>>11490588
>Modern Algebra and the Rise of Mathematical Structures
https://www.amazon.com/Modern-Algebra-Rise-Mathematical-Structures/dp/3764370025
Corry is a historian

Although I kind of argued against reading such texts, didn't I? Things have changed so much after around 1880 that I don't think there's good definite views I'd agree with. In any case, plato.stanford.edu has enough content for a long while

>>11487996
let me shill this
https://youtu.be/PrF5eMXr0Bc

>>11490093
making it is a myth, the only people I would grant having made it are those which decidedly stopped trying to make anything

>>11490705
The pion vertex in QED [math]\propto\int_M F^2[/math], the chiral anomaly [math]\propto \int_D F^-[/math]. Anomalies in general come about when your partition function [math]Z[\psi^g] \neq Z[\psi][/math] in the thermodynamic limit, where [math]g\in\operatorname{Map}(M,G)[/math] is a gauge transformation. In terms of the normalized invariant functional measure [math]d\Psi[/math], the partition function [math]Z[\psi^g] = \int d\Psi e^{-S[\psi^g]}[/math] picks up an extra functional Jacobian [math]|J_{g^{-1}}(\psi)| = e^{-S_g[\psi]}[/math] corresponding to the gauge transformation [math]g[/math] when you send [math]\psi \rightarrow \psi^{g^{-1}}[/math] in the functional integral. This functional Jacobian can be cancelled only if you can renormalize your action such that [math]S \rightarrow S
+ S_\text{ren}[/math] where [math]S_\text{ren}[\psi^g] - S_\text{ren}[\psi] = S_g[\psi][/math]. When you can't, there's typically a topological reason for it: for instance, if [math]S[/math] is an NLSM wth [math]\psi:M \rightarrow \Sigma[/math] then an anomaly occurs if holomorphic bundles [math]V\rightarrow \Sigma[/math] encounters an obstruction to a spin structure in the pullback; i.e. [math]\psi^*V \ominus \mathcal{S}[/math] is non-trivial, where [math]\mathcal{S}[/math] is the spinor bundle on [math]M[/math].

Undergrad here, explain to my why the Riemann Hypothesis is so hard to solve

I'm taking a course in matrix theory using the text book "Matrix Analysis for Scientists and Engineers" by Alan J. Laub, and the textbook is somewhat lacking. My education in linear algebra focused almost entirely on computations, we did not do a single proof the entire class(mediocre state school), hence my understanding of theory is severely lacking. I should note I am comfortable with analysis and proofs, just not in linear algebra/matrix theory. I struggle even with finding pseudoinverses of arbitrary matrices, and finding more technical SVDs, especially for arbitrary matrices.
What books would yall recommend on the subject for someone in my position?

>>11490711
It makes sense that books on PoM treat the subject as a dogma. I'm specifically interested in learning about PoM as a whole and the way we conceptualize modern mathematics. I have started with "Mathematics and it's History" by John Stillwell, to get an insight on the development of mathematics up until recently. I was thinking of reading a book on the overview of PoM, which I ended up choosing "Think about Mathematics" by Stewart Shapiro, followed by reading "Concepts of Modern Mathematics" by Ian Stewart. Would you suggest Corry's book as a substitute for Shapiro, or Stewart? Or would you suggest reading all three? I'm a little lost on what would give me a nice overview and introduction to PoM.

>>11490739
Bellman
halmos
horn&johnson
lancaster
meyer
ortega

>>11490726
I'm seriously starting to think you [math]dothisshitonpurpose[/math].

>>11490757
Also prove all the theorem in lay only reading definitions and theorems

>>11490765
Nah, 4chan TeX fucks up if there are too many [math]math[/math] environments in one line.

>>11490747
Haven't read those books, sorry.
But it sounds like you already got a few great options. I'd think that if you wrote those books or the first order them, you have a general birds eye on the common topics.
Again, as I said, I feel doing formal logic is almost a substitute and it's also sort of the playground where the non-human aspects of math may be decided.
Why are you into it, what are you looking for?

>>11490780
Try spacing things out, indead of [mathtag]\frac{1}{2}[/mathtag], do [mathtag] \frac {1} {2} [/mathtag]. Some anon hinted at this a few years ago and I've since not had any problems.

