/sci/ - Science & Math

i forgot to include the official /mg/ curriculum in OP.

>>10711651
>i forgot to include the official /mg/ curriculum in OP.
Mathematicians use "we", not "I".

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

This seems like some good shit to me

>>10712186
>wikipedia

What [math]\LaTeX[/math] setup do you guys use for writing mathematics? I am interested in all aspects from editor, compiler, macros, templates, snippets, keyboard layouts etc.

Mine:

vim with 800 lines of .vimrc
vim plugins: ultisnips (https://castel.dev/post/lecture-notes-1/)), vim-airline, vifm.vim, vimagit, vim-surround.
Some of my snippets uses this pure .vimrc solution by Luke Smith: https://github.com/LukeSmithxyz/voidrice/blob/archi3/.config/nvim/init.vim
texlive compiler
zathura pdf-reader
Microsoft Natrual Ergonomic Keyboard 4000
Python script for inserting the preamble. I write my files as .kex file without the preamble which this script translate into a .tex file which then gets compiled.

>>10712274
Oh, and

Caps Lock as Escape in vim is crucial. Using Xcape to use Caps Lock as Ctrl at the same time. Using Esc as Caps Lock. And in vim buffers are better than tabs. Using vim-airline to list the buffers on the top.

>>10712274

I thought people just paid someone else to do it

>>10711657
we think youre retarded

>>10712308
I don't think it's common. I think if you are collaborating, it's better that only one person does the first draft, then the others can comment and edit afterwards.

>>10712274
>>10712289
>>10712335
please dont spam this thread with /g/-tier discussion again thanks :)

what THE FUCK is an opposite category?

>>10712427
Sorry

>>10712438
apology not accepted you passive bitch
/mg/ is a vitriolic thread
>>10712431
what the fuck do you think it is arr-tard

>>10712431
>what THE FUCK is an opposite category?
take any category, then turn all the arrows around

as a simple example, consider the category of axioms, what is it's opposite category?

>>10712431
What happens when you put a little "op" on top of the category. It means nothing, its just flair.

>>10711631
No no no fuck this meme, fuck the sad einstein poster and fuck the Feynman diagram. We can't the most stupid meme on 4chan that as op.

>>10711651
Also fuck the meme chart.

>>10712453
how does this make sense? do you assume that for every morphism going one way there should be another morphism going the other way?

Why is it that (1 + 2 + 3 + ... + n)/n is different than { [ ( 1+2 )/2 + 3 ]/2 + 4 ... }/2 + n}/2 ?

>>10712461
No. For each arrow you consider its head as its ass and its ass as its head. In other words you just reverse all arrows.

>>10712463
Because the two expressions yield different results.

>>10712463
Why not?

>>10712448
>apology not accepted you passive bitch
I know you're secretly checking out my LaTeX setup dude. No need to hide it. I know you're yearning for that .vimrc.
>/mg/ is a vitriolic thread
Hm, I've always thought /mg/ was pretty chill. People helping each other with surprisingly little fuckery. Always a miserable asshole or two though.. I guess that's usually you then.

>>10712509
but the arrow is something. how do you reverse an arrow in grp without an isomorphism?

>>10712518
You create a new category where objects are the same, but arrows are reversed. It's contrary day. This is used mainly to describe situations when directions get swapped, i.e. contravariant functors.

Lads, my autism is too weak. Please give me properties of general finite groups generated by two elements.

>>10712554
dihedrals, right? use the presentation with the reflections and rotations, that works I think.

>>10712599
No, all finite groups generated by two elements.

>>10712518
You know the reversed arrows aren't required to be inverses to the original arrows right?

>>10712289
>not using superior software

>>10712815

should I point out the basedness of this post

>>10712815
I am going to switch to emacs and run evil mode. Then I still will need to use the configs described in that post. Modal editing with vim bindings is clearly superior.

>>10713009
Epic, thanks for the blogpost.

>>10713013
>>3 short sentences
>thanks for the blogpost
wat

>>10711631
I just want to go math monk mode to have something to get me through my depressed days and to entertain me during my loneliness, could one of you kind anons direct me towards the best resources to learn math (inforgraphics books, and links much appreciated), my dream is to go study in some french research school in the future!

>>10713138
Bourbaki, from the beginning. You can actually get the first few books at greatly reduced prices through the end of June.

>>10713138
see >>10711651

>>10712186
From Brouwer's own uni:
http://festschriften.illc.uva.nl/D65/terwijn.pdf

>>10712274
>vim with 800 lines of .vimrc
>snippets
I'm curious. What does all that do?

What's the mathematics behind advance wars?

>walked into the most productive researcher's office
>The board is clean, there are not piles of paper scattered, it's just him sitting on a chair with crossed arms looking out the window

based

>>10712463
distribute it out retard

>>10714012

>algebra hacks

what's the intuitive/visual explanation

>>10714015
theyre totally fucking different
thats like asking for the intuitive/visual explanation of 0 =/= 1

What do they mean by "coordinates as functions on Y" in this proof?

----------------------------------------------------------------------------------

Theorem: A finite mapping between affine varieties [math]f:X\to Y[/math] is epimorphic.

Let [math]y\in Y.[/math] Let [math]m_y[/math] be the ideal of [math]K[Y][/math] consisting of functions that vanish at [math]y[/math].

If [math]t_1,...,t_n[/math] are coordinates as functions on Y and if [math]y=(a_1,...,a_n),[/math] then [math]m_y =(t_1 - a_1,...,t_n - a_n)[/math]. [...]

>>10714120
The coordinates of the ambient affine space restricted to Y. If [math]Y[/math] lies in [math]\mathbb A^n[/math] and you write [math]K[\mathbb A^n] = K[t_1, \dots, t_n][/math], then you could take as coordinates functions the [math]t_1, \dots, t_n[/math]

>>10712815
Jesus, even the logo looks like chad, I can see the hair, nose, and massive jaw. Virgin chad image when?
>>10713138
Why a french school specifically?

>>10714142

The proof goes on to state that

"The set [math]f^{-1}(y)[/math] is empty iff [math](f^*(t_1)-a_1, ..., f^*(t_n)-a_n) = k[X][/math]."

where [math]f^*[/math] is the pullback, [math]f^*:k[Y]\to k[X]; t\mapsto t(f)[/math].

If the LHS of the equation denotes the generators of an ideal, what property does each generator have that causes this to be true?

>>10714240
This is more or less the statement of the Nullstellensatz. The Nullstellensatz tells you that a closed subset [math]Z[/math] of an affine variety [math]X[/math] is empty if and only if [math]I(Z) = k[X][/math]. The thing to understand is that the ideal of definition of [math]f^{-1}(y)[/math] is [math](f^*(t_1) - a_1, \dots, f^*(t_n)-a_n)[/math], do you see why that is ?

What textbook/s would you use to consolidate the fundamentals in preparation to undegraduate courses?

>>10714261

I now understand that it's a statement of the Nullstellensatz because you're saying "If they all share a common zero at y, the ring doesn't contain one and it forms some ideal. If they don't share a common zero, the ideal is all of [math]k[X][/math] because one is a generator."

They're also saying that [math]f^*(t_1)-a_1 = 0, ..., f^*(t_n)-a_n = 0[/math] are equations for [math]f^{-1}(y)[/math] as a variety.

what is the reason for this? I can't figure it out.

>>10714303
>what is the reason for this? I can't figure it out.
Just spell it out: an element x is in [math]f^{-1}(y)[/math] if and only if [math] f(x) = y[/math], ie. if and only if each coordinate of [math]f(x)[/math] is equal to the corresponding coordinate of [math]y[/math], ie. if [math]t_i(f(x)) = a_i[/math] for each i

>>10714202
>Why a french school specifically?
A professor of mine studied at eth zurich and told me about the quality of mathematics done at european universities, I really have only ever heard about there being math dedicated research institutes in france

>>10714202
>Why a french school specifically?
Why not?

