MATH GENERAL

>>9002320
>What are you studying?

I'm trying to do a few things (probably too many at once desu...). I'm reading through Alfhor's complex analysis book, Brin and Stuck's Dynamical Systems book, and I'm giving multivariable analysis a second pass using Multidimensional Real Analysis by Duistermaat and Kolk.

>Cool theorems?

A Besicovitch cover of a subset [math]A \subset \mathbb{R}^n[/math] is a collection of balls [math]\mathcal{B}[/math] such that for each point [math]x \in A[/math], there is a ball in [math]\mathcal{B}[/math] which is centered at [math]x[/math].

Besicovitch covering theorem asserts that given a bounded set [math]A \subset \mathbb{R}^n[/math] and a Besicovitch cover [math]\mathcal{B}[/math], there is a constant [math]c[/math] depending only on the dimension [math]n[/math], and subcollections [math]\mathcal{B}_1, \dotsc, \mathcal{B}_c[/math] of [math]\mathcal{B}[/math], such that each [math]\mathcal{B}_i[/math] is pairwise disjoint, and [math]A \subset \bigcup_{i=1}^c \bigcup_{B \in \mathcal{B}_i} B[/math].

The constant [math]c[/math] given in the proof is [math]5^n[/math]. Interestingly, optimising this constant is open in general, we don't even really know the rate at which it grows.

>Problems

I have two simple problems relating to this theorem.
1. Show that in the case n=1, we have c = 2.
2. Show that in the case n=2, c > 4.

>References
If you're interested in this, check out Geometry of Sets and Measures in Euclidean spaces by Mattila.

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>>9002320
Very bad OP.
>what are you studying?
Same shit as always.
>any cool problems?
As I've elaborated previously, I may be able to construct a correspondence between TQFTs a la Atiyah with CFTs by cutting up and assigning link components in decorated 3-manifolds to marked points the space conformal blocks are on. However the process of "cutting up" these decorated 3-manifolds wasn't precisely defined in the context of space structures. Turaev was able to precisely define a version of this cutting up called "excision" on 2-manifolds instead, and I intend on working to extending this to 2-surfaces.
>any cool theorems or remarks?
Wentzl's limit for the quantum invariant of unitary TQFTs over a semisimple category [math]\mathscr{V}[/math] with fundamental object [math]V_l[/math]: [eqn]\tau(M) = \Delta^{\sigma(L)}\mathscr{D}^{-\sigma(L) + m -1} \lim_{N\rightarrow \infty}\frac{1}{N^m}\left[\sum_{\lambda \in \{1,\dots,N\}^m}(\operatorname{dim}(l))^{-|\lambda|} F(L^\lambda_l)\right]. [/eqn]
It's very interesting that a mostly algebraic quantity can be calculated with analysis. It has a very nice proof too.
>reference suggestions?
Nayak's paper is very nice for quantum braiding algebras.

>Reading through Liu's Algebraic Geometry and Arithmetic Curves and reviewing commutative algebra
>Completely unrelated questions but things I have been asked/have had to think about: 1) Let [math]\phi[/math] be Euler's totient function. Prove that [eqn]\phi(n) = \sum_{k=0}^{n-1} gcd(n,k)e^{2\pi i k}{n}[/eqn]. 2) Prove that the Fourier transform [math]\mathcal{F}: L^1(\mathbb R) \to C_0(\mathbb R)[/math] is not onto. 3) Prove that the convex hull of a compact set is compact in a finite-dimensional real vector space
>Noether's lemma: If A is a finitely-generated K-algebra (for a field K), then you can find an algebraically independent family [math]t_1, \dots, t_n \in A[/math] such that A is integral over [math]K[t_1, \dots, t_n][/math]. Using this, you may answer the apparently dumb question: why can't I find two algebraically independent polynomials in K[T] ? (which I just realized today is absolutely non-trivial)

>>9002146
Try to get the best approximation of a sphere with the material you have. I recommend the icosahedron, because it's very easy to make triangular tilings on boards.

>>9002394
>[math]\mathcal{F}:L^1(\mathbb{R}) \rightarrow C_0(\mathbb{R})[/math]
>[math]L^1[/math]
Did you mean [math]L^2[/math]?

Is it necessary to understand fields and rings to properly understand linear algebra ?

We've never touched the subject in class, but then again the teacher simply expects us to take what she says for granted.

