/mg/ - maths general

[Anonymous No.11928516]

Formerly >>11918605
Ya Boy Eliashberg edition.

Talk maths.

[Anonymous No.11928519]

infinity is a real number

[Anonymous No.11928586]

>>11928519
only when he puts his hat on

[Anonymous No.11928598]

>>11928586
big hat or little hat?

[Anonymous No.11928615]

infinity doesn't exist

[Anonymous No.11928651]

>>11928516
>Elliptic curves

[Anonymous No.11928664]

>>11928519
numbers aren't real

[Anonymous No.11928666]

Which one sucks more, combinatorics or topology? I know I hate them both almost equally but I have to pick one to take and would like to minimize how much bullshit I have to put up with.

[Anonymous No.11928671]

>>11928666
>topology
bro all you have to do is draw a fuckin picture lol

[Anonymous No.11928683]

>>11928666
All of those are really cool and you are an absolute brainlet for thinking otherwise.

[Anonymous No.11928690]

>>11928666
Combinatorics is annoying but it's nice to know, since it can pop up unexpectedly literally anywhere.
But why do you hate topology, satanic trips?

[Anonymous No.11928691]

>I hate combinatorics

[Anonymous No.11928716]

>>11928666
Point-set topology is the comfiest math class you can take

[Anonymous No.11928722]

>>11928664
You're not real.

[Anonymous No.11928748]

>>11928716
Comfiest math class is definitely commutative algebra.

[Anonymous No.11928751]

>>11928722
Yes, and?

[Anonymous No.11928759]

>>11928516
Is this list good?
https://www.youtube.com/watch?v=pOThNItNuqE

[Anonymous No.11928778]

Why does it seem like so few unis offer courses on Ramsey theory?

[Anonymous No.11928784]

>>11928691
Why is it so easy to piss off CS fags and discrete math niggers? Could it be that you're aware what you do is hideous and anti-life, that it constitutes the deconstruction of everything ennobling in the human endeavor to structure and beautify the world? Is it an innate realization that within the kernel of progress you claim to be nurturing for the future there is the undoing of our entire civilization, western culture, the enlightenment, and humanity? I wonder honestly what could compel a group of people to hate themselves so thoroughly, to wish to replace themselves with a false image of intelligence, and to attempt to automate human creativity. Its really something that makes me think, desu.

[Anonymous No.11928795]

>>11928784
What if I told you that CS is math?

[Anonymous No.11928806]

>>11928795
I wouldn't care because I didn't claim it wasn't though it isn't and should never be considered math on purely aesthetic grounds.

[Anonymous No.11928812]

>>11928784
why is it so easy to trigger you into writing a seething essay with a three-word shitpost?

[Anonymous No.11928817]

>>11928784
>Is it an innate realization that within the kernel of progress you claim to be nurturing for the future there is the undoing of our entire civilization, western culture, the enlightenment, and humanity?

Math doesn't kill people, people kill people

[Anonymous No.11928835]

>>11928812
I don't feel emotions most of my waking life, I just wanted to bring to attention the tendency among the inferior to try to make a fuss about their inferiority and to beg for approval from the superior. I've succeeded as always, you always respond.

[Anonymous No.11928837]

>>11928778
>tfw the trannies have even infiltrated Ramsey theory
https://en.wikipedia.org/wiki/Structural_Ramsey_theory

[Anonymous No.11928884]

>>11928837
Ramsey theory is too based for that to happen.

[Anonymous No.11928957]

Ok bros I need some help. I can either take algebraic topology OR complex analysis in the winter, but I can't take both (this year, no matter what both will be taken). Which would you recommend? Intro to topology is in the fall, so I don't know how I feel about it yet, but I like algebra so far and hated every moment with baby Rudin. The other option is that I push ODEs to my graduating year but that will change my minor a little for better or for worse (don't really have an opinion on that yet). Let me know if you need any more info. Thx for any replies! =)

[Anonymous No.11928989]

>>11928957
Wait, you're a junior and haven't taken ODEs?

[Anonymous No.11929000]

>>11928989
Nor number theory, weird huh?

[Anonymous No.11929003]

>>11928837
Based Leo Harrington is cited on that page. He and Jeff Paris demonstrated an actual Ramsey theoretic problem that is independent of PA. I have been at conferences, and several times someone puts an open question on their slides only for a hand to go up to comment that it was already proven by Leo.

[Anonymous No.11929010]

>>11928516
Cann u guyz xplain 2 mii whyyy 2+2=5???
Thks u guize, nerdzz rool xDDD <3

[Anonymous No.11929017]

>>11928989
>>11929000
But I'm taking 7 courses this year to catch up lol I'm gonna die

[Anonymous No.11929034]

>>11929017
It's better to take an extra year than to overload on courses and end up failing some because of the work load. Anyways, I would personally go with CA between the two.

[Anonymous No.11929064]

>>11929034
>It's better to take an extra year than to overload on courses and end up failing some because of the work load
Yeah, I know, but I'm already doing a 4 year degree in 5, so I'd be starting the Ph.D. at 24/25, which is unacceptable as we all know.

>Anyways, I would personally go with CA between the two
So CA and ODE instead of Alg Top? Any reasons? My one worry is that I decide that that's what I want to do but I have to apply for masters at the start of my senior year, so I wouldn't have done it yet.

[Anonymous No.11929075]

>>11929064
Algebraic topology, relative to those other two, is a pretty specialized field. Unless that's something you want to pursue your master's in, then I'd say the other two are more beneficial. It also depends on what exactly the ODE class consists of; unfortunately, a lot of introductory ODE classes are exclusively geared towards engineering majors and as a result they are almost entirely computational and don't really cover the interesting aspects of ODEs.

[Anonymous No.11929099]

>>11928957
Both ComplexAna and AlgTopo are shit, but AlgTopo is less shit than ComplexAna.

I know you'd be studying nice functions in ComplexAna but the idea of taking a fifth analysis course (at least for me) is fucking retarded. I just want to do comfy algebra, for god's sake.

And ODEs can be fun, but as >>11929075 said, check the material first. If it's mindless computation best avoid it.

[Anonymous No.11929120]

>>11929075
>Algebraic topology, relative to those other two, is a pretty specialized field. Unless that's something you want to pursue your master's in
Well, I watched some of the OATS seminars. Like I said, I don't know if I want to do the masters there, but it's definitely on my list of possibilities.

>It also depends on what exactly the ODE class consists of
Theory and applications of ordinary differential equations; existence and uniqueness of solutions, linear systems, simple nonlinear systems. This course is theory-based and is intended for students in mathematically rich disciplines

Seems like they actually separated them. And y'know, I was also (deep in the back of my mind) considering dynamical systems, but that's a (semi)graduate course (like alg top and a course on rings and modules I wanna take next year) so that can wait until the masters.

>>11929099
>Both ComplexAna and AlgTopo are shit, but AlgTopo is less shit than ComplexAna.
I hear so many people talking about how great complex anal is tho. Is it really that bad? (H8 real anal so far though)

[Anonymous No.11929153]

>>11929120
My school also separated ODEs and linear algebra for engineering majors and math majors, but some don't. In your case, I'd definitely recommend taking that ODE class.
CA is definitely a neat field, with lots of cool stuff in it. It's a lot more interesting than Analysis 1 and you'll be able to make a lot of connections to past things that were never rigorously shown.

[Anonymous No.11929157]

>>11929120
>Is it really that bad? (H8 real anal so far though)

I had the same problem. I am on my third year of a math degree and I wanted to take complex analysis so badly, mostly because I was fucking obssesed with how colorful the plots of complex functions were.