>>11490780
[math]No[/math] it [math]doesn't[/math] you [math]fucking[/math] retarded [math]motherfucker.[/math]

>>11490780
just use \text{}

>>11490729
There's nothing to be 'solved' in the RH you idiot, what we need is to PROVE wether or not it's true. It most likely is, but with current mathematics it's impossible to logically demonstrate it, it's proof will most likely be part of a whole new area not discovered yet where the RH will be a simple particular case, similar to what happened to Fermat's Last Theorem

>>11490726
That's not layman language, I couldn't understand a single thing about your post, what does it say about the universe in general? What are the anomalies in reality? Explain without using formulas ou technical terms please

>>11490808
>it's proof will most likely be part of a whole new area not discovered yet where the RH will be a simple particular case
t. Jacob Lurie,
>similar to what happened to Fermat's Last Theorem
t. knows literally nothing about what he's talking about.

>>11490808
>with current mathematics it's impossible to logically demonstrate it
The dumbest thing I have ever hear
Either a statement is decidable or it is undecidable. Are you claiming that RH is undecidable and that we are going to add an axiom? Are you retarded?

>>11490814
Looks like you're the one completely lost here, FLT was a particular case of an elliptic curve, an area that was just discovered several years after Fermat

>>11490822
I'm saying it's literally impossible to prove its veracity, Gödel already showed that some problems will never be proved unless we keep adding more axioms, thus creating more areas of math. The RH will probably be proved in the far future where a new area with new axioms and perspectived have been discovered.

>>11490822
Since he made the comparison to FLT, I think he meant to say that with the current collection theorems that we have, RH wouldn't be a direct consequence, and that we'd have to make some logical leap like Taniyama-Shimura.

>>11490834
Exactly.

>>11489178
i dont usually listen to music with math but this shit is really cool:
https://www.youtube.com/watch?v=EmkX6Yoml8E

>>11490828
>an area that was just discovered several years after Fermat
Do people really do that? Just go post about things without knowing absolutely basic stuff, like Fermat doing research on elliptic curves, and being the first to classically give their group operation?
At the absolute least read Conics and Cubics, you autist.

>>11490849
I seriously have no idea why anyone would think "nonsingular cubic" would be something that it took time for people to find out about.

>>11490857
*non-singular cubic of this form,

>>11490830
>>11490834
>>11490839
I wasn't part of the conversation before, but why does the last post here says "exactly" when the first and the second are different answers (new axioms necessary vs. no new axioms nececessary).

Anyway, I doubt anybody would want to add axioms to math to solve an analysis question

>>11490884
So do you think that the RH will be proven using number theoretic tools or complex analytic tools? Robin's inequality seems to have great promise for a proof, we just have to work with it more.

>>11490811
>what does it say about the universe in general
It suggests that ground states might be topologically degenerate.
>What are the anomalies in reality?
Cosmic string in cosmology, monopoles and the AHE/QHE in condensed matter, etc.
>Explain without using formulas ou technical terms please
Think of fields as a big bed sheet and anomalies are the clumps of cum your bf left on it.

>>11490896
Ok, that's a little better, bur what is a ground state? What does topologically degenerate means?
What is a cosmic string and why's it important in cosmology?

>>11490884
>>11490888
Can the proof of the RH be in Several Complex Variables? It's a fairly recent area.

>>11490938
>SVC
>fairly recent
No.jpg

>>11490757
>>11490774
Thanks lad.

>>11490933
Sure let's turn this into /pg/.
>what is a ground state?
Ground states are quantum states of the lowest energy.
>topologically degenerate
Degeneracy of the ground state means you have multiple states with the same lowest energy, and topologically degenerate means the distinct ground states are labeled by a topological invariant. The pion vertices, the chiral anomaly and more generally the obstructions I mentioned are such examples of these topological invariants.
>What is a cosmic string and why's it important in cosmology?
It's a string of defects in spacetime, like a 1D blackhole/2-brane. It's important because they show up when studying possible mechanisms for spontaneous symmetry breaking in order to explain e.g. DM, DE, strong CPT violation, and other things in cosmology and HEP.

If you're interested it'd be good for you to read up on intro QFT from e.g. Weinberg or Peskin-Schroeder, it'll give you a more solid understanding.

>>11490955
You're very patient and kind, and although I didn't understand much I'm still thankful for the explanation and recommendations.

>>11490884
> why does the last post here says "exactly" when the first and the second are different answers
idk man, I just like trying to pretend to understand what people write here

>>11490975
You're quite welcome sweetie.

How can I succeed in mathematics if I'm such a dumbshit?