>>10714480
>>10714202
also alizee

>>10712274
I was using Vscode, but you (or someone else) in the last thread memed me into looking at https://castel.dev/post/lecture-notes-1/ and trying to configure vim for myself.

I really want to try it in the next lecture to see if I can keep up better, need a bit of training before that I assume.

>>10711631

>>10712274
ShareLaTeX

>>10714502

it's the chance of getting at least 4 heads (1/2)^4

Here's some thoughts of my angle to set/category/set-theory and in the last 30 minutes a start of a book review

https://youtu.be/IR0GkYoRzeE

>>10714480
I can see that, the French have a lot of respect for mathematics so I can see why people might want to work there. Though positions at places like IHES seems really tough to secure. There are places in the US like MSRI which are first class as well.

Can you even formulate math without making implicit use of set theory?

>>10714944
You can use category theory which supplants set theory

>>10714956
But categories are usually defined with sets and classes

>>10714309

> I couldn't figure that out in a few hours

>>10711631
I backtraced it it's coming from a runescape server somewhere in southern Poland

>>10714969
>usually
and?

>>10714944
Try the constructivisrs, of course you can be really pedantic about this and try to rephrase their definitions in set theory, but that kinda misses the point.

>>10711631
One billion

>>10712815

>>10714278
bump

>>10714502
Is it: {1-[(6Choose3)/2^6]}/2 ?

>>10715136
>and?
And how do you define categories without sets and classes?

>>10715672
It's provably impossible. All math is inherently based on set theory, as we know.

>>10715579
>slightly heavier
(E)IGHT (M)EGS (A)ND (C)ONSTANTLY (S)WAPPING

>>10715579
Fuck you! Fuck you, you dumb nigger! I'll C-c C-w C-d your ass, and C-x M-x C-a C-a to your face. If you never planned your own birthday in org-mode, you don't know what love is.

>>10715679
>All math is inherently based on set theory
Factually incorrect, how did mathematics exist before the end of the 19th century?

>>10715872

>how did mathematics exist before the end of the 19th century?

It was naive mathematics ;P)

>>10715872
It didn't. Set theory is synonymous with "mathematics" these days.

>>10712274
>kex
kek

>>10716060

type theory is actually used a lot in practice

Year numbers are ordinals or cardinals?

Don't know if this is the place to ask if not in /sqt/,
what's a good book for first year calculus?

I kinda failed my second semester math and I have a removal exam in July. I did pretty well in the first semester though since it was differentiation and the basics of integration.
The book we used is The Calculus 7 by Leithold, but the only pdfs online are in Spanish.

Is Calculus by Strang all right?
Should I just be watching Khan Academy videos instead?

>>10716245

>>10716245
>what's a good book for first year calculus?
Spivak

>>10716548
That book is dead 'ard

>>10712274
I use an ebic Vim + Emacs + Latex + Word combination

>>10716706
Yes, but it's a great book nonetheless.

If you want something easy, go with Stewart.

Any tips for applying to PhD programs fellas?

I'm a rising senior and am slated to graduate with both my bachelor's and master's in Spring 2020. My school has shit for resources when it comes to this stuff and while I'm starting to reach out to professors at my university and others, I'm looking for help wherever I can get it.

As I understand it, pretty much everything from research experience to grades is shit compared to recs, so I'm trying to feel the waters for those already, but I'm mostly looking for general advice, guidance, and tips if any of you have it.

>>10712431
Think of category theory as archery. But instead of shooting the targets, you are now shooting yourself. Similar to retards and the mentally insane, that's how opposite categories work.

>>10717705
>Any tips for applying to PhD programs fellas?

Don't

>>10717755
>but instead of shooting the targets, you are now shooting youself.
kek, gonna use this. thanks.

>>10717755
>But instead of shooting the targets, you are now shooting yourself.
Extremely accurate.
>>10716235
Remind me, was there a 0 A.D. or a 0 B.C.?

>>10717705
Make friends with tons of people so you can get free travels.

>>10717705
You need to provide a little more info if you want more specific answers. In which country do you want to do your phd, which field are you working in, do you have infinite money available?

>>10717948
I'm probably aiming to stay in the USA, but I'm willing to travel.

I'll also have a BA in German and would be willing to put in the effort to get language certification, so the most likely place outside of the US would be Germany.

I'm most interested in algebra, in particular group theory, but that's part of my problem. As passionate as I am about the field, I don't know if I'm well-versed enough in what the "modern" field entails to know if this is what I want to study, and there are major fields (algebraic topology, algebraic geometry) which I have almost no familiarity with.

My math education was decent and as far as I can see pretty much standard for these things in the US, so I don't think my knowledge is severely lacking compared to other people in my position, but I guess I could be wrong.

I'm getting in contact with various professors and it's going well but I don't know how acceptable it is to own up to my ignorance about these things.

I don't feel like (in the US at least) the level of education I have gives you much indication as to what is currently happening in the relevant fields, but in any case, I'm looking into strong algebra programs primarily.

I also certainly do not have an infinite well of money by any means.

>>10717813
>0 B.C.
This

>>10717813
>Remind me, was there a 0 A.D. or a 0 B.C.?
Yes.

>>10712463
lol

>>10718044
Just curious whats your gpa, gre, and research experience

>>10718122
My overall math GPA is a 3.9 if I remember correctly. My cumulative GPA is slightly higher, so it's not I'm throwing up red flags anywhere else in my course work.

I'd have to pull up my transcript if you wanted an exact breakdown, but if I'm remembering correctly I got a B in one undergrad class, an A- in one master's class and all A/A+s (both are 4.0 in our system) in all my other math courses.

At this point, I've completely finished all my undergrad course work and in the next year have to take 6 grad courses to leave with my master's.

I took my math subject GRE this past spring and did very mediocre, it was somewhere between 45-55th percentile, I don't remember exactly where. I'm retaking in the fall though.

I haven't taken the GRE general yet.

My research experience is pretty much nothing of note. I essentially "shadowed" a student at our grad center and did some very little stuff with him, but I don't have an REU or anything like that. The professor who set this up for me is someone who I could ask for a rec, though.

I also have an independent study in this upcoming semester with another professor, but this isn't on research-level stuff by any means.

Many of my professors have encouraged me to do this, but I always feel like a very underwhelming candidate. My research experience is minimal, I'm pretty much just grades and recs, but our department supervisor is usually pretty good about shutting people down, so I don't know.

Thanks for taking the time to respond, in any case.

>>10717259
Please tell us more

>>10716060
>It didn't.
???
>these days
And?

>>10712274
sudo apt-get install texstudio
If you do anything more you're an autistic /g/tard who spends (read: wastes) too much time on "muh personalized config" instead of actually doing math. And I bet the main reason you do it is so you can dick wave at your local autism circlejerk (like /sci/), rather than actually getting any sort of convenience out of it. Hence why made this post, rather than actually talking about math.

Data Science or Mathematical Finance (quant)?
In terms of skills they are very similar. I'm not sure what to focus on.

>>10718310
>rather than actually getting any sort of convenience out of it
Are you retarded or what?
The point of the whole exercise is to be more efficient.
People who just "use the defaults" spend an enormous amount of time in their lives on things that they didn't need to do.

>>10717813
No, apparently before 1AD there was 1BC. Years AD are the age of Christ (Year nAD ends when christ turns n, the same way your nth year of life ends when you turn n), so it makes sense that he was born in his first year of life (although he was actually born between 6BC and 4BC). And years before christ count the years before christ, so at the end of 1BC there were no more years before christ and Christ was born.

>>10718310
Basic arithmetic
If you do anything more you're an autistic mathtard who spends (read: wastes) too much time on "muh abstract algebra" instead of actually doing science. And I bet the main reason you do it is so you can dick wave at your local autism circlejerk (like /sci/), rather than actually getting any sort of convenience out of it. Hence why made this post, rather than actually talking about science.