But then I had an obligatory real ana course and I hated almost everything there. I didn't care about Fourier analysis (it was just rushed through), curves and surfaces which were cute but again, rushed through and vector fields were just mentioned casually. We didn't even get to line integration.

So I can't really think that a complex ana course would be better, simply because how broad of a subject you're dealing with and how actually unprepared we are.

On the other hand, everything relating to algebra was building up on itself, so taking four algebra courses and now a fifth one never felt like a chore. Analysis was just.. all over the place.

inb4 smoothbrain, my analysis grades were pretty good overall.

[Anonymous No.11929168]

>>11929157
Fourier analysis is peak applied math.

[Anonymous No.11929172]

>>11929168
The way we did it, it never felt that way.

[Anonymous No.11929182]

Well, I shall shitpost for the remaining 10 minutes and then I will try this time planning schedule thing, hope I get my life fully in order.
From 8am to 10am I shall be doing Topology, no one @ me.

[Anonymous No.11929191]

>>11929168
you mean dynamical systems and probability.

[Anonymous No.11929192]

>>11929153
The one final caveat for CA I should throw in there is that I had a horrible multivariable calc class, so I'm worried theres a lot (or even some) of that in CA

>>11929157
>Fourier analysis
>Differentiable maps from [math] \mathbb{R}^n
[/math] to [math] \mathbb{R}^n [/math]
Lol that's RA 3 for us, I don't think I'll even be taking that one

[Anonymous No.11929199]

>>11929192
You sound like a woman lol

[Anonymous No.11929209]

>>11929192
>Differentiable maps from [math] \mathbb{R}^n \to \mathbb{R}^n [/math]
bro our second half of ana3 was this shit, literally the most unintuitive, dry and annoying part of my education I had to suffer through.
I think the major problem is how unstandardized the notation is.

[Anonymous No.11929213]

>>11928516
>Eliashberg
Reminds me to finally work through "Introduction to the h-Principle". Why is that kind of topic, or related diffiety, so rarely studied anyway? Pleb filter?

[Anonymous No.11929221]

>>11929192
The only multivariable stuff that will show up in CA is partial differentiation and from vector calculus, parametrization.

[Anonymous No.11929239]

>>11929221
>parametrization
I think you’re underestimating how shit that class was by a little. That’s at least a little reassuring though

>>11929199
Well, while I am an anime tranny, I’m not a woman, so swing and a foul ball?

>>11929209
>unstanderdized notation
Oof, sounds like a pain and pretty uninteresting. Glad I’ll probably get to skip it

[Anonymous No.11929262]

>>11929239
>skipping vector analysis
Don't, like absolutely don't. If notation keeps you from understanding stuff than that simply means that you didn't properly understand the preliminaries. Notation in vector analysis (read: broadly everything involving multivalued functions on something looking like an open set in [math]\mathbb{R}^n[/math]) are always adapted to the more specialized field you are working with. E.g. in DG you will most see [math]\tfrac{\partial}{\partial x^i}[/math], while in more analysis focus PDE theory you use [math]\partial_i[/math]. Understanding real multivalued aka vector analysis is crucial for anything that's not pure algebraic unintuitive nongeometric bullshit desu. Sure, you don't need it for applied (lol) category theory, but there's a reason this field is overrun by trannies by which I mean real life ballchopping flip-from-the-bridge trannies.

[Anonymous No.11929273]

>>11929262
I mean even if I don’t go the route of alg top/cat theory, I sure as hell am not doing DG or PDEs (not even taking a class in PDEs), I’d definitely be going down an algebra route (or maybe topology)

[Anonymous No.11929274]

>>11929239
No, I mean everything you complain about and seem averse to requires strong logical reasoning skills and visual-spatial intelligence. You sound like you are cognitively female. Being afraid of analysis is embarrassing

[Anonymous No.11929281]

>>11929274
>visual-spatial intelligence
Nah dude I’m hyped for topology, I just had a bad class with baby Rudin and a prof who was bad at explaining things or making anything interesting. I’m not afraid of analysis it just doesn’t interest me.

[Anonymous No.11929285]

>>11929273
>route
So you're an undergrad? As someone who heard this kind of "fixing" of a goal from undergrads maybe many many times with nearly all of them either failing (because turns out limiting yourself early is retarded, who would have thunk) or becoming bitter (because turns out that another topic would have suited them better, but muh pure algebra). All of the latter quit academia.

So yeah, being able to deal with basic abstract algebra like groups, non-[math]\mathbb{R}^n[/math] problems, is important. But a mathematician who's not able to understand real multivariable calculus + analysis is at least equally dumb as one not able to deal with abstraction.

[Anonymous No.11929294]

>>11929285
>asking about which undergrad courses to take
>so you’re an undergrad?
It’s not like I’m taking zero analysis, I just am pretty certain I don’t want to specialize in it. I’m taking what I have to and moving on. Hell I have one course I haven’t decided on yet, it might be that RA one. Sorry I’d rather be an elliptic curve gigachad instead of an analysis megaultraüberchad.

[Anonymous No.11929299]

>>11929285
Is there a bigger trap than the pure algebra meme?

[Anonymous No.11929303]

>>11929294
Take a course in stochastics.

[Anonymous No.11929306]

>>11929294
Welp just don't close your mind off completely from analysis then. Also I'm not too familiar with the burger system, but are you allowed to simply participate without exam in other courses? Like, for free? If not consider moving to the eu (or Russia, unironically), for all the bullshit we got here, the math departments are pretty neat. In particular in Germany, Italy, Cechia, Romania, Denmark. NOT France or Bongland though.
>>11929299
No.

[Anonymous No.11929316]

>>11929306
It's called auditing a class. You can usually do it with consent from the professor.

[Anonymous No.11929329]

>>11929303
Lol we don’t have one (I don’t think at least)

>>11929306
>burger system
Please don’t lump me in with them, I’m Canadian. We can (sit in lectures, though I think it still cost a small amount), so depending on how things go I might do that, though I don’t mind self teaching myself things that I won’t use often (I did some set theory over the summer for instance). Too bad for you I only speak English and French well so while I could struggle in Deutschland I don’t think it’s be a very smart idea to go to Europe

>>11929316
This

[Anonymous No.11929337]

10-15 min break from Topology

>>11929239
>Oof, sounds like a pain and pretty uninteresting. Glad I’ll probably get to skip it
As some anon pointed out, don't. You need to know this stuff. I am not avoiding complex analysis because I don't want to know it, but simply because I have too many courses already, and I could definitely pull a self-taught course over a summer, simply because I already took all the previous analysis (analysii?). Similarly for DiffEq; I could do it on my own, and I will at some point, but I don't want it to be main focus.

On the other hand, I am taking a lot of algebra and algebra related courses so I will be doing Topology and Functional Analysis just to break the monotony and get a broader understanding of math in general. Undergrad courses are all mostly doable, and I see no reason not to get a wide curriculum.

[Anonymous No.11929348]

I know that this is a platinum post, but is there any push for multi-letter variables in math and math-adjecent fields?

[Anonymous No.11929353]

>>11929348
what

[Anonymous No.11929356]

>>11929329
Are you sure? I'd imagine stochastic processes would be a required course for any applied math major.

[Anonymous No.11929359]

>>11929348
I've literally never seen someone use a multi-letter variable in mathematics. Any minor increase in readability by "naming" variables like a programmer is going to be far counteracted by the fact that any remotely complicated equation is going to balloon into an absolute monstrosity.

[Anonymous No.11929367]

>>11929359
Physics is already does this through subscripts.