>>9002782
>is it necessary
no
but you should learn about fields and rings anyway because they're fuggin cool

>>9002320
>started reading functional analysis
>confused by some things so started reading measure theory
>confused by some things from that, so started reading topology
current reading a textbook so i can read another textbook so i can read the book I actually want to read.

classic /math/

>>9002852
>wanted to know why you can't divide by zero
>almost to phd and still don't know
i want off mr bones' wild ride

>>9002852
>I tried to read functional analysis without knowing real analysis or topology
>it's math's fault
ask someone who knows what he's doing for advice

>>9002852
That can sometimes actually be nice. General Topology and Measure Theory are fairly dry topics, knowing why you need the results and a vague idea of where they need to applied will help motivate you.

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https://www.youtube.com/watch?v=m6rfpQXzXu0
>yfw the trend of physics driving new mathematical research is only going to increase

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>>9002320
Mathematics is much too hard for mathematicians.

>>9002552
Nah, I meant L^1, that's the first place you define the Fourier transform (if you look at the formula for it, it doesn't really make sense if the function is only L^2). You can then check that it's continuous and goes to 0 at infinity and I'm asking whether every continuous function that goes to 0 at infinity is the Fourier tranform of some L^1 function.
You can then extend the Fourier transform to L^2 (and it's nontrivial) but the Fourier transform of an L^2 function is in general not continuous.

>>9002146
To be fair, that's a very difficult problem if you impose no condition on the shape of the box (and even if you look for a rectangular box, it's not completely obvious)

>>9002854
It's actually how trivial, and it goes back to the definition of division. It's even simpler to understand in rings (think about divisibility).

>>9002878
>linking to numberphile unironically
Kill yourself.

>>9002944
It's actually trivial. You want a solid which has the lowest possible surface area to volume ratio, i.e. you want to approximate a sphere. It all boils down to the fact that
[math] area(S) \geq 3 (\frac{4 \pi}{3})^{\frac{1}{3}} vol(S)^{\frac{2}{3}} [/math]
where equality holds only when [math] S [/math] is a ball.

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Has anyone here tried studying magic?

>>9002377
>Very bad OP.
There's literally nothing wrong with the OP, you autist.

>>9003021
Yes and how do you prove that inequality ?

>>9003163
See section 2, here: http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf

>>9002320
I am doing Calculus1. Learned pretty cool stuff about derivatives. For example y=x^2 -> y' = 2x
So I am really good at math.

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>he's not studying outside

>>9003270
As a general rule, if you need to refer the other party to a paper, it's most likely non-trivial.

>>9003338
I disagree. "Isoperimetric" inequalities are as old as math. I linked the paper because it proves the general, n-dimensional case w.r.t. volumes of (n-1)-dimensional hypersurfaces. You don't need anything beyond middle-school math to prove the 2D case, or high-school math to prove the 3D case.

Not sure if this is the correct thread to ask in, but having just finished "forallχ" by PD Magnus, what should I read next?

>>9003351
I'm really interested in seeing a proof, even of the 2D case, that only relies on HS math (I mean the mere formalization of the problem requires calculus)

>>9003373
>implying calculus isn't high school math

>>9003373
The isoperimetric inequality was proven in antiquity, by Zenodorus and Archimedes. Calculus? No, like I said: middle-school math: elementary Euclidean geometry in the plane. Here: http://link.springer.com.ololo.sci-hub.io/article/10.1007/BF03024397

>>9003407
Another elementary proof: https://arxiv.org/pdf/1609.08458.pdf

>>9003423
another http://www3.nd.edu/~andyp/notes/Isoperimetric.pdf

>>9003407
>>9003423
Heh, that's really neat. Thanks !

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>>9003534
Kein Problem. Geometry is underrated.

>>9002854
a common feel, friend
>>9002857
i wasn't blaming math. just expressing the well known feel of going down a rabbit hole of sources.
>>9002875
this is what I meant. Reading topology after taking multiple courses in linear algebra is a very good motivation. My Topology book has a whole chapter on function spaces. I'm excited to get to it.

>>9003056
yes, currently in program for a Master's in Magic. My degree title will be "Masterful Magician"

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>>9003056
Magic, math, what's the difference?

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>>9003666
I think there is a possibility that "strong" forms of magic can be both complete and consistent.