>>11490884
There's no contradictions between my posts (First and third) and his (second), he just said what I was trying to convey into words but might have been confused

>>11491009
Algebra more specifically algebraic topology more specifically homological algebra more specifically category theory

i want to talk about math with someone. does anyone feel like having a math chat? post something youre into!

>>11491036
I like adding and subtracting whole decimal numbers

>>11491038
For example 6 + 87 = 93 and 77 - 66 = 11

>>11491036
to start us along, im currently working on understanding the intuition behind taylor series in real and complex fields, and next up laurent series..
.
.
i think its really interesting the simple algebraic expansion of 1/(1-n), ive intuited mostly and i see how it relates to polynomial matching and then i thought, well it crops up in complex functions due to fitting in with the cauchy integral formula. i think some others couldve worked but they mightve had a similar form. the other curio is, why is the proof in complex so much more straightforward seeming than in reals?im gonna go back and read the real proof right now! then ill ponder the differences, ponder reals harder because they always bugged me, and move on to laurent

next up im interested in drawing some complex graphs in a variety of ways (though i might lazy out) to understand conformal maps geometrically

>>11491034
Isn't cat theory supposed to be really hard? I think the easy fields are graphs, statistics, dynamical systems etc

>>11491038
>>11491041
ooh. that one is tough and interesting. first off why decimal? i always thought it was interesting how 0+5 = 5 but the set of ints from [0,5] contains 6 elements. always made compsci harder. but its interesting to work out a full adder in my brain, following the loops and moving the symbols. is decimal more efficient? seems faster than binary cause fewer loops and works cause i can store more symbol types.

>>11491041
Prove that any whole number ending in seven when subtracted by 6 gives a number that ends in a 1
>>11491047
>t. harry gindi

>>11491047
hes memeing you because anons here hate philosophy and category theory is philosophy. statistics is gay, if you suck at math why not be a physicist? or perservere

>>11491049
Apparently humans can hold 7 objects in their head at a time so maybe base 7 in the best. But I like decimal because I learned it first and I only later practiced with hex and binary outside a school setting.

>>11491050
7-6 = 1. a number ending in 7 is of the form n(10^k) + 7. because n10^k are the only numbers ending in 0 and so adding 7 makes them end in 7. n(10^k)+7 - 6 = n10k + 1. so it ends in 0 then you add one now it ends in 1.

>>11491058
i dont know if thats true. i frequently memorize 8 digit codes by grouping them into two of fours

>my theorem of choice?
> [math]\pi : \mathbb{Z} \rightarrow \mathbb{Z}_n [/math] being a ring homomorphism, of course, never found a result more natural, concise and useful

>>11491061
>n(10^k) + 7
not true consider 117

>>11491066
>i frequently memorize 8 digit codes
Not everyone has autism desu

>>11491073
oh yeah. then define an array of various 10^ki summed up with each ki being any positive int

>>11489178
>music you listen to when you're studying math?
https://www.youtube.com/watch?v=YoFsuoHBA58

>>11491069
can you post a proof that the pi function from Z to Zn is a ring homomorphism? and what is a ring homomorphism?

>>11491082
>and what is a ring homomorphism?
Out

>>11491085
just explain it retard

>>11491069
>>11491082
>pi from z to z
pi = 3

>>11491052
Physics is harder than maths.

I'll persevere again this year, if I keep failing then I'll end myself.

>>11491087
how is pi = 3 a proof of that? by the way your proof is false because pi is defined not to equal 3.

>>11491090
dont end yourself anon. even if you fail, theres industry jobs you can get by finishing your degree with easier courses or moving to compsci. youd become rich, and get to do comfy math at slower pace. life is worth it

>>11491094
I don't want 'industry jobs', it's not about jobs or money, it's about amounting to something in math, it's about leaving at least a very small contribution in some field, proving an unproved theorem. Mathematics has been my life since I was 15, I made a vow to become someone respected in math, to do great contributions, it hurts when nothing goes as you expected and if I really suck at marh then I rather not keep living such a worthless and boring life anymore.

I was so focused this year, studying for hours every single day, but then that shitty chinese virus came and the classes got canceled, now I'm back to being a NEET, a worthless trash.

>>11491111
are you retarded anon? you now have extra time to study. just delve into a field and learn it deeply, im sure youll find something to prove

>>11491118
I can't concentrate on home, there are too many distractions. When I'm on college at least I study on the library where it's peaceful and quiet, but now I have to stay home 24/7, with loud people watching TV and the computer right by my side, tempting me all time.