[Anonymous No.11929370]

>>11929337
I’d be skipping it for the same reason you’re skipping CA, not enough courses. My uni is pretty slow with the material (I feel like) so I’m scared if I don’t rush through all the things I wanna go into I’ll be behind during the masters and have to catch up. Idk, I probably should talk to some of the profs and get their opinions (on that Ra course specifically)

>>11929356
>applied
Nah you’ve got the wrong guy, I’m in pure math. Even then I couldn’t find one. What do you think it would be grouped under?


Anyways I’m going to sleep, I’ll catch you guys in the morning

[Anonymous No.11929373]

>>11929370
Night, lad.

[Anonymous No.11929486]

>>11929370
Night animefag

[Anonymous No.11929774]

I fucking hate anything related to analysis, I wish all math was just algebra, geometry and topology.

[Anonymous No.11929788]

>>11929774
>geometry, topology
>not related to analysis
?

[Anonymous No.11929823]

>>11929239
L O N D O N

[Anonymous No.11929828]

>>11929788
Geometry isn't, except analytic geometry which is boring shit. Topology is more vast than analysis.

[Anonymous No.11929834]

>>11929828
>what is differential geometry

[Anonymous No.11929839]

>>11929774
algebra is analysis on finite fields
geometry is analysis in different planes than IR^2
topology is general analysis

[Anonymous No.11929847]

>>11929774
>Hating analysis

Why?

[Anonymous No.11929858]

I'm about to finish my bachelors and I have never had a class on complex analysis.

[Anonymous No.11929863]

>>11929858
You should take it if you can. Complex is a beautiful; and comes up in a lot of places you would not expect.

[Anonymous No.11929865]

>>11929863
[math] \text{based} [/math]

[Anonymous No.11929866]

>>11929281
>>11929774
all interesting problems can be solved with analysis and geometry. all useful problems are solved with analysis and geometry. all great problems will be solved with analysis and geometry. sorry!

[Anonymous No.11929875]

>>11929847
algebra: basic object of study is a group. one operation, three axioms.
topology: basic object of study is a topological space. one property, three axioms.
analysis: basic object of study is probably real numbers. this is an extremely complicated structure, it's an ordered field, it's a topological space with many distinguished properties, the basic notion (limit) is a logical formula which is a total pain in the ass to work with etc.

you read a definition of a group and you can start proving theorems on the spot. it takes more time to truly get into analysis. many people resort to hating on it because it doesn't go as neatly and it makes them feel like brainlets, basically without giving it a proper chance.

>t. had to relearn analysis because I was like that in undergrad

[1d central extensions No.11929892]

If you're bored this morning

https://youtu.be/IYBY0qe01x8

[Anonymous No.11929901]

>If I'm good at combinatorics? Oh, no, far from it. I am horrendous at combinatorics, which is why I have so much respect for the field.

[Anonymous No.11929907]

>>11929892
Do you lift? Post your routine

>>11929901
>Am I good at combinatorics? Of course not, that's why I simulate all problems.

[Anonymous No.11930004]

/gdmg/

>>11928957
Algebraic topology is cool. I recommend taking that if you want to do abstract non-sense. It is very algebraic, too, so I would assume you will like it.

>>11929370
Nighty night, mon ami.

[Anonymous No.11930008]

>>11928666
Combinatorics is trash, that's for sure, sad thing is, it's one of the most active areas of research right now, IT needs the finite stuff, Topology is absolute bae tho, algebraic is fun, difftop is really hard to wrap your head around (like tangent vectorfield is a differential equation and sruff like that), but what I've learned was pretty interesting and useful (for Riemann geometry for example)

[Anonymous No.11930014]

>>11929182
2 hours uninterrupted sounds a bit too much, how did you manage?

[Anonymous No.11930022]

>>11930014
Fine. I did take a 10 minute break somewhere after the first 50 minutes to make a coffee. I covered Metric and Quotient topologies in Munkres, so I feel accomplished. Later today I will do the exercises as well (at least some), but the schedule says I have break up until 5pm and then an hour and half to work.

[Anonymous No.11930025]

>>11930014
>2 hours of unbroken focus is too much
Just lol, try 6 hours

[Anonymous No.11930033]

>>11930025
6 hours seems really a lot, Idk how I could do that without uppers

[Anonymous No.11930034]

>>11930022
whats the 10-5 filled with

[Anonymous No.11930042]

>>11930034
I had a break from 10 to 11 to breakfast, then I had another 3h of working. So from 1pm to 5pm I'm taking a long break

[Anonymous No.11930049]

>>11930033
>without uppers
Pure anger and frustration at yourself for being a retard, but usually its just a lot of uppers desu

[Anonymous No.11930164]

>>11930042
gj anon, keep it up

[Anonymous No.11930249]

>>11929858
I'm finishing up my dissertation and neither have I

[Anonymous No.11930434]

>>11929866
https://www.youtube.com/watch?v=oHC1230OpOg

[Anonymous No.11930442]

F-friends, say a have a domain [math]\Omega[/math] in [math]\mathbb{R}^n[/math], and a compactly supported function f in the sobolev space [math]W^{1,1}(\Omega)[/math]. If the support of f is a subset of Omega I know that for small enough r I have
[eqn]\partial_i f_r(x)=\partial_i f* J_{(r)}(x),[/eqn]
where [math]J(x)=Ce^{\frac{-1}{1-|x|^2}}\chi_{B(0,1)}(x)[/math] is the standard mollifier and [math]J_{(r)}(x):=\frac{1}{r^n}J\left(\frac{1}{r}x\right).[/math] Now let's say that the support of f is not contained in Omega, I extend f to [math]\overline{f}[/math] outside of Omega by 0 and do the same mollification. I know that the new function is not necessarily weakly differentiable but can I get a similar relation as above? In particular I'm interesed if I can write something like
[eqn]|\partial_i \overline{f}_r(x)|\leq C_1|\partial_i f* J_{(r)}(x)|?[/eqn]

[Anonymous No.11930487]

how do people come up with new mathematical ideas
for instance to use auxiliary functions

[Anonymous No.11930509]

>>11930442
seems trivial
what's your IQ?

[Anonymous No.11930510]

>>11930487
>think what you want
>think what you need
>try to construct what you need
>fail a myriad of times
>succeed

[le tooky no1 fan No.11930514]

>>11930510
>>fail a myriad of times
This is the most important step

[Anonymous No.11930518]

>>11930509
85,
so can you do it?

[Anonymous No.11930528]

>>11930514
Unironically yes. That's a good way to find vulnerabilities in the construction.

[Anonymous No.11930530]

>>11930518
>so can you do it?
yeah, but I'm leaving it as an exerecise to (You)

[Anonymous No.11930535]

>>11930442
Remind me for a second. If [math]\Omega = [0, 1][/math] and [math]f = 1[/math], what happens with the derivative at the border?

[Anonymous No.11930539]

>>11930510
>>11930514
tthanks :<

[Anonymous No.11930550]

>>11930535
Thank you!

[Anonymous No.11930775]

Are algebra, topology, and analysis the courses you'll be taking during your first semester in grad school?

[Anonymous No.11930784]

>>11930775
not THE courses as in "that and nothing else", but that's the fairly standard core sequence for first-year grad students in the USA

[Anonymous No.11930813]

>>11930784
Will your other classes usually be up to you to choose with those three I mentioned being the required ones?

[Anonymous No.11930835]

>>11929774
Just give it a chance. Analysis can be beautiful
>Geometry
Two words: differential geometry

[Anonymous No.11930843]

>>11930813
I don't know about the US, but we had a list of courses for each specialisation that one was required to pass in order to get the master's degree. There could be some compulsory stuff like Sobolev spaces if do anal or algebraic topology if you do topo etc. Otherwise I would assume it is pretty free to choose whatever you like.

[Anonymous No.11930844]

>>11930835
Two words: Planar geometry.