Excellent pasta.

>>10718310
Windows XP
If you do anything more you're an autistic /g/tard who spends (read: wastes) too much time on "muh personalized and free OS" instead of actually doing math. And I bet the main reason you do it is so you can dick wave at your local autism circlejerk (like /sci/), rather than actually getting any sort of convenience out of it. Hence why made this post, rather than actually talking about math.

>>10718103
No anon. No, there wasn't.
>>10718453
Thanks lad.
So, if there was no 0, but there was a -1, it clearly can't be cardinal.

>>10718453
>age of Christ
>Christ was born
He did not exist.

What textbook/s would you use to consolidate the fundamentals in preparation to undegraduate courses?

>fiddles with higher cateogories and shit
>forgets basic undergrads results

>>10718717
>implying higher categories are not basic undergrad results
Not going to make it.

>>10718679
Depends on the subject lad, and I'm not answering for every one.
>>10718717
Me too tbqhwyf.

>>10715679
Shut the fuck up.
>>10715672
Uh, axiomatize via category theory principles?? Obviously?

>>10718723
>Shut the fuck up.
Insecure undergraduate detected.

>>10718727
>it's provably impossible!!!!
find me a proof then

>>10714944
What is type theory?

>>10711651
It's kind of uncanny to see Misha Verbitsky's pasta here, after all these years.
http://imperium.lenin.ru/~verbit/MATH/programma.html

>>10718769
Verbitsky posts here and translated our curriculum to Russian.

>>10718775
>Verbitsky posts here
considering he's an eccentric shitposter, I wouldn't be surprised actually

>>10712274
>>10718310
>>10718500
A typewriter (preferably operated by your faculty's secretary) and a pen.
If you do anything more you're an autistic /g/tard who spends (read: wastes) too much time on these fancy "personal computers" instead of actually doing math. And I bet the main reason you do it is so you can dick wave at your local autism circlejerk (like /sci/), rather than actually getting any sort of convenience out of it. Hence why made this post, rather than actually talking about math.

>>10718860
/g/tards btfo once again. Seriously, go make a new /latex/ general and discuss this shit there.

>>10718874
I use texstudio btw

>>10718723
Categories are still a "collection" of arrows. You could take categories ad primitive objects and consider sets as categories where there's only identity arrows, but that isn't very different from set theory.

>>10717813
>>10718453
It's AD 1, not 1 AD.

>>10718154
fuck off with your reddit spacing.
best of luck otherwise

Erdos and Rado are the Patrick Swayze of mathematics

>>10719019
...which is what i suggest you do. it's totally different.

>>10719019
>build set theory on top of category theory by considering the class of objects of a category
Genuinely disgusting. What even is the point?

>>10719394
>build set theory
>What even is the point?
Indeed.

>>10718310
>If you do anything more you're an autistic /g/tard who spends (read: wastes) too much time on "muh personalized config" instead of actually doing math.
I mean, when I post my setup here it means that you'll reap the benefits of my "wasted time".
>And I bet the main reason you do it is so you can dick wave at your local autism circlejerk (like /sci/)
I wanted us to learn from each others configs so we could reduce the time spent as much as possible. I accept that this turned out to be the wrong thread for this, and I fucked off with a simple "Sorry" at the very fucking moment anybody indicated this. I didn't argue, I didn't complain and I was a utter delight without any indication of asshattery. By acting like such a miserable faggots in aftermath of this, you are basicly begging me to start posting about this in every /mg/ from now on. For perspective: There are now far more posts associated with this "conflict" about the existence of LaTeX posts than actual LaTeX posts in thread. How retarded is that?

>>10719394
someone wanted to know how to make math without set theory. it's stupid but works.

>>10719527
You're still defining a class and working over a class of objects, you're just detouring through categories.

>>10719538
the class is itself a category where the morphism "sets" are actually singletons. that's easy enough to get around.
i'm not seeing what the issue here is. of course they're similar but in terms of the axioms and elementary constructions they're rather disjoint.

>>10719538
Do you genuinely have a reading disability?

listen you fucking morons, i (the guy who says it's easy enough to define math without set theory) fucking hate category theory and everyone who uses it unironically. i'm just making a fucking point. go masturbate over your fundamentals in some fucking philosophy thread on /lit/.

>>10719539
You literally went "a class is the class of objects of a category."
It's either circular or bloated.

any tips for applying for phds in the UK?
i don't have any specific questions
sorry

>>10719550
>fucking hate category theory and everyone who uses it unironically
How can someone be this mentally ill?

>>10719554
>It's either circular or bloated.
As all philosophical circlejerking inherently is. I suggest you discuss this at >>>/lit/.
>>10719539
See the above message.

>>10719645
i'm not mentally ill, i'm an analyst (i.e. a real mathematician)
you may have meant to reply to one of the many unabashed algebraic geometers in this thread, they are some of the most mentally unstable people i've ever had the displeasure to meet. goodness, alg geo is a blight on the gentle and elegant world of geometry. especially grothendieck's modern abomination. it's like tearing down an italian villa and replacing it with a monolithic brutalist cube.
>>10719554
no, a category is defined to HAVE objects and morphisms. it does not have a class of objects.
listen you fucking idiot, the difference between sets and categories is not in their internal structure. it's in how the axioms for sets allow sets to interact, and how the axioms for categories allow categories to interact. people's issue with set theory is often that they don't like how some of the axioms let you work with sets (yada yada naive paradoxes) but categorical axioms might be more appealing to some, in the foundational sense. don't ask me why. i will fucking condemn categoryshits to the grave. the whole point is that it still fucking works.

>>10719690
i fucking agree with you on this which is why i literally posted the EXACT SAME FUCKING THING here >>10719550 and it's obvious that you're responding to the person who posted that.
i bet you thought you were being so clever though. giggling in front of your bluelight computer screen, blinds closed, door bolted shut to keep your mother's cats out.

>>10719645
i'm not mentally ill, i'm an analyshit (i.e. my work is the most applicable and least pure of all math)
you may have meant to reply to one of the many handsome algebraic geometers in this thread, they are some of the most mentally stable people i've ever had the pleasure to meet. I cannot believe that they are kind enough to let me share a world with their godlike intellect and insight. goodness, analysis is a blight on the gentle and elegant world of mathematics. especially tao's modern abomination. it's like tearing down an italian villa and replacing it with a gay bar which is my favorite place to hang out.

Since I discovered Wildberger I've never even ACKKNOWLEDGED a """real""" number or set theory again

>>10719703
>analyst
>mathematician
These words don't go well together.

>>10719690
Fuck off.
>>10719703
>axioms allow categories to interact
Name one axiom in category theory about interactions between categories.

>>10719731
analysis is the soul of mathematics.
algebra is at best a brittle skeleton.
>>10719732
when i say axioms i mean the basic categorical facts which you obviously have to take in lieu of set theory
what are you trying to argue? i don't think you know what we're talking about.

>>10719732
>axiom
Refer to >>>/lit/ for philosophical circlejerking.
>>10719740
>axioms
Refer to the above message.

>>10719740
>which you obviously have to take
Give an example. I have a set that was constructed in your categorical technique, so it's a category. Give me an axiomatization or a sketch of one that lets me do basic set theory.

>>10719748
>axiomatization
>set theory
Refer to >>>/lit/. We don't discuss such garbage here.

>>10719740
>soul
see >>10719751

>>10719732
>>10719747
>>10719751
>>10719759
ironic shitposting is still shitposting
>>10719748
i could not possibly care less about your stupid fucking category
ALL I DID WAS SAY THAT I WANT TO SEE PROOF THAT YOU CANNOT PROVIDE SUCH AN AXIOMATIZATION.
I'M NOT FUCKING RECONSTRUCTING MATH FOR YOU. I'M NOT NORMAN J WILDBERGER. FUCK OFF FOUNDATIONTARDS!!!!!!!!!!!!