>>11491126
Stop making excuses.

Haa hha math fun numbers hit eachother n go splat n make more numbrrs haha bing bong number hit 1 2 3 four 5 hehe work count eas e heehee haha

>>11491161
i laughed a bunch

>>11491090
>>11491111
>>11491126

Look I don't know if you'll read this but please understand there is a life beyond math. It may seem like math is the one true pursuit, that mathematical immortality is something you want but at the end of the day, none of us are Euler, Newton, etc. You may spend your entire life pursuing a field to write some seminal paper but all you will be is some book in some dusty annals in some old forgotten academic library. That is what you fight so hard for. If you want to keep doing math, do it because you love math. Because you want to learn more, not because you want the respect of others.

>>11491047
category theory is easy. it's abstract, yes, but that don't mean it's hard

Let [math]\omega[/math] be a [math]C^1[/math] differential k-form on some open [math]U\subset\mathbb R^n[/math]. If [math]f: \mathbb R^n \to \mathbb R[/math] is a continuously differentiable function such that [math]d(f\omega) = 0[/math], I want to show that [math]\omega \land d\omega = 0[/math] IF k is odd.

This is my proof:

[math]d(f\omega \land \omega) = d(f\omega) \land \omega + (-1)^k+1 f\omega \land d\omega
= 0 \land \omega + f\omega \land d\omega
=f(\omega \land d\omega)[/math]

Now, [math]f\omega \land \omega = f(\omega \land \omega) = 0[/math], for [math]\omega \land \omega = 0[/math]. Then [math]f(\omega \land d\omega) = 0[/math] which much mean that [math]\omega \land d\omega = 0[/math], for [math]f[/math] is never 0.

Even if k were even, I'd end up with [math]-f(\omega \land d\omega) = 0[/math], and if [math]f[/math] never vanishes then [math]-f[/math] never vanishes, so for the same reason as above, [math]\omega \land d\omega = 0[/math]. What am I missing here?

>>11491358
>(−1)k+1fω∧ω
This is supposed to be [math](-1)^(k+1) f\omega \land d\omega[/math], sorry

>>11491082
A function between rings which preserves ring structure.

>>11491358
[math](f\omega)\wedge d\omega\neq f(\omega\wedge d\omega)[/math]

IF YOU LOVE MATH SO MUCH WHY DON'T YOU JUST MARRY IT

>>11486156
Consider the integers modulo p, for p a prime number.

>>11486889
Urysohn is actually useful for partitions of unity, which I see discussed in my physics books.

>>11491373
Really? Given a constant c and alternating k-tensors [math]\omega[/math] and [math]\eta[/math], [math]c\omega \land \eta = c(\omega \land \eta)[/math]. This is a fact.

Is it not true that [math](f\omega \land d\omega)(p) = f(p)\omega (p) \land d\omega (p)[/math], so that [math](f\omega \land d\omega)(p) = f(p)(\omega (p) \land d\omega (p)[/math]?

>>11491388
>f(p)(ω(p)∧dω(p)
f(p)(ω(p)∧dω(p))*

>>11490955
>Weinberg QFT
My nigga.

>>11491378
This is not arbitrarily large cardinality

>>11491370
what does preserving ring structure entail? what traits must be preserved?

>>11491358
Sorry latex is rusty, but how does f(w ^ w_ = 0 imply f(w^dw) = 0?
hypothesis: d(fw) = 0
fact: d(fw) = df^w + (-1)^(k+1)fdw
k is odd
iff fdw = df ^ w
iff f(w ^ dw) = w ^ (fdw) = w ^ (df ^ w)

I think you can figure out the rest.

>>11491433
>fact: d(fw) = df^w + (-1)^(k+1)fdw
Shouldn't it be d(fω) = df ^ ω + fdω, since f is a 0-form?
>how does f(w ^ w_ = 0 imply f(w^dw) = 0?
Well, fω ^ ω = f(ω ^ ω) (I think so, at least), and the wedge product of any tensor (or differential form) with itself is 0, so that d(fω ^ ω) = d(0) = 0.

Since d(fω) = 0, we have that d(fω ^ ω) = ((-1)^(k+1))fω ^ dω.

>>11491416
Doing ring operations in the first ring then using the homomorphism is the same as using the homomorphism first then doing ring operations in the second ring.