[Anonymous No.11930849]

>>11929294
>Sorry I’d rather be an elliptic curve gigachad instead of an analysis megaultraüberchad.
I'm sorry that your goals are informed by internet memes

[Anonymous No.11930855]

>>11930844
Wow dude, analysis rendered useful to geometry?? Planar geometry is not even an active field of research

[Anonymous No.11930857]

>>11930855
Useless*

[Anonymous No.11930866]

>>11930843
Thanks, animeposter.

[1d central extensions No.11930867]

>>11929907
Squats to Deadlifts, and then what I feel like that day. My brother is a physician and gymrat, tho, I should have all the means to do more.

Hey does anybody know what to formally do with ones imposed initial condition when passing from the Euler-Lagrange equations to the stationary action principle.
I'm a tad confused right now, since I don't recall anyone ever discussing the procedure of how to search for the stationary trajectory under [math]\delta S=0[/math] while imposing initial condition on [math]q'(t_1)=v_{t_1}[/math].

[Anonymous No.11930875]

>>11930813
Pretty much. Even the core classes are usually "up to you". Most US schools I know do not have REQUIRED courses. If you can pass the analysis qualifying exam they don't really care where you learned your analysis.
some of it is essentially locked in by your specialization though. there isn't a formal list of courses you have to take like the other guy described for Europe, at least not at any school I've seen, it's just implicit that if you want to specialize in e.g. topology you will take the available topology-related courses unless you already know the content of the course.
The US analog of a "list of required courses" is probably closest to the oral exam you take with a few members of your group where they grill you about more specialized things they think you should know for a while.

[Anonymous No.11930895]

>>11930875
Thanks. That makes sense, I just remember seeing a video where the person said that typically a first year grad student will take algebra, topology, and analysis.

[Anonymous No.11930900]

>>11929823
Why do you keep spamming this?

[Anonymous No.11930903]

https://bicornum.neocities.org/JuliaSet/JuliaSet.html

[Anonymous No.11930912]

>d00d fractals are so beautiful!!
>literally just a blob
explain this analshits

[Anonymous No.11930918]

>>11930912
https://en.wikipedia.org/wiki/Burning_Ship_fractal

[Anonymous No.11930966]

>Whether or not the proof's technicalities are messy, the idea is as simple as "draw a transverse line with nontrivial intersection, and segments of this line alternate between lying inside and outside of S^n." Sure, checking that these things are two different connected components is nasty. But the point of topology is that one relegates nasty things like that to elementary theorems and lemmas so that one may focus on the geometric picture.
Can someone explain this proof to me? Firstly, to be able to talk of transversality it seems to me like we need to be talking about smooth things. I understand we can find a smooth approximation of the embedding but why must an approximation preserve separation? Surely one being close enough to the other is not enough to guarantee it?
Assuming we've taken care of this (or restrict to only smooth embeddings) surely we want the number of intersections of the sphere with the line to be finite to be able to talk about alternating segments. How can we ensure this?
>Sure, checking that these things are two different connected components is nasty
How nasty is it? I've no idea even how to approach this. What does the proof of this look like?
>But the point of topology is that one relegates nasty things like that to elementary theorems and lemmas so that one may focus on the geometric picture.
Does /mg/ agree with this? For me it seems like anon simply made the whole thing more complicated by intruducing more objects into the picture without actually simplifying anything, i.e. just complicated the picture. In what way is this more elementary than the question being asked ( S^n separating R^(n+1))?
Do most topologists think like this? This is so bizarre to me, but then again, I'm only a noob in topology.

[Anonymous No.11930975]

>>11930966
he's talking out his ass
did you not see literally anything in the thread after that post

topologists do have bizarre thought patterns, but this is not one of them. this is just smugness+stupidity

[Anonymous No.11930988]

>>11930975
Idk, there seem to be other anons like >>11928744
defending him in that thread

[Anonymous No.11930995]

>>11930988
what do you mean? he's not defending him at all, he's just spending multiple hours embarrassing himself arguing over what the word transverse means

[Anonymous No.11931012]

>>11930995
Look guy, the whole thread there is just people bickering and sniggering. It's not clear whether or not there is substance behind what anon said, so don't be giving me a hard time, ok? All I want to know is can the "technical details" (i.e. the proof) actually be worked out like he said it can.

[Anonymous No.11931029]

>>11928516
Serious question ahead:
Is it possible to an "humanities" kind of person to learn and be prolific with mathematics? I want to go back to the Uni. I have a job so I'd going on mornings and maybe take more years to end the maths career. I have the time and money, but I've never been good on mathematics in my childhood (though I never was on other things that later I could do).

What's a good starting point with maths to know if I will enjoy it?

[Anonymous No.11931031]

>>11930966
>>11930988
As far as I know about topology, insisting on rigor has been *extremely* historically fruitful.
Being handwavy is convenient as a preliminary tho.

[Anonymous No.11931045]

>>11931031
Yes but anon is implying this can be extended to a rigorous proof. I'm asking how.

[Anonymous No.11931055]

>>11931029
Yes. Read about Fermat.
>What's a good starting point with maths to know if I will enjoy it?
Khanacademy
>>>/sqt/

[Anonymous No.11931057]

>>11930966
it's hilariously pathetic that you're asking /mg/ for arguments against me.
>>11931031
obviously it's been historically fruitful. there is a massive difference between proving extremely fundamental point-set and manifold topology facts versus doing modern low dimensional topology. we're talking about how a modern topologist thinks and does things, not about how a proof in a topology textbook should work.
>>11931045
i'm telling you, it's a long and complicated and messy argument from there, with a lot of ugly technicalities. but once they're done, they're done. look it the fuck up, no one here is going to regurgitate the steps to you.

[Anonymous No.11931065]

>>11931045
There are some very strong smooth approximation theorems out there, so assuming smoothness is probably not an issue.
The remainder of his proof is basically a non sequitur tho.

As far as I recall, proving that S^(n-1) separates R^n into two connected components is usually done as follows:
>if an element has norm smaller than one, it's path connected to zero by the convex path
>if an element has norm larger than zero, it's connected to some previously chosen vector through explicit computation in polar coordinates
>the vectors with norm smaller than one aren't connected to the vectors with norm larger than one because of intermediate value theorem

Proving that S^k doesn't separate R^n is hard as shit tho.

[Anonymous No.11931067]

>>11931057
>look it the fuck up
Where?

[Anonymous No.11931070]

>>11931065
*larger than one, not zero.

[Anonymous No.11931086]

>>11930966
ok, now that i'm reading your post i can answer some of your basic questions.
1. yes, whitney approcimation says we can find an approximation which is arbitrarily close. using the tubular neighborhood theorem and compactness of the embedding of S^n, we find an isotopy of the two objects, which is enough to say the separation is preserved (look at the complement).
2. we get finite number of intersections with the sphere due to the fact that the sphere is compact and that the line is transverse. hence the intersection is a compact zero-manifold which must have finitely many points.
>how nasty is it? i have no idea even how to approach this
it uses signed intersection theory (which is what i'm talking about here) to do some fancy stuff with rays sweeping out paths on the inside between points. look up a proof of the jordan curve theorem for a special case. not hard, just long and messy to formalize.
>For me it seems like anon simply made the whole thing more complicated by intruducing more objects into the picture without actually simplifying anything
if you don't know the tools people use in topology all the time, i.e. intersection theory and transversality, then i don't know why you're so upset about them. it's almost like you just don't like topology.
>>11930975
i'm really not talking out of my ass at all

[Anonymous No.11931088]

>>11931067
ANY FUCKING UNDERGRAD DIFFERENTIAL TOPOLOGY TEXTBOOK! HOLY FUCKING SHIT!