>>9003679
Grandwizard Kurt Gödel found that no complete spellcasting anthology may exist without contradiction in construction of the initial library of spells. Look other places for magic research, friend.

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You should be able to solve this.

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>>9003714
http://web.stanford.edu/~danlass/esslli2011stus/petrovic.pdf

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>>9003110
>There's literally nothing wrong with the OP, you autist.
The OP is a disaster.

>dumb pic of ducks instead of something math-related
>no /math/ or /mg/ in subject (people should fight over which of these should be used)
plus the current format is horrible, what's the point of just randomly asking for reference suggestions?

should just be something like

>a math pic
>a link to a page about the math pic
>what are you studying?
>interesting problems, theorems, proofs, textbooks, papers?
>a couple other interesting recent links

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>>9003728
I have no math pics, and those are swans.
t OP

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>>9003709
You have much to learn.
http://www.corelab.ntua.gr/studygroup/Tarski_ElGeom.pdf

>>9003679
Back off! Patchouli is mine.

>>9003723
That doesn't solve it.

>>9003739
>elementary geometry is complete, consistent and decidable
Yet more proof that geometry is underrated.
Stop underrating geometry faggots.

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>>9003741
No but that does imply all Euclidean geometry (including your homework problem) belongs on /g/, not here.

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>>9003745
>faggots.
Can we like, leave out the homophobia from this thread?

>>9003746
You're free to fuck off anytime.

>>9003756
No, you faggot.

>>9003757
>>>/g/

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>>9003759
If anything, you're the one who belongs somewhere else. No one who paraphrases grandmaster Gödel this poorly >>9003709 has any business being here.
Tone down the mod LARPing.

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>>9003776
I'm not that person. Please take your meds next time.

>>9003779
Blow me anyway you incurious fag.

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>>9003786
>not wanting tedious uninteresting shit here makes me incurious
So this is what being a freshman again feels like. Can't say I miss it.

>>9002377
>>9002878
>>9002903
>>9003056
>>9003570
>>9003666
>>9003679
>>9003723
>>9003739
>>9003746
>>9003776
>>9003788
stop with your pedo cartoons

I don't care about your tedious, inelegant, half-baked physics sketches either, so I guess that makes us even.

I swear that I will s___ _p this retarded thread on a regular basis.

>>9003799
>talking about elegance when he posted an elementary geometry problem
LMAO

Why would anyone want math in a math thread anyway? The airs this guy is putting on...

>>9003804
>something must be complicated for it to be elegant

what a cancerous opinion

>>9003807
Who are you quoting?

>>9003804
Kill yourself you fucking plebeian.
Or get a trip so I can filter you.

Physics should be banned and all physicists should be hanged.

>>9003814
Amen.

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>>9003809
Wow talk about math anytime.

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>>9003801
>s___ _p

>>9003788
>>9003819
>reddit frogs
Subhuman garbage belongs in some other thread.

>>9002320
>what are you studying?
Developing a new field of mathematics to help create weapons which will destroy physics and hang all physicists.
>any cool theorems
Yes. It's a theorem which states that physics should be prohibited.

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>>9003820
The last word is "up".

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>>9003828
>trying to hang the ones that gets you funding
What do they do to disobedient dogs again?

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>>9003844
It might be Ip.

>>9003851
I have unlimited funding. I wouldn't be needing any money from dead people.

>>9003851
Ooops.......... Wrong thread! Look for a "Phyzicks" thread somewhere on this "Board"!

>>9003788
>>9003746
>>9003723
>>9003666
Hello... Please see the attached post below:
>>9003865

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>>9003865
>>9003869
Funny how it's you who brought up physics in the first place. See @9003799.

>>9002377
>>9002878
Hey... Please see the attached posts:
>>9003865
>>9003869
>>9003871
Hello.
Please look at the attached post:
>>9003865
This is not an appropriate thread for so-called "Phyzicks"

>>9003739
Hey... Please see the attached message:
>>9003872

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Reminder to report all spammers.

Jesus Christ, I just found out about the Stacks Project.

I think reading that gave me autism. Send help.

http://stacks.math.columbia.edu/browse

>>9003804
>>9003819
Oooops... Seems like you made a mistake! This is not a thread for "Phyzicks".
see the attached message below:
>>9003872

>>9003879
hello... it appears you are using the wrong thread.
please use the so-called "catalog" function on this website to find a "Phyzicks" thread.