>>10719763
>AXIOMATIZATION
>FOUNDATION
Refer to >>>/lit/ for your philosophical circlejerking needs. We don't discuss such garbage here.

>>10719763
H O W W O U L D Y O U E V E N P R O V E T H A T

This kid is so easy to bait that I'm tempted to believe some retard is doing it on purpose because he thinks it's ""funny"".

>>10719779
You fundamentally misunderstand the situation.
Me and this other lad are pretending to bait analysis autist, and analysis autist is pretending to be baited.
We do it out of love.

>>10719763

>fuck foundatationtards

I bet you believe in a religion too kek

>>10719792
see? this guy gets it. cute bp btw <3
>>10719798
i accept that foundations are important, but i don't care about them at all.

>>10719855
>foundations are important
see >>10719765

>>10718722
Isn't there anything specific to aid people who want to attend an undergraduate degree in mathematics? I'm open to pretty much anything.

>i don't care about them at all
How do I unsubscribe from your blog?

>>10719862
Start learning set theory. It's the only field in mathematics that truly matters and gives us a greater insight into the truth.

>>10719798
>I bet you believe in a religion
Refer to >>>/lit/.
>kek
Refer to >>>/b/.

>>10719865
>set theory
>truth
See the above message.

>>10719863
just press the "unfollow" button on the top right. but fill out the survey first about why you're leaving! we'll be sad to see you go :(
>>10719865
stop ruining people's lives
>>10719862
watch gilbert strang's linear algebra lectures on MIT OCW to get a good linear algebra foundation which is part computational and part theoretical.
if you like that and want more, read axler's Linear Algebra Done Right.
if you want something different and liked calculus, read pugh's analysis and do exercises.

>>10719798
I do. Problems?

>>10719873
>watch gilbert strang's linear algebra lectures on MIT OCW to get a good linear algebra foundation which is part computational and part theoretical.
>if you like that and want more, read axler's Linear Algebra Done Right.
>if you want something different and liked calculus, read pugh's analysis and do exercises.
Unrigorous Garbage. They don't even define rigorously define what a Class is.

>just press the "unfollow" button on the top right.
Where? I can't see it. Can you take an image of your screen?
>fill out the survey first
No. You can fuck right off.

This is what happens when people study math locked in their rooms instead of observing nature like the ancients. They create deities like set theory and infinty.

>>10719873
>computational
>liked calculus
Where did he say that he was a subhuman brainlet?

>>10719019
>Categories are still a "collection" of arrows.
Really? Are they now? What's a ``collection''? Please be rigorous with us.

What is the best way to study math (calculus, linear algebra) with full rigor? I'm enrolled in an undergrade course right now, but they don't properly explain or define most objects they work with, instead they use what they call "intuition". My intellectual thirst isn't satiated and I keep wanting for more. What rigorous book should I read to remedy this?

>>10719905
>I'm
We are*
>I keep
We keep*
>I read
We read*

Am I existing right now? I mean, mathematically speaking, as in in a mathematical sense. Am I truly existing at this moment? Or is this just some set-theoretic nonsense a bunch of abstract settards dreamt up? I am inclined to believe it is the former, but I am not certain.

>>10719884
can you fucking press the number of my post if you're replying to it or go back to wherever you came from?
>>10719892
he asked for undergrad prep materials, anyone who needs to prep for undergrad math classes is such a person.

>>10719905
rudin and axler

>can you fucking press the number of my post if you're replying to it or go back to wherever you came from?
I can't see it. Can you take a picture of your screen?

>>10719922
[math]x\vdash y[/math]
[math]x \doteq y\frown y [/math]
[math]y \frown x \vdash y \smile y[/math]
[math]y\parallel x \parallel y \doteq (y\bowtie y) \vdash y \vdash (x\frown x \frown y\smile x) [/math]

>Can you take a picture of your screen?
what screen? i can't see it. can you take a picture of your screen?

>>10719922

you're a set

>>10719932
I just skimmed through those and they seem to use a lot of intuition too. I want something with pure rigor and no traces of this low-quality type of math that is taught these days. I'm so sick of it from my classes already, hell, I can't even be bothered to attend them anymore because of this. Does it get better on your second year?

>>10719941
no ur mom is a set

>>10719945
you don't understand what mathematics is, do you?
you start with axioms and derive facts from these axioms. both rudin and axler do this without reference to outside material.
what exactly do you mean by "intuition"? there is no such thing in a pure math book, at least unaccompanied by rigor.

>>10719937
The extended proof goes as follows:

Let us deeply and truly assume:
[math]x^{\flat \sharp} \vdash^0 x[/math]
[math]x^{\sharp \flat} \vdash^0 x[/math]
[math](x \vdash_{meta} x) \vdash^0 x^\natural[/math]

And so
[math](x \vdash_{meta} y) \vdash^1 x^\natural[/math]
But then
[math](y \vdash_{meta} y) \vdash^2 y^\natural[/math]
So by induction
[math]((x \otimes \infty) \vdash_{meta} \infty) \vdash^n \infty[/math] for all [math]n \in \mathbb{N}[/math], which is a deep contradiction.

>>10719972
>[math]n \in \mathbb{N}[/math] , which is a deep contradiction.
No shit, sherlock.

>>10719951
As I see it, it's the rigorous study of unraveling the truth by doing some syntactic/logical manipulations. I really want to fully understand what they mean by "group" when they introduce it by saying that it's a "set" with some "operation". They don't even rigorously define what it fully means to be a set and an operation and such. In analysis, they just draw a picture and tell us that something is continuous without rigorously defining it in a way which leaves my mind satisfied. This is the low-quality math I'm talking about. I hope it gets better later.

Maybe I can take some extra classes for the more intelligent/advanced students so the professors can see that I already have this thirst for rigour at an early part of my education? Should I just approach someone about it?

>>10720038
people who care about rigor are usually mathematically immature and have this romantic view of what they consider to be "beautiful pure logic". they have not seen enough rigor to understand why it is boring and worthless, a necessary evil. you are someone who knows very very little about mathematics. if you really, really want to learn exactly what those things are, then you need to find a foundational text which goes from the very start. however, even this text will make certain assumptions, the axioms of their mathematics. you need to learn how to accept that something can be without justification if you want to do mathematics, because otherwise there is nowhere to start.
i cannot point you to a text that would "satisfy" you because i'm not autistic.

>>10719703
>i.e. a real mathematician
LARPing undergrad confirmed

>>10720056
>they have not seen enough rigor to understand why it is boring and worthless
I agree with some of your post and the guy you're responding to is retarded, but don't tell me that rigor is "worthless"

>>10720056
The thing is, the point isn't rigor, but a general confidence that it could be made rigorous if needs be.
Wiles didn't do the full proof in logical terms and without skipping a single step, but you can reasonably pick it up and throw a couple years in the trash writing the entire thing out.

>>10720068
yes of course, i was being hyperbolic
>>10720077
i agree with this. don't worry about how precisely one defines a set when you pick up a group theory text. accept that it can be defined as the text asks you to do and everything from there on will be rigorous.

>>10720038
>Maybe I can take some extra classes for the more intelligent/advanced students
you wouldnt belong anyway

>As I see it, it's the rigorous study of unraveling the truth
leave platonism at the door if you want to be a serious mathematician

>They don't even rigorously define what it fully means to be a set and an operation and such.
how fucking retarded are you if you cant fill that in on your own
set theory is trivial

>>10720038
They didn't give you lads the definition of continuity?
That's actually weird. Would you particularly mind naming the uni, or your country?