[Anonymous No.11931091]

>>11931065
we're talking about arbitrary embeddings of S^n-1 into R^n, not just the typical S^n-1.
proving that (the typical) S^k does not separate R^n is not hard at all, what could you possibly be on about? literally just exhibit any path from 0 to some point with norm more than 1.

[Anonymous No.11931110]

>>11931088
Looked up Guillemin&Pollack and Milnors Difftop books, neither have the proof.

[Anonymous No.11931112]

>>11931091
>proving that (the typical) S^k does not separate R^n is not hard at all, what could you possibly be on about? literally just exhibit any path from 0 to some point with norm more than 1.
There seems to be a misunderstanding.
I thought that you were proving "for some embedding of S^n-1 in R^n we have separation and for no embedding of S^k into R^n" rather than "for any embedding of S^n-1 in R^n and for some embedding of S^k into R^n".
Eihter works, I suppose.

[Anonymous No.11931114]

>>11931091
Shouldn't it be sufficient to show it for the usual embedding and then just use the fact that any other embedding gives you a homeomorphic image of the sphere. Whatever happens to connectivity in the standard case will then be transferred to the general case via the homeomorphisms.

[Anonymous No.11931133]

>>11930900
I need a cute anon to come be my house husband.

[Anonymous No.11931142]

>>11931057
>it's hilariously pathetic that you're asking /mg/ for arguments against me.
I asked for someone in the previous thread to explain it to me. No one responded, so I asked here. All I want is to understand the proof because I'm a noob. Your assumption of hostility says more about you than it does about me.
It seems clear that you refuse to explain your proof, so perhaps you could point me to a resource that does?
I've read differential topology books before and not one of them talked about this proof. Where could I find it? I'm genuinely interested. A book, a paper, a MSE post, anything.

[Anonymous No.11931147]

>Pure mathematics is, in its way, the poetry of logical ideas.

[Anonymous No.11931157]

>>11931133
hmm, depends if it's London Ontario or london Britbongland

>>11931029
>What's a good starting point with maths to know if I will enjoy it?
At my uni the "discrete math" course was the first proof based course (ish). That'll let you know if you like it

[Anonymous No.11931171]

Maybe someone else has read an undergraduate topology book which covers the proof that topologist-sama is talking about here?

[Anonymous No.11931184]

>>11930995
The whole point was that in the context of a handwavy argument you expect handwavy terms and the word transverse is not exclusive to differential topology, well in the context of a line or hyperplane dividing a sphere, it is trivial and you don't need that definition ffs, it's just a line crossing a sphere,i,e not tangent to it. What I vent off on is that I know this autistic shit mathematicians do with languange in being retardedly inflexible with terminology. For example what is the answer to
>what is the definition of a curve in math?
A reasonable answer is to say
>well it dependes on the context really
But still people start sperging out because in some book they read that the term was used for
>A continuous function from an interval of the reals to the plane
>A continuous function from an interrval of the reals to any euclidan space
>A smooth variation of either of these
>A one manifold (in whatever category the autists knows)
>A one manifold embbeded into some n dimensional euclidean space
>A continuous function from an interval of the reals to a topological space
Etc. And saying "x is trasnsverse to y" is not exclusive to differential topology and for most cases people use the term to refer as somthing crossing through another thing. I wasn't denying there was a definition of a property called "trasnversality" that needs that technical detail in order to be a generic property, but that thw word transverse has an obvious unambiguous meaning in the context the other guy was presenting because it is also used everywhere else in the sense I was thinking about. It's pretty retarded to get worked up about that, but it pisses me off how autistic people start to sound when doing higher level math .

[Anonymous No.11931185]

[math]\mathbb{Z}[/math] is Noetherian but not Artinian. What's an example of a ring that's Artinian but not Noetherian?

[Anonymous No.11931187]

>>11929875
>algebra: basic object of study is a group. one operation, three axioms.

[Anonymous No.11931198]

>>11931184
lmao
absolute sperglord

[Anonymous No.11931202]

>>11931185
None. Artinian implies Noetherian.

[Anonymous No.11931217]

STOP FUCKING BLUEBALLING ME AND TELL ME WHERE I CAN FIND THE FUCKING PROOF YOU KEEP REFUSING TO EXPLAIN!
EITHER THAT OR ADMIT THERE IS NONE YOU UTTER MONGOLOID!!!

[Anonymous No.11931226]

>>11931202
Why do we make a distinction for Artinian rings, then?

[Anonymous No.11931233]

>>11931226
Holy shit! That's a great point. How stupid we mathematicians have been not to notice this! I will include your name in the paper.

[Anonymous No.11931238]

>>11931226
...
...
...
Because noetherian doesn't imply artinian?

[Anonymous No.11931250]

>>11928835
>Im a psychopath Xd

Fucking cringe

[Anonymous No.11931254]

>>11931184
>>11931086
>>11931057
Respond to my posts, fool.

[Anonymous No.11931258]

>>11931233
Thanks.

[Anonymous No.11931272]

>>11931250
C'mon now, it's kind of cute. Let's just give him a chance to grow more edges.

[Anonymous No.11931276]

>>11931250
>I'm a psychopath

[Anonymous No.11931292]

>>11931226
>if square implies rectangle why have both concepts

[Anonymous No.11931299]

>>11931292
This but unironically.

[Anonymous No.11931330]

Any books on the social nature of topological truth?

[Anonymous No.11931342]

Is the princeton analysis series good to learn analysis and complex analysis after tao?

[Anonymous No.11931351]

>>11931342
Yes.

[Anonymous No.11931363]

>>11931342
No.

[Anonymous No.11931366]

>>11931363
>>11931351
reddit intensifying

[Anonymous No.11931389]

>>11931351
>>11931363
The reason I ask is because I was wondering whether Papa Rudin, or Needham's Visual Complex Analysis/Ahlfor's Complex Analysis was worth picking up instead.

Essentially, I wanting to get to a graduate level of analysis from where I am now.

[Anonymous No.11931400]

I fucking hate topologists...

[Anonymous No.11931405]

>>11931226
I feel a bit guilty when pointing this out, but you yourself gave a non-Artinian Noetherian ring to us as an example. It's OK, anon. We love you anyway.

[Anonymous No.11931423]

>>11931400
>hate topologists...
woah everybody
looks like we got a homeomorphiphobe in the thread

[Anonymous No.11931439]

help me boys, how would i go on about checking if two given operators have all of their eigenvectors in common?

[Anonymous No.11931441]

>>11931400

[Anonymous No.11931449]

>>11931389
If you went through Tao I and II understanding most of it and doing the exercises, then you're in a good position to move onto to grad level analysis which the Princeton series is a good introduction to.

[Anonymous No.11931454]

>>11931439
figure out the eigenvectors of first operator
figure out the eigenvectors of second operator
compare the sets

[Anonymous No.11931460]

>>11931449
thanks

I assume that it's only an intro to grad analysis because reading articles becomes necessary eventually; would I need to supplement the series with ahlfors and rudin, etc because I've heard Stein's quite conversational?

[Anonymous No.11931467]

>>11931439
Wasn't that equivalent to the Lie bracket zeroing or something?

[Anonymous No.11931468]

>>11931439
this should do it :)

[Anonymous No.11931489]

>>11931460
You could look into those, or you could keep going with Tao's books. His Intro to Measure Theory book is really good and his other analysis books are just as good and his stuff will take you up to about a second year grad student level.

[Anonymous No.11931492]

>>11931468
Kek

[Anonymous No.11931497]

>>11931467
iirc if two operators commute then they share the same set of eigenvectors, so this should be enough, assuming Lie bracket is the commutator [A,B] = AB - BA

[Anonymous No.11931507]

I have a question to all the PhD students here. When you give a talk about your progress do you feel like anyone beside your supervisor cares about your results?