>>11491047
you're delusional. there aren't "hard" and "easy" fields in math; they are all hard if you achieve a certain level. also you shouldn't pick a field of study based on how "easy" it is

>>11491526
how does this compare with an invertible linear map? the way i intuit those is that all the vectors are simply relabeled. i think that property you mentioned would hold, it seems the same as simple linearity + the condition that all the vectors exist, dont go to 0 after operation

>>11491395
retard. do you even know how big prime numbers can become

>>11491581
certainly not bigger than aleph-zero

>>11491577
homomorphism means that the map preserves the algebraic structure. ring homomorphism preserves ring addition and ring multiplication, vector space homomorphism (which is just a linear map) preserves addition and multiplication by scalars etc. but a homomorphism is not necessarily invertible, that's not part of the definition. if a homomorphism is invertible, then it's called an isomorphism and in that case it really is just a relabelling of elements.

>>11491598
would non injective/surjective linear maps be homomorphic? i ask because theyre not invertible but i think the ring structure might work even given that lots of vectors go to 0.

whats the intuition for non invertible homomorphisms? that its just like invertible with some elements getting got in some kind of trap, such as a 0 trap?

>>11491630
example of non-surjective linear map is an embedding of a plane into space as the (x,y)-plane

f(x,y) = (x,y,0)

example of non-injective linear map is an orthogonal projection of a space onto the (x,y)-plane:

f(x,y,z) = f(x,y)

>>11491567
I'm pretty certain triangle geometry is simpler than Riemannian geometry.

>>11491636
except no one would say "triangle geometry" is a "field of math"

>>11491637
Trust me, triangle geometry is thriving. See Encyclopedia of Triangle Cenetrs https://faculty.evansville.edu/ck6/encyclopedia/ETC.html

>>11491647
Some people still do it. High school students preparing for olympiads still do it. It is not "thriving", there are no people doing a PhD in "triangle geometry", it is not an active field of research.

>>11491635
would those be homomorphic? what about my second question?

>>11491676
>>11491630
>whats the intuition for non invertible homomorphisms?
injective + surjective = invertible. relabelling of elements

injective + non-surjective: it's a fact that the image of a ring (group, vector space) homomorphism is a subring (subgroup, subspace). this is an embedding of a smaller algebraic structure into a larger algebraic structure. the point is that if you restrict the codomain to the image, the map becomes an isomorphism. example: [math]f\colon \mathbb{R} \to \text{2}\times\text{2 Matrices}, f(x) = \big(\begin{smallmatrix} x & 0\\ 0 & x \end{smallmatrix}\big)[/math]. clearly this is just an "implementation" of the real numbers as a subring of the matrix ring.

surjective + non-injective: the domain is partitioned into sections where each section consists of elements that are mapped to the same element. in particular, the section containing zero is precisely what is called the kernel. intuitively all sections (called cosets) look the same and they are neatly stacked next to each other. what the mapping does is that it squishes every section to a single point. example for vector spaces: [math]f \colon \mathbb{R^3} \to \mathbb{R^2}, f(x,y,z) = (x,y)[/math]. this is the orthogonal projection onto the (x,y)-plane. the kernel is the z-axis and the cosets are affine lines parallel to the z-axis (check that these are precisely the sets that are all mapped to a single element). in this example you can see that if you stop considering individual points and you work with these whole lines instead (i.e. two points lying on the same line will be the same thing to you), the map becomes injective. it will actually become an isomorphism, this is the content of so-called first isomorphism theorem.

non-surjective + non-injective: combination of the two. the domain is partitioned into cosets, cosets are squished, and the resulting algebraic structure is embedded into some larger structure.

>>11491676
>would those be homomorphic?
a map either is or isn't a homomorphism. "being homomorphic" doesn't mean anything. (maybe you've heard "homEOmorphic" somewhere, but that's topology, a different thing entirely.)

>>11491681
sounds a lot like linear algebra intuitions, except that in lin alg inj and surj go together. are all linear maps homorphic?

what does non homomorphic look like? is homomorphic basically just a word for linear, and non homomorphy is non linearity?

er...homomorphic iff linear?

>>11491726
wait, i kinda negated myself because linear algebra does a thing homomorphy does not. but im thinking maybe theres other types of linearity than the kind expressed by linear maps on vectorizable objects.. btw im sleepy so dont jduge my stupids rn

>>11491787
>>11491787
>>11491787
>>11491787

you go hear nao

>>11491393
What's wrong with it?