>>10720200
> leave platonism at the door if you want to be a serious mathematician

low level bait
this is the only good thread on /sci, so maybe try not to shit it up with your braindead discussions and garbage posts

>be me
>pick up a ring R on which I'm doing SDG
>set up a function that's 0 everywhere, but 1 at d
>apply Axiom 1
>inverse constructively given lmao
>I literally have an element that gives zero when multiplied by any element other than a specific one, but is invertible nonetheless
>it doesn't matter since no third excluded
I really like the subject, but I genuinely can't take it seriously.

>>10720200
>set theory is trivial
anon... I...

>>10720268
formalism is the only right approach you petulant child

>>10720300
I dont care about formalism, platonism or whatever you brainlets think is the right approach, just stop shitting up the thread

>>10720298
Who are you quoting?

>>10720298
>I really like the subject, but I genuinely can't take it seriously.
Well, at least now you know how most of your peers feel about you.

>>10720429
They really like me?

>>10720451
but they can't take you seriously.
you're like a little puppy!

>>10720298
What is SDG?

>>10720486
Synthetic differential geometry.

>>10720497
*Symplectic.

I want to start relearning mathematics. I gone through normal high school curriculum up to business calculus at uni (which i'm assuming is a gimped/more business-applied version of cal 1).

I have some strengths in algebra but I know my geometry skills/knowledge is garbage. I have a few questions that I would be very grateful if someone could answer or guide me:

1. What is the best path of self learning? Should I just start from algebra 1 and move up? How much geometry is necessary to know in order to move into trig?

2. What books or online resources do you recommend? Youtubers? I know nobody except khan academy and professor dave

What I'm most worried about is dumping my time into areas of math that aren't exactly necessary. I want to build on a core foundation in order to process to higher maths (not implying I can or want to rush it, just that I don't have much free time and I want to spend it well).

>>10720561
>literally tho?
I mean, not really. Unless you're actually interested in rigor and such. There's no point in studying this shit. Also, are you on that picture?

>>10720298
>still believing that things in math aren't just there for you to have fun with
Insecure undergraduate detected.

Just finished precalc and loved it. Took a peek at my calculus textbook, saw a bunch of three dimensional shapes. I tried drawing a cube, but I just can't.

Summer calculus starts in 5 days. Is it possible to learn how to draw in that time? Am I going to have a bad time if I can't draw?

>>10720706
different anon here, I love the subject but the higher level math courses you take, the concepts become more abstract. Which is why i'm a physics major, I usually understand the concept in physics then the methodology becomes easy in math (the only case this wasn't true for me was learning damped oscillations, for some reason physicists cant teach that for shit), which is why I enjoy subjects like differential equation but not linear algebra (I know it has its applications in physics like thicc lenses but I haven't been taught that yet) Call me whatever, but I still love the subject enough to currently be pursuing a major in it

>i'm a physics major
Stopped reading right there.

has everyone here proved all of Euclid's props?

Trying to learn p-adic field basics and I'm fucking up somewhere. I have an example that if I take the field Qp and append the (p^n-1)th root of unity the end result has ramification degree 1 and residual degree n.

So I'm trying to understand this via an example, with the really easy case p=5 and n=1. We want to append the primitive 4th root of unity, which is conveniently i. The valuation of i needs to be 1 so our valuation group is identical to the valuation group of Q5, meaning the ramification degree is 1, as expected.

But looking at the residual degree, I don't get 1. Firstly the ring of integers O is just going to be Z[i] which is easy to work with. The uniformizer I can take to be 5, so the maximal ideal is 5Z[i]. Elements look like 5a+5ib where a and b are integers. Now the residue field is O/M which I get to be 25 elements of the form a+ib where a is in [0,4] and b is in [0,4]. As the original residue field was just F5 and has 5 elements I get the residue degree to be 5, not 1 as claimed.

What am I doing wrong? Thanks.

What is the most ridiculous word made up by mathematicians? I think it's cobordism.

>>10721365
I think the thing you overlooked but that solves all your problems is that Q5 already has a primitive 4-th root of unity (F5 does and then you apply Hensel's lemma), so that you were talking about Q5 all along (and obviously the extension Q5/Q5 is unramified with residual degree 1).
Now, there are still a few things that need correcting:
>The valuation of i needs to be 1 so our valuation group is identical to the valuation group of Q5, meaning the ramification degree is 1, as expected.
That is not how you compute the ramification degree. You should compute the valuation of 5 in your extension, not i.

What's some cozy corner of mathematics that mainstream academia doesn't care about?

>>10721564
Fuck. Thanks for that. It makes sense.

The reason I took the valuation of i was to show that the valuation group is identical, and the ramification degree is just the index of the valuation group of the extension over Qp, which is 1 here.

In fact, for any p we will have a (p-1)th root of unity so the residual degree will always be 1, which is what we wanted.

Now, looking at n=2, so for example Q3 and I'm adding in an 8th root of unity. This is a number x with fourth power equal to 2. x has valuation 1 so it will be present in O.
The ring of integers is therefore:
Z[x]/(x^4-2)
The maximal ideal M is still going to be given by uniformiser 3 so we get:
M=3Z[x]/(x^4-2)
I cannot see how I'm going to get a residue field of index 2 over F3 here...

>>10721638
>The reason I took the valuation of i was to show that the valuation group is identical,
Yes but what I mean is that i is invertible in the ring of integers of the extension, so its valuation is always going to be 0. It does not yield any information about the valuation group.
What you do to compute the ramification index is look at the decomposition of a uniformizer of the bottom field, here 5 in the larger field K. There, it can be written [math]5 = \varpi^e[/math], with [math]\varpi[/math] a uniformizer for K and e is the ramification index. Alternately, you see that the valuation group of [math]\mathbb Q_5[/math] has index e in the valuation group of K, which ties the two descriptions together.

(cont.)

>>10721806
>The ring of integers is therefore Z[x]/(x^4-2)
No, that is not possible and you had kind of got to the contradiction on your own. x cannot have degree 4 if we believe in the result we are trying to prove and recall that the degree of an extension is the product of the ramification index and residual degree.
What we are trying to prove suggests that we should expect x to have degree 2, and indeed the polynomial x^4 - 2 splits over F3, and therefore over Q3 (by Hensel) as (x^2+x+2)(x^2+2x+2) (thanks Wolfram).
So we choose one of these, say x^2+x+2, as the minimal polynomial for our extension.
Now what is the ring of integers ? It is not a priori clear that it should be Z3[x]/(x^2+x+2). If you have never done this before, I suggest you try to compute the ring of integers of Q[x]/(x^2 - d). It is not always Z[x]/(x^2-d).
Fortunately, in this case, because the arithmetic of Z3 is very nice, it is not very hard to check that it really is Z3[x]/(x^2+x+2).
Therefore, the discriminant of the extension K/Q3 is just the discriminant of x^2+x+2, ie. -7, which is coprime to 3, hence the extension is unramified. Or alternately, you may check that 3 generates the maximal ideal in Z3[x]/(x^2+x+2), which also proves unramifiedness.
Moreover, you get that the residual extension is [Z3/(x^2+x+2)]/(3) ~ F3[x]/(x^2+x+2), which does have degree 2 over F3.

>>10721812
You're a lifesaver. The annoying thing is that I've learned p-adic and field extension stuff completely separately, so when you say to look at the ring of integers of Q[x]/(x^2-d) I know that this is Z[sqrt(d)] or Z[(1+sqrt(d))/2] based on d mod 4. This comes from taking the ring of integers to be the elements whose minimal polynomial has integer coefficients.
But then as soon as I'm working p-adically I think "the ring of integers is the elements with value less than or equal to 1" and it feels like a world away. You've given me a great idea of how to work with these structures though. I'm going to run a few more examples for myself. Thanks.