[Anonymous No.11931509]

>>11931439
Well, it's trivial if we are talking finite vector spaces. What kinda operators are we talking about

[Anonymous No.11931517]

>>11931507
Piggybanking off of this: Do PhD students see TAing as nuisance or do you actually enjoy doing it (barring the grading, which obviously sucks)?

[Anonymous No.11931522]

>>11931517
>>11931507
kys

[Anonymous No.11931523]

>>11931441
what hole?

[Anonymous No.11931531]

>>11931522
No.

[Anonymous No.11931536]

>>11931531
Then at the very least post your retarded faggot questions in /sqt/ instead of shitting up this place.

[Anonymous No.11931539]

>>11931507
Most likely not. My work is, as my supervisor put it a week ago, highly esoteric. I am starting to see possible connections to other stuff, but I am also used to being disappointed at myself and my inability to actually do something worthwhile, so no high hopes yet.

>>11931517
Giving tutorials was hell. There were like 3 active students in the group of more than 60 people and they were complaining about me not answering their questions. I do admit I didn't, though. It is hard to read people's minds if they don't ask.

[Anonymous No.11931542]

>>11931536
Ok.

[Anonymous No.11931548]

Suppose I have some monotonic decreasing function f(x) where f(0) = 1, and as x goes to infinity f(x) -> 0. Is the fourier transform of this function also a monotonic decreasing function?

I feel like I should try the Riemann Steljes integral but I'm not sure how to use it here.

[Anonymous No.11931561]

>>11931489
Ok thanks. Tao's real analysis was quite nice so I'll supplement and then follow Shakarchi with Tao's measure theory and then his epsilon. I'll probably supplement ahlfors and needham for complex too anyway if time allows.

[Anonymous No.11931569]

>>11931561
Tao's measure theory picks right up from where II left off, so it'll follow nicely from what you already know. Cohn's Measure Theory is also really good, if you want a more in depth look on the topic.

[Anonymous No.11931570]

>>11931539
>Giving tutorials was hell. There were like 3 active students in the group of more than 60 people and they were complaining about me not answering their questions.
We split into smaller groups than that, maybe 20-30. Still only 3 active people per group. I think it's not bad that the rest is whining. It just shows that they are seething brainlets who don't belong.

[Anonymous No.11931572]

erklaeren sie konforme Karten

[Anonymous No.11931578]

>>11931570
>That one student that always is engaged with the material and asks genuinely good questions

[Anonymous No.11931582]

>>11931572
enjoy your summer

[Anonymous No.11931583]

Does going to university actually help one become a better mathematician over self study?

[Anonymous No.11931589]

>>11931582
ich liebe mathematik und sprachen unter der Sonne

[Anonymous No.11931593]

>>11931583
Unless you're Abel-tier, studying in an academic context will always be preferable to doing it by yourself regardless of the subject.

[Anonymous No.11931595]

>>11931583
>is interacting with people more skilled than you helpful in developing an ability
yes

[Anonymous No.11931597]

>>11931570
Yeah, I had 3 groups but still only 3 people in total. It was a waste of time for both most of them and me, but I don't care. At least they paid like 15 pounds a session or something like that.

>>11931578
He was nice, and also pretty handsome.

>>11931583
It gives you access to experts for question asking purposes, and so on. Suppose you are stuck with something. Then it helps to be in uni, or to go to /sqt/.

[Anonymous No.11931602]

>>11931330
I fucking hope not, that sounds like a schizo dumpster fire.

[Anonymous No.11931606]

>>11931330
ask any /mg/poster, topologists do not use rigor and therefore there is no such thing as topological truth

[Anonymous No.11931609]

>>11931572
>>11931589
God didn't invent the beautiful english language for people to speak whatever the fuck that monstrosity is.

[Anonymous No.11931612]

>>11931330
Dune by Frank Herbert

>>11931593
>>11931595
>>11931597
i considered this, but i just wonder why it's not the same to just ask people online when you get stuck

[Anonymous No.11931614]

>>11931578
god bless.
>>11931597
The best part is always when answering a question broadens ones own horizon.

[Anonymous No.11931616]

>>11931609
English has its roots in Germanic languages.

[Anonymous No.11931626]

>>11931609
German. English is concise, but accurate descriptions are inelegantly short-breathed. German is long-winded, and is more elegant for complex constructions.

[Anonymous No.11931628]

>>11931593
studying in an academic context would've been preferable for Abel too. he was gigabrain enough to succeed despite being isolated with shit resources, but he would've done better if he lived around people who were on or above his level. having professors and colleagues who know things is incredibly helpful no matter how brilliant you are

[Anonymous No.11931637]

The only relevant languages for math are English and French. If you speak both, you have access to all the literature.

[Anonymous No.11931639]

>>11929157
CA is VERY different. Nigger, you don't have the slightest clue what different it makes for a function to either be [math]C^0[/math] or [math]C^\infty[/math], NOTHING inbetween.

[Anonymous No.11931655]

>>11931637
ich sprache englisch und franzosiche, aber ich musse deutsche fur das universitat lernen

[Anonymous No.11931657]

>>11931637
This.

[Anonymous No.11931660]

>>11931628
True.

[Anonymous No.11931664]

>>11931655
>universitat
universität retard

[Anonymous No.11931667]

>>11931612
Face to face it is easier for your conversation partner to help you ask the question you just can't get out the way you wanted to. Online you and your slightly malformulated question will not enjoy that luxury.

>>11931614
Indeed it is. If there is live tutoring this autumn, I hope I get some 3rd year stuff so I can see him again.

[Anonymous No.11931677]

>>11931548
Bro,
[eqn]\cal F[f(x)](k) = \int_{-\infty}^\infty \frac{{\rm d}k}{\sqrt{2\pi}}e^{-ikx}f(x)
=\int_{-\infty}^\infty \frac{{\rm d}k}{\sqrt{2\pi}}\cos(kx)f(x)
+i\int_{-\infty}^\infty \frac{{\rm d}k}{\sqrt{2\pi}}\sin(kx)f(x)[/eqn]
your function isn't even guaranteed to be REAL. How can it be monotonic
if we don't even know it's real? But alright, suppose we continue the function
such that [math]f(-x)=f(x)[/math].

[Anonymous No.11931683]

>>11931677
My bad. The function is indeed symmetric with f(-x) = f(x).

[Anonymous No.11931686]

>>11931655
super
aber /em-gay/ habe eine sprache
deine lehrerin ist ein taschenrechner, wir sind ein tischrechner

[Anonymous No.11931689]

>>11931637
>forgetting about the russkies
I think you could almost argue Russian is more important than French is
Russia is somewhat detached from the western-European math community, especially comparing to France, so there is a lot more untranslated stuff, especially from the Soviet era but even today there's a lot.
There's also the consideration that French is similar enough to English that you can easily bumble your way through a French math text knowing almost literally no French. This is much, much harder to do in Russian.

[Anonymous No.11931693]

Können die Piefke bitter aufhören den thread zuzuspammen? Danke.

Then again, this thread is so shit even without it, nothing happens here. Does anybody of you even still do math?

[Anonymous No.11931697]

>>11931689
Name one relevant work in Russian who who wasn't translated in English.

[Anonymous No.11931698]

>>11931609
>>11931616
>tfw realizing English probably looks and sounds just as hideous to non-English speakers as German looks to us
Anglos were a mistake
the Britons should have won

[Anonymous No.11931704]

>>11931697
Everything by Cantor and Kolmogorov.

[Anonymous No.11931706]

>>11931698
>tfw you don't live in the timeline where britonic became the world's most spoken language
Not fair bros.

[Anonymous No.11931707]

>>11931698
The french should have won

[Anonymous No.11931710]

>>11931616
Nuh uh. God invented English long before germanic languages even existed, and I have irrefutable proof.
In Genesis 1:3 God says, and I quote, "Let there be light." What language is "Let there be light" in? English. Now, how could God speak English when there weren't even people yet? This is day one of all time, see. The conclusion is thus that, before even inventing light, the LORD invented the glorious English language, God bless.

[Anonymous No.11931711]

>>11931698
I can confirm this. It is an easy language, but it can be extremely ugly.

[Anonymous No.11931715]

>>11931711
I thought English was actually one of the harder languages to learn? Unrelated, but Norwegian is surprisingly easy to learn for a native English speaker.

[Anonymous No.11931719]

>>11931710
The Bible was written in Hebrew.

[Anonymous No.11931723]

>>11931704
>Cantor
Published in German, don't know why you even thought that he published in Russian
>Kolmogorov
Everything (relevant) is translated, mainly in Selected Works of A.N. Kolmogorov.
Don't know of you're a russiaboo or a delusional ruskie, but either way you're wrong.

[Anonymous No.11931734]

>>11931715
English is easy to learn, it's extraordinarily hard to learn to a near-native level.
The basic structure of English is actually quite simple compared to many languages, the reason people say it's hard is because of the massive amount of exceptional cases (I could guess this is due to how English has committed a much larger amount of borrowing than other languages, but I can't be sure). Exceptional cases usually won't cause people to not understand you, you'll just sound silly.
English pronunciation/spelling/grammar simply does not make any sense a lot of the time, and there's no way to get it perfect except by internalizing all the endless piles of unpredictable exceptions over years.

[Anonymous No.11931738]

>>11931683
Okay, let's just hope for a second that [math]\tilde f\!(k)[/math] is nice and differentiable. What then?
Hm.
[eqn]\frac{{\rm d}\tilde f}{{\rm d}k}
=\frac{{\rm d}}{{\rm d}k} \int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\cos(kx)f(x)
=-\int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\sin(kx)xf(x)[/eqn]
is the obvious identity. And in particular, this expression may never change it's sign.

[Anonymous No.11931739]

>>11931184
see >>11929669

[Anonymous No.11931740]

>>11931715
I don't know. It was easy for me with internet and games and stuff. Although, because of that my language may seem a bit old fashioned. I really liked all sorts of fantasy and medieval stuff. If you want a tricky one, try Hungarian. I think it has even worse word forming rules than the crazy Estonians and Finns have.

>>11931719
The new testament was written in Greek.

>>11931734
>English pronunciation/spelling/grammar simply does not make any sense a lot of the time, and there's no way to get it perfect except by internalizing all the endless piles of unpredictable exceptions over years.
This very much. I thought for years that gauge is pronounced as if it was French but then I found out it is just all about gayz.

[Anonymous No.11931742]

>>11931738
>works in the preview
>breaks on page
thanks 4chan

[Anonymous No.11931744]

>>11931742
Use [math].

[Anonymous No.11931745]

>>11931738
>[eqn]\frac{{\rm d}\tilde f}{{\rm d}k}
>=\frac{{\rm d}}{{\rm d}k} \int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\cos(kx)f(x)
>=-\int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\sin(kx)xf(x)[/eqn]
Absolute yikes.

[Anonymous No.11931747]

>>11931740
>retarded anime tranny thinks genesis is in the new testament

[Anonymous No.11931752]

>>11931719
No, it wasn't, the Bible was written in english.
The catholics over at the Vatican hate the american people's proper worship of the Lord, so they hatched an evil plan to convince people that the bible was written in hebrew. They lied to you, even though the truth is clearly in front of you.
In fact, Mussolini and Hitler were supported by the Vatican to exterminate the Chosen people of the Lord for knowing the truth, and this was known as the Holocaust, but America single-handedly stopped them.

[Anonymous No.11931755]

>>11931747
Why do I not remember claiming this?

[Anonymous No.11931758]

>>11931744
[math]\displaystyle\frac{{\rm d}\tilde f}{{\rm d}k}
=\frac{{\rm d}}{{\rm d}k} \int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\cos(kx)f(x)[/math]
[math]\displaystyle
=-\int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\sin(kx)xf(x)[/math]

[Anonymous No.11931764]

>>11931738
peak wew.
[math]\displaystyle\tilde f'(k)
=-\int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\sin(kx)xf(x)[/math]

[Anonymous No.11931765]

>>11931758
Dude just... Stop. Write your stuff in latex, compile it, take a screencap and post it. It will be easier that way.

[Anonymous No.11931771]

>>11931765
>>11931738
>>11931765
very strange. Never had 4chan fuck up anything TeX like that before.

[Anonymous No.11931776]

>>11931764
>[math]\displaystyle\tilde f'(k)
>=-\int_{-\infty}^\infty \frac{{\rm d}x}{\sqrt{2\pi}}\sin(kx)xf(x)[/math]

[Anonymous No.11931778]

>>11931771
If I had to hazard a guess, there are line breaks in your original eqn tag. pretty sure 4chan can't handle line breaks in the middle of a LaTeX tag, you have to squish it all on one line.

[Anonymous No.11931780]

>>11931776
like I said, the preview works. Reeks of a bug.

[Anonymous No.11931782]

>>11931771
>>11931776
>starting the integral with the dx
I think I see the problem, the thread is cursing you and preventing the Latex from working.

[Anonymous No.11931792]

>>11931782
Well yeah, [math]\cal F[/math] behaves like a unitary operation, so I like to write the integral like a multiplicative operator.

[Anonymous No.11931818]

>>11931792
Retarded reason.
If you wanna emphasize the fact that it's an integral operator do [eqn]K(k, x) = \frac{ \sin (kx) x}{\sqrt{ 2 \pi }} [/eqn] and [eqn]\tilde{f}'(k) = \int _{- \infty} ^{\infty} K(k, x) f(x) dx[/eqn], as our ancestors have taught us.

[Anonymous No.11931866]

>>11931818
>retarded reason
I wouldn't call writing a linear operator such that the notation emphasizes it's multiplicative behavior "retarded". However, I am also the kind of person that writes [math]\partial_x f[/math] for a partial derivative w.r.t. x.

[Anonymous No.11931909]

What are the applications of Chaitin's difference function [math]f(a,b)=|a-b|[/math]?

[Anonymous No.11931914]

>>11931739
I asked to clarify what the terms mean and I got meme answers and some faggot saying I don't understand "trasversality". I called out on people one being either memeing or autistic, because in the particular case the guy was refering there is no fucking need for such a definition. The onyl example people gave me as to why defining it such way is important had nothing to do with the problem at hand.

[Anonymous No.11931923]

>>11931914
no one cares
let it go you spergmaster

[Anonymous No.11931927]

>>11928666

[Anonymous No.11931928]

>>11931923
He replied, so he cared :^). Though yea it is a pretty retarded thing to bring up. I just don't wanna work.

[Anonymous No.11931930]

>>11931914
>I got meme answers and some faggot saying I don't understand "trasversality"
but you don't, anon

[Anonymous No.11931933]

>>11928666
Shame

[Anonymous No.11931976]

>>11931110
Guillemin and Pollack DOES have the proof, because that's the book where I read it the first time. FUCKING IDIOT!

[Anonymous No.11931980]

>>11931114
>via the homeomorphisms
not quite, you need some kind of ambient isotopy to preserve the connected components of the complement.
>>11931142
some dude in the other thread gave you a proof sketch
>>11931184
there's a huge, huge difference between someone asking "what's a curve" without there being a pre-existing context, versus someone asking "what's transversal mean" to people who are already talking about topology of manifolds. and then arguing about what that means. we told you it can mean disjoint, and you got really, really angry about it.