>>10721923
No problem ! I should thank you for making me go through this number theory stuff I had not touched in quite a bit.
So yeah, you really should think of p-adic fields as number fields, but with only one interesting prime. Moreover, because the rings of integers are always DVR, a lot of the complexity that occurs in number fields disappears. In particular, all the ramification, decomposition, etc. theory of Galois extensions that you studied for number fields has a simpler analogue for p-adic fields. It is the equivalent of looking at one prime p on the ground field of a Galois extension of number fields L/K, then focusing on one prime of O_L above p, and it is one way you can study number fields "locally" (so this gives at least some motivation for studying these local fields).

>>10721812
Woops I’m just realizing that this was a bit sloppy
>and indeed the polynomial x^4 - 2 splits over F3, and therefore over Q3 (by Hensel) as (x^2+x+2)(x^2+2x+2) (thanks Wolfram).
x^2+x+2 does not really divide x^4-2, but what Hensel’s lemma says is that x^4-2 has a monic factor of degree 2 that reduces to x^2+x+2 mod 3. I don’t think it changes much of what I was saying, but the factorization I wrote is just patently false in Q3

I am just starting to use Latex, and I'm wondering if it matters whether I use spacing.
For example are \sin \theta, \sin\theta the same thing?

>>10722095
doesn't matter

>>10722102
Thanks

>>10713696
Thanks for this

>>10722095
doesn't matter if it's two \ commands.
if it's one command and one letter, like \sinx instead of \sin x, it will throw an error

Have you ever met a philosopher of math in the wild?
What were they like?

>>10722273

he was a fields medalist

>>10721512
I'd say clopen

>watching emeritus interviews
>they all talk about self-learning from books all by themselves naturally
>meanwhile I'm here struggling with baby math with the whole internet and pirated books at my disposal

IQ is a hard and bitter pill

>>10722380
I've heard of professors who refuse to read proofs in textbooks, they force themselves to come up with them themselves after reading the theorem

>>10722273
Usually they are mathematicians that like to think about logic, from what I can tell. They tend to have a math background

>>10722626
I’ve had a professor who insisted on coming up with all statements and proofs at the blackboard, although he always had a complete set of lecture notes with him that he had fully typed up himself.
It made for entertaining lectures, but they did tend to be pretty messy.

>those 4 guys that always appear at the conferences in your field that use techniques nobody else understands or uses
and every talk I tell myself to at least look up chebotarev's theorem for next time, but I never do

Just passed my quals

Does anyone else feel like their training as a mathematician has changed their perspective on cannibalism? I think it makes a lot of sense.

>>10711631
Can someone help a brainlet with this? The proof that B can not be an element of A makes sense to me, I am convinced by the proof. What I don’t understand is how this implies “nothing contains everything”.

>>10722967
it's always the russians in my experience
sitting in a seminar with russian students/profs is quite an experience

> "there's hamming codes, reed-muller codes, hadamard codes, golay codes, reed-solomon codes, goppa codes, repetition codes, [...]"

>>10723144
because if you claim to have a set A which contains everything, then i can produce an object B which is the set of elements of A which do not contain themselves, and we have proven that B is not in A. So in fact A does not contain everything and you were wrong.

>>10723148
>there's Bergman spaces, Lebesgue spaces, Banach spaces, Fock spaces, Frechet spaces, Morrey spaces, Sobolev spaces, Hardy spaces, [...]

>>10723158
Don't forget Hilbert spaces! Heh!

>>10723158
>there's compacts, pseudocompacts, anr-compacts, local compacts, compactifications, [...]

>>10723322
>there's topological manifolds, formal manifolds, smooth manifolds, Kahler manifolds, Mazur manifolds, hyperkahler manifolds, symplectic manifolds, complex manifolds, Banach manifolds, C^k manifolds, Calabi-Yau manifolds, Riemannian manifolds, analytic manifolds, Einstein manifolds, pseudocomplex manifolds, [...]

>>10722639
>>10722639
I prefer seeing an original proof from the professor than just a regurgitation of what I already read in the textbook. The best professors will give some remarks about their approach vs the book's approach and the strategic advantages of each.

>>10723153
but what about objects in A that do contain themselves?

>>10723525
Why does this picture of a mentally retarded anime girl make me so fucking angry every time I see it? It's just so stupid looking

Someone please help me with this differential geometry question, I'm going mad over here. Given two atlases [math]\mathcal{A}_1[/math] and [math]\mathcal{A}_2[/math], I want to show that if [math]\mathcal{A}_1\cup\mathcal{A_2}[/math] is an atlas then [math]\mathcal{A}_1[/math] and [math]\mathcal{A}_2[/math] are equivalent. I found this answer https://math.stackexchange.com/questions/2930261/two-atlases-are-compatible-if-and-only-if-their-associated-maximal-atlases-are-e ,
but I don't see how [math]\mathcal{A}_2 \subset \mathcal{A}_1^+[/math] implies those two are compatible.

>>10722976
Congrats anon.

>>10723748
fuck off rapcak go watch your toystory 3 and kill yourself

>>10721598
Good luck finding it

>>10719716
Nice

should I consider a job with 25K€ starting with my MSc in physics ?

>>10724469
>physics
I heard call centers look for physicists, try there

>>10721598

try to find a lower bound for the number of "unhappy cycles" in an arbitrary base.

For example, base 10 has two unhappy cycles and they are known as the happy and unhappy numbers.

>>10724471
dude, this is the biggest meme ever, if anyone says this it means he doesn't know shit about the job market

a professor proposed this job for me and told me that 25K isn't much, but while I work my PhD will be financed by them

>>10723748
thanks anon

>>10724145
please keep /mg/ a polite place. professionally studying math is already makes people think about suicide. you don't have to force the issue.

What's a frobenius norm projection onto a set of semi definite matrices?

>>10724540
The point in the set of semidefinite matrices which minimizes distance to a given matrix with respect to the norm inherited by a standard inner product. If I recall correctly, inner product in this case is

[math]trace(A^T B)[/math]

>>10721598
I'd say combinatorics, it's autistic as fuck but generally a cozy environment of counting hyperplane arrangements and solving problems no one but you and your mates care about.
>>10723148
>>10723158
>>10723363
>there's 1-categories, 2 categories, 3 categories, 4 categories
>>10724145
Im-fucking-plying, the brave little toaster is the one you watch and then kill yourself.
>>10722380
It's not like they do it in a day, it usually requires intense study. I mean, it's not so different then reading papers really, in both your forced to digest and integrate new material, really it's par for the course if you want to get to the bleeding edge of research.
>>10722273
I was once in the presence of Gromov and I lost the ability to understand what a manifold is.

>>10725178
FUCK combinatorics. maybe its because i had russian jew teacher who would throw the class huge curveballs on tests, that was the only B i received as an undergrad.

>>10724598
Ah ok. So there's no closed form expression

How do I rewrite a^1/2(2a^2 - 4/a)?

My problem says the - 4/a becomes-4a^-1, but how does this happen? They don't explain it

>>10725436
1/a is the equivalent of a^-1

Proof: 1/a = a^-1, multiply both sides by a and a^0 = 1 which is true

>>10725484
so the negative exponent doesn't ever apply to the constant terms?

>>10725504
What?

>>10725504
4/a is the same as 4*(1/a)

Looking to get these de-obfuscated.
https://en.wikipedia.org/wiki/Free_motion_equation
https://en.wikipedia.org/wiki/Non-autonomous_mechanics

I wish to see if they can apply to induce massively broad and certain wave function collapse.
I could probably handle defucking Non Materiae sui iuris, if I'm given a codec.

Yes I'm well aware of my status as a faggot. Thanks for asking.

>>10725504
Division is defined to be multiplication by the multiplicative inverse of the divisor. To give an example, x/y is the multiplication of x by the multiplicative inverse of y, in other words it's x*y^-1.

In this case, you're dividing -4 by a, ergo you're multiplying -4 by a^-1.

>>10720038
>They don't even rigorously define what it fully means to be a set and an operation and such.
They don't define this kind of stuff because there's such a thing as too much information (in a lecture because of time constraints and in a book because of page count, avoiding redundant information and sticking to a topic).