[Anonymous No.11931985]

>>11931548
Absolutely note. What about the characteristic function on [-1, 1]?

[Anonymous No.11931995]

>>11931980
>not quite, you need some kind of ambient isotopy to preserve the connected components of the complement.
Care to explain why? If we have embeddings [math]f, g\colon S^k \to \mathbb{R}^n[/math], shouldn't [math]fS^k[/math] be homeomorphic to [math]gS^k[/math]? If they are, then why would it not be the case that [math]\mathbb{R}^n \setminus fS^k \approx \mathbb{R}^n \setminus gS^k[/math]? Maybe a stupid question, but my point set topology is very rusty.

[Anonymous No.11931997]

>>11931995
your logic would imply that knot theory is trivial

[Anonymous No.11932004]

>>11931997
Not him, but knot theory is trivial for n>3.

[Anonymous No.11932007]

>>11931997
Yeah but why?

[Anonymous No.11932013]

>>11931997
you're asking why some principle doesn't work and the answer is because counter examples exist

[Anonymous No.11932014]

>>11932013
meant for >>11932007

[Anonymous No.11932018]

>>11932013
Thank you very much.

[Anonymous No.11932021]

>>11932004
Is it also trivial for n<3?

[Anonymous No.11932024]

>>11932021
of course not, but the real question is whether that means the validity of the statement depends on n, which I doubt.

[Anonymous No.11932033]

>>11932024
>of course not
WAIT WHAT
THERE ARE NON-TRIVIAL KNOTS IN THE PLANE?
WHAT THE FUCK

[Anonymous No.11932035]

>>11932018
see http://www.math.tifr.res.in/~mahan/knots.pdf
you can use the fundamental group to show that R^3 \ loop is not the same as R^3 \ trefoil knot

[Anonymous No.11932040]

>>11932033
>knots
>in a plane
But in all seriousness. I do not see that the initial statement has any inductive properties. So "it's wrong for n=3" leads nowhere by itself.

[Anonymous No.11932043]

Also, I misread what you said as n<=3.

[Anonymous No.11932049]

Hey math nerds, why dont you try to disprove... THIS

[Anonymous No.11932051]

>>11932049
What, exactly, am I disproving?

[Anonymous No.11932055]

>>11932051
Not enough for ya, eh... well how about... THIS

[Anonymous No.11932102]

>>11932055
Your punches form a quasigroup since you have no identity and no one wants to associate with you, and is thus useless.

qed

[Anonymous No.11932110]

>>11932102
Based and [math]\nexists e[/math]pilled.

[Anonymous No.11932128]

>>11932102
You've defeated me...

[Anonymous No.11932161]

>>11932102
If you think all quasigroups lack an identity I think you'll be thrown for a loop.

[Anonymous No.11932162]

>Complex analysis course is called "Complex analysis and special functions"
>Didn't cover special functions at all due to the pandemic
Feels good, I really don't give a shit about them.

[Anonymous No.11932165]

>>11932162
Based.

[Anonymous No.11932173]

>>11932161
nice

[Anonymous No.11932210]

>>11928615
what's the last number?

[Anonymous No.11932211]

I don't understand why e^ix=isiny+cosy
I would say they just made it up but there's that taylor thing
But I hate that taylor thing

Can a big strong man explain it for me?

[Anonymous No.11932223]

>>11932211
It just werks.

[Anonymous No.11932224]

>>11932210
[math]TREE(3)^{TREE(3)^{TREE(3)}}+2[/math]

[Anonymous No.11932225]

>>11932211
Start doing the Taylor expansion for exp(x) and see if you notice anything familiar about it.

[Anonymous No.11932226]

>>11932223
there HAS to be a mathematical STRUCTURE fitting it into PLACE
if there isnot god will rain hell down upon our meagre EXISTENCE

[Anonymous No.11932229]

>>11932211
[eqn] e^{ix}\neq i \cdot sin(y)+cos(y) \\
e^{ix}= i \cdot sin(x)+cos(x) [/eqn]

[Anonymous No.11932230]

>>11932225
i just told you i hate that taylor thing

[Anonymous No.11932234]

>>11932229
y=x you bureaucrat

[Anonymous No.11932239]

>>11932211
I mean breh, you have [math]D^2 \exp (ix) = - \exp (ix)[/math] and [math]D^2 \sin x = - \sin x[/math], [math]D^2 \cos x = - \cos x[/math],
The rest is interpolation.

[Anonymous No.11932243]

>>11932239
what?

[Anonymous No.11932244]

>>11932226
It's the exponential map from the lie algebra to the lie group in the case of the unit circle group. Faggot

[Anonymous No.11932248]

>>11932244
proof?

[Anonymous No.11932249]

>>11932243
Second order ODE has bidimensional solution space.
Interpolation.

[Anonymous No.11932251]

>>11932211
you're changing from polar to rectangular coordinates
e^{ix} is the point on the unit circle marked off by the angle x. the rectangular coordinates of this point are (cosx, sinx)

[Anonymous No.11932252]

>>11932211
I can't build muscle.

[Anonymous No.11932259]

>>11932248
What the fuck do you mean proof bro, you just need to check the constant.

[Anonymous No.11932261]

>>11932249
how do the ODEs get equated? There are two ODEs that equal themselves on second bet, right, why does this make them equal? Uniqueness of analytic continuations?

>>11932251
that assumes that e^ix works to begin with, how do you prove that x is the argument without sin?

>>11932252
eat protein and work out harder

[Anonymous No.11932263]

>>11932259
i dont know anything about lie algebras or groups

[Anonymous No.11932269]

>>11932261
>how do ODEs get equated
The ODE is [math]D^2 f(x) = - f(x)[/math].
Let Qiaochuan spoonfeed you: https://math.stackexchange.com/questions/2191485/must-any-nth-order

[Anonymous No.11932270]

>>11932261
>NOOOOO I FUCKING HATE FIRST YEAR CALCULUS NO CALCULUS ALLOWED
>NOOOOO YOUR EXPLANATION ISN'T RIGOROUS ENOUGH PROVE YOUR CLAIMS FORMALLY

[Anonymous No.11932275]

>>11932270
first of all, the taylor thing is completely unaesthetic. second of all, it's proved with reals in mind, how can we be so sure it will hold true for complexes other than "just werks"

[Anonymous No.11932280]

This lad has to be trolling us, right?

[Anonymous No.11932290]

>>11932269
so they come from the same ODE and are two dimensions thus. so we can pick two as bases, say sin and cos, and e must be a linear combo of them? that is.... interesting

[Anonymous No.11932303]

Take the Gröbpill.

[Anonymous No.11932304]

>>11932290
Yes.

[Anonymous No.11932308]

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

[Anonymous No.11932315]

>>11932308
Unexpected /mu/.

[Anonymous No.11932335]

neuer: https://boards.4channel.org/sci/thread/11932327#bottom

[Anonymous No.11932358]

>>11931157
It’s actually East Coast, USA. I just like to post the London meme. That location may change because I will be applying for PhD programs for Fall 2021.

[Anonymous No.11932480]

>>11931985
Can you elaborate? Isn't that function not monotonic? The thought process I have right now is that it seems to be true for simple functions like exponentials or gaussians and I remember someone saying something among those lines, but I'm having trouble deriving it.

[Anonymous No.11933011]

>>11932335
Why did you do a new thread already?

[Anonymous No.11934218]

>reading "Basic Mathematics" by Serge Lang
>tfw no complete answer sheet anywhere on the internet
How do I check my work, lads? The book only contains a selected number of answers. I am trying to finish the book completely.