The book will either assume that the reader already knows enough about those topics or stick that stuff in an appendix; either way, it's obvious that you should refer to a book about those topics if you need more information.

Also you sound like a tool.

>>10720211
The impression I'm getting from his post is that those are first year classes, and in my uni those are the same for math, physics and engineering students (i.e. very basic and hardly rigorous), in which case using a simple (or really, a non delta-epsilon) notion of continuity is not very surprising (in my case, it was that the limit exists and is equal to the function evaluated at that point).

>>10713158
>Bourbaki, from the beginning.
How good is Bourbaki and how should you optimally utilize it? I've read that they're somewhat controversial and not all of them are good, but I like what I've read of them and intend to go through them eventually.

>>10725670
>in my case, it was that the limit exists and is equal to the function evaluated at that point
That's standard, tho.
>inb4 no, the standard is the open ball contained in some open ball
That's a rewriting of the one you gave. Emphasis on "a rewriting", not "an equivalent definition".

>>10723611
Please help me, I'm losing sleep over this.

>>10725685
>That's standard, tho.
Oh no argument there, but I'd call it implicitly "loose" when it's introduced in calc I because the concept of limit isn't rigorously defined at that point.

>>10725689
>>10723611
Let A, B and C be three charts on a manifold.
Theorem One: If A has a smooth transition map to B, and B has a smooth transition map to C, then A has a smooth transition map to C.
Proof: take compositions, and apply the fact that smooth functions compose to form smooth functions.
Theorem two: if an atlas A is compatible with an atlas B, and B is compatible with C, then A is compatible with C.
Proof: follows trivially from theorem one.
Theorem Three: the one in your post.
Proof: follows immediately from theorem two and the definition of maximal.

>>10725695
Compatibility of charts is not a transitive property, otherwise I wouldn't be asking this question.

>>10725698
It is when they have nonempty intersection.

>>10725701
It isn't, when you take the composition, compatibility is only guaranteed on the intersection of all neighborhood.
https://math.stackexchange.com/questions/110576/is-c-infty-related-an-equivalence-relation

>>10725711
Yah lad, but you have an entire fucking atlas.
Repeatedly apply transitivity on every single fucking triple intersection until you prove it's smooth everywhere.

>start Calc1
>lims are easy as fuck and *every thing comes super easy
>start doing something that required algebra
>dumbfounded and fumble my way around and cant solve

this is hell, the worst part is that i dont know exactly what im struggling or how far my holes in algebraic knowledge goes. Any good way of finding out for sure withouth reading 500+ pages of Stewart, lang or engineering math shit... ?

>>10725714
>Yah lad, but you have an entire fucking atlas.
Let [math](U_\beta, \varphi_\beta) \in \mathcal{A}_2, (U, \varphi) \in \mathcal{A}_1^+, W = \varphi_\beta(U_\beta)\cap\varphi(U)[/math]. We want to prove that [math]\varphi_\beta^{-1}\circ \varphi \colon \varphi^{-1}(W) \to \varphi_\beta^{-1}(W)[/math] is smooth. Since [math]\mathcal{A}_1[/math] is compatible with both [math]\mathcal{A}_1^+, \mathcal{A}_2[/math], for any chart [math](U_\alpha, \varphi_\alpha) \in \mathcal{A}_1[/math] we have that the map [math]\varphi_\beta^{-1}\circ\varphi_\alpha\circ\varphi_\alpha^{-1}\circ\varphi\colon\varphi^{-1}(W') \to \varphi_\beta^{-1}(W')[/math] is smooth, where [math]W' = W\cap\varphi_\alpha(U_\alpha) \forall \alpha[/math].

Since [math]\mathcal{A}_1[/math] is an atlas, we have that [math]\bigcap\varphi_\alpha(U_\alpha) = M[/math], and we conclude that the initial transition map is smooth on [math]W\cap\bigcap\varphi_\alpha(U_\alpha) = W\cap M = W[/math]. Is this what you're suggesting? Sounds reasonable, I'll think about it some more.

>>10725752
The intersections in the last paragraph are supposed to be unions, sorry for the typo. [math]\bigcup\varphi_\alpha(U_\alpha) = M[/math]

>>10723611
The maximal atlas of [math]A_{1}^{+}[/math] of [math]A_1[/math] is by definition the union of all atlases compatible with [math]A_1[/math]. So if [math]A_2[/math] is compatible with [math]A_1[/math] then by definition [math]A_2[/math] is in the union of all atlases compatible with [math]A_1[/math] and thus in [math]A_{1}^{+}[/math] . By a similar argument [math]A_1[/math] is in [math]A_{2}^{+}[/math] . Since they are maximal atlases [math]A_{1}^{+}=A_{2}^{+}[/math].

>>10725759
I already know that [math]\mathcal{A}_2 \subseteq \mathcal{A}_1^+,\quad \mathcal{A}_1 \subseteq \mathcal{A}_2^+[/math], but I don't see how that implies [math]\mathcal{A}_1^+ = \mathcal{A}_2^+[/math]

>>10725763
Oh, well since [math]A_{1}[/math] and [math]A_{2}[/math] are compatible, this means that [math]A_{2}[/math] and [math]A_{1}^{+}[/math] are compatible. Since [math]A_{2}^{+}[/math] is defined as the union of all atlases compatible with [math]A_{2}[/math] this implies that [math]A_{1}^{+}[/math] is in [math]A_{2}^{+}[/math]. By similar reasoning on that [math]A_{2}^{+}[/math] is in [math]A_{1}^{+}[/math] thus [math]A_{1}^{+}=A_{2}^{+}[/math]

>>10725769
>[math]\mathcal{A}_1, \mathcal{A}_2 [/math] compatible [math]\implies \mathcal{A}_2, \mathcal{A}_1^+[/math] compatible
The problem is that this isn't true a priori. [math]\mathcal{A}_1^+[/math] contains all charts compatible with [math]\mathcal{A}_1[/math], they're not required to be compatible with [math]\mathcal{A}_2[/math] which is why I followed this argument >>10725752

>>10714502
The probability of landing on heads or tails is 1/2, therefore we can assume the probability of there being more heads than tails is equal to the probability of there being more tails than heads.

This means we just need to subtract probability of having an equal number of heads and tails, 6C3 x 1/2^6 = 0.3125, from 1, then divide by 2 to get 0.34375

>>10725775
>The problem is that this isn't true a priori
I thought that was true by definition. If [math]A_1, A_2[/math] are compatible that means [math]A_1 \cup A_2[/math] is also an atlas. Since [math]A_{1}^{+}=\bigcup_{i} B_{i}[/math] where [math]B_i[/math] are all the atlases compatible with [math]A_1[/math] then by our assumption one of those Bs has to be [math]A_2[/math] since it is also compatible with [math]A_1[/math], let's just say it's [math]B_1[/math]. But that also means that [math] A_2 \cup A_{1}^{+} = A_2 \cup (\bigcup_{i} B_i ) = A_2 \cup B_1 \cup (\bigcup_{i \neq 1} B_i ) = B_1 \cup (\bigcup_{i \neq 1} B_i )= A_{1}^{+}[/math] so that the union [math]A_2 \cup A_{1}^{+}=A_{1}^{+}[/math] which is an atlas.

>>10725797
I'm trying to prove that if the union of two atlases is an atlas, then they're compatible. If I claim that [math]\mathcal{A}_2\cup\mathcal{A}_1^+[/math] is an atlas and therefore they're compatible, wouldn't I be invoking the same result that I'm trying to prove?

>>10725807
>I'm trying to prove that if the union of two atlases is an atlas, then they're compatible
https://math.stackexchange.com/questions/3093079/cant-understand-the-definition-of-equivalence-of-topological-atlas?noredirect=1&lq=1
https://en.wikipedia.org/wiki/Manifold#Atlases
I don't know if there's some massive miscommunication going on here, but the very definition of two atlases being compatible is that their union is compatible.

>>10725820
The definition I'm using is that two atlases are equivalent if they introduce the same differential structure (they define the same maximal atlas). This is an exercise on Godinho & Natario's Introduction to Riemannian geometry, but sadly it doesn't include the solution.

>>10725820
>>10725825
Honestly I think my brain melted because I don't even know what I'm trying to prove anymore. Pic related is the exercise.

>>10725825
Alright, now I see what the issue is, basically the problem is showing that two different definitions are equivalent. Now this makes way more sense. I also found the book so this may streamline things.

This shit is annoying to type up, so let my try explaining in words what's going on. First we'll start with the assumption that the union of the two atlases in question is itself an atlas. Basically, if the union of two atlases is an atlas, that means property ii in definition 2.1 is satisfied. Now, in order to prove the statement we need to show that the maximal atlas of [math]A_{1}[/math] is the same as the maximal atlas of [math]A_{2}[/math]. How can we do this? Well note that in remark 2.2 he says "In fact, we can take the set [math]A_{1}^{+}[/math] of all parameterizations that satisfy (ii) with every parameterization on [math]A_{1}[/math].". BUT if order for the union of [math]A_{1}[/math] and [math]A_{2}[/math] to be an atlas, all the parameterizations on [math]A_{2}[/math] must satisfy ii with all of the paramertizations on [math]A_{1}[/math], so that by remark 2.2 [math]A_{2}[/math] is in the maximal atlas of [math]A_{1}[/math]. By the same logic [math]A_{1}[/math] is in the maximal atlas of [math]A_{2}[/math]. Since the maximal atlas is unique, the two maximal atlases must the same, and thus [math]A_{1}[/math] and [math]A_{2}[/math] are equivalent. If fact, since I only used the definitions of various properties, you can pretty easily reverse the arguments to prove the other direction.

>>10725825
>but sadly it doesn't include the solution.
Hey wait a second you faggot, the book does have the solution to the problem, I just checked it.

>>10725864
I swear to god it doesn't, if I'm wrong I'll kill myself.

>>10725867
https://www.math.tecnico.ulisboa.pt/~jnatar/books/geometria_secret.pdf
Scroll down to the bottom of the table of contents on page 2. Literally says "solutions to exercises" starts on page 301. I am satisfied knowing that the solution I gave is basically exactly the same as the book, but I'm a little mad that this could've been resolved instantly if you'd just check the fucking table of contents.

>>10725856
>Since the maximal atlas is unique, the two maximal atlases must the same
Every time people bring up this as if it's obvious, but I can't see it. Still, I think >>10725752 proves the non-trivial implication.

>>10725871
Goddamnit I have another edition. Fuck you Springer you fucking niggers.

>>10725873
>Every time people bring up this as if it's obvious, but I can't see it.
It's not that you're necessarily wrong, you aren't, it's just that there are shorter arguments. If you look at the solution on page 303 it spells it out a bit more, basically, since [math]\mathcal{A}_2^+ \subseteq \mathcal{A}_1^+,\quad \mathcal{A}_1^+ \subseteq \mathcal{A}_2^+[/math] this implies that [math]\mathcal{A}_1^+ = \mathcal{A}_2^+[/math] and to see this is really to use the definition of maximal, if [math]\mathcal{A}_2^+ \subseteq \mathcal{A}_1^+[/math] but [math]\mathcal{A}_1^+ \neq \mathcal{A}_2^+[/math] then [math]\mathcal{A}_1^+[/math] would fail to be maximal, there would be parameterizations in [math]\mathcal{A}_2^+[/math] that satisfied property ii with all of the parameterizations in [math]\mathcal{A}_1^+[/math] yet not in [math]\mathcal{A}_1^+[/math]
>Goddamnit I have another edition. Fuck you Springer you fucking niggers

Well anon...at least you have it now.

>>10725881
Whoops, fucked up a but, it should be "if [math]\mathcal{A}_1^+ \subseteq \mathcal{A}_2^+[/math] but [math]\mathcal{A}_1^+ \neq \mathcal{A}_2^+[/math] then"

>>10725881
>it's just that there are shorter arguments.
I know, which is why this is driving me nuts. I know this is one of those "conclude in two lines" statements and I can't figure it out. My problem is not the double inclusion, that one is obvious and doesn't really need the maximal property; I have trouble understanding why [math]\mathcal{A}_2 \subseteq \mathcal{A}_1^+ \implies \mathcal{A}_1^+ \subseteq \mathcal{A}_2^+[/math]. It follows from my other post but I know there's a simpler argument.

>>10725889
>It follows from my other post but I know there's a simpler argument.
If [math]\mathcal{A}_2 \subset \mathcal{A}_1^+[/math] then since [math]\mathcal{A}_1^+[/math] is an atlas it must satisfy property ii, which means that every parameterization of [math]\mathcal{A}_1^+[/math] must satisfy property ii with every parameterization on [math]\mathcal{A}_2[/math], which by definition means that [math]\mathcal{A}_1^+ \subset \mathcal{A}_2^+[/math]

>>10725924
It's so obvious, how could I miss this? Thank you for putting up with my stupidity anon, time to hang myself.

>>10725929
>It's so obvious, how could I miss this?
Don't worry anon, I've also felt that a lot too. When I first took smooth manifolds and transitioned into grad courses I finally understood what my undergrad advisor told me about "mathematical maturity". Basically there a bunch of arguments that were kind of subtle and in some sense easy to miss and skip over, but you need to be keen and well versed enough in the material to be able to spot gaps in proofs and fill them in. Usually this requires chaining a few definitions, but it can make some proofs almost seem like black magic if you aren't yet familiar enough with the definitions. The upshot is that once you are you can proceed more rapidly, since you're thinking and writing up proofs "modulo" these sort of short arguments that follow from applying the definitions without explicitly mentioning them. If it makes you feel better, even terry tao had issues with this sort of thing when he was a grad student but later grew to appreciate them.

>>10725929
The thing is lad, you can prove the transitivity of the compatibility between atlases by an instantaneous appeal to the existence of the maximal atlas functor.
But I was seriously getting the impression you hadn't seem that proof either.

>>10726334
I think it really boils down to the definitions, even when you appeal to category theory you're kind of doing the same thing

Can anyone give any recommendations for numerical methods texts?

I'm specifically interested in numerical solution of ODEs, Dormand-Prince method etc.

>>10726957
It isn't category theory.
It's really just "Let A, B and C be atlases. Assume A is compatible with B, and A is compatible with C, but B and C are not compatible, i.e. have non-smooth transition.
Then AUB and AUC are atlases, and admit maximal atlases. Then A is contained in two different maximal atlases, which is a contradiction to the uniqueness.

differential geometry is a mental illness

>>10727560
Differential anything is usually a mental illness.

>>10727571
fuck off algebrashit
the hierarchy goes diff top > alg top > diff geo >>> shit >>> alg geo

>>10727575
>>
>>>>
What do you mean by this?

>>10727575
>can take derivatives
>big boy
>can't take derivatives
>oh no where am i where are my pants

>>10727576
The distance between diff geo and shit is about four times greater than that between diff top and alg top for example

>>10723748
weeaboo trash

>>10727579
>>can take derivatives
>>big boy
>>can't take derivatives
>>oh no where am i where are my pants
Who said this?

>>10727576
">" means is better than
">>>" means is much better than
>>10727579
>can take homology
>big boy
>no reason to take homology
>oh no where am i where are my pants
>>10727587
*3 times greater

>>10727592
Greentext doesn't have to be a quote. Come on now... newkid. I'm not saying the f-word this time but watch out.

>>10716245
thompson - calculus made easy
stewart's ok. strang is what I used.