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Prove that any (n, q)-regular planar space with q > n which has at least one projective line and in which the intersection of any two planes is non-empty is isomorphic to PG(4, n).

>>9003880
What's wrong with Stacks project? It's a pretty noble effort considering the lack of other published, streamlined resources

>>9003880
Can't help those afflicted with brainletism.

>>9003889
Don't get me wrong. What they are doing is magnificent.

It's just that it really does a good job at showing me that even after years of academic mathematical education I know so little.

Also it looks really autistic.

>>9003871
Just fyi, no, he isn't the one who brought up physics. I did. And, speaking of Stacks project (schemes specifically), this reminds me that this isn't the first time you sperg out over someone posting math in the math general.

Fag.

>>9003906
>phyzicks
Ooppss... Wrong thread, my mate...
Seems like you should search for "Phyzicks" on this website on the board /sci/ (this is a math thread)

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>>9003876
>>9003911
You too pissbrain, I allow you to choke on a cock.
Get on it.

>>9003920
This is a very cute image. Please don't defile it by saying such garbage.

>>9003819
>>9003820
>>9003844
>>9003851
>>9003879
>>9003886
>>9003920
see >>9003793

>>9003927
see >>9003927

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>>9003927
Anime site.

>>9003927
see >>9003934

>>9003934
>>9003935
You should really see a professional and figure out why you are obsessed with cartoon girls.

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How does one define an internal natural transformation?

If [math]\mathscr{E}[/math] is a category with finite limits, an internal category [math]\textbf{C}[/math] of [math]\mathscr{E}[/math] consists of a pair [math]C_0, C_1[/math] of objects, the object of object and the object of morphisms, resp., and morphisms [math]s, t\colon C_1\to C_0, i\colon C_0\to C_1, c\colon C_1\times_{C_0}C_1\to C_1[/math] satisfying equations some equations. An internal functor [math]f\colon\textbf{C}\to\textbf{D}[/math], for some internal categories [math]\textbf{C}, \textbf{D}[/math] of [math]\mathscr{E}[/math], is a pair of morphisms [math]f_0\colon C_0\to D_0, f_1\colon C_1\to D_1[/math] commuting with all those morphisms above.

Then I'd like to define an internal natural transformation [math]\tau\colon f\to g[/math]. Is it just a quadruple [math]\tau^X_i\colon X_i\to X_i[/math], with [math]X=\textbf{C}, \textbf{D}[/math] and [math]i=0, 1[/math], such that [math]\tau^\textbf{D}_i\circ f_i=g_i\circ\tau^\textbf{C}_i[/math]? Pls halp!

>>9003960
[math]\tau^\textbf{D}_i \circ f_i = g_i \circ \tau^\textbf{C}_i[/math] pls work

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>>9003960
Natural with respect to the internal structure? Just draw some diagrams dude lamo.

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>>9003966
>Natural with respect to the internal structure?
I don't know which structure exactly.
>Just draw some diagrams dude lamo.
That's the only way I could think of such that I only have those functors there, but then I'd have those C-morphisms too. This sucks because I'd only like to have D-morphisms, as is the case with normal natural transformations.

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>>9003984
>I don't know which structure exactly.
I mean the "internal" structure that makes a category internal.
>>9003984
>This sucks because I'd only like to have D-morphisms
Maybe the equations that the internal morphisms satisfy can make some of these dependent?

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>>9003997
They just represent source, target, identity and composition. Maybe I should approach this using generic elements. Since I have finite limits in [math]\mathscr{E}[/math], I have a terminal object. Then I would just do stuff like define [math]\tau_x \colon f_0(x)\to g_0(x)[/math], for each such generic element [math]x\colon 1\to C_0[/math]. This would give the ordinary commutative square of natural transformations: for generic elements [math]x, y\colon 1\to C_0[/math], and an internal morphism [math]\varphi\colon x\to y[/math] in [math]C_1[/math], [math]\tau_y \circ f_1(\varphi) = g_1(\varphi)\circ \tau_x[/math].

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>ban finally expires
hello fellows, how's the thursday /math/ going?

>>9003960
>>9003966
>>9003984
>>9003997
>>9004034
seek help

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>read DiFrancesco
>extremely powerful CFT, describes all critical phenomenon
>dry as fuck
>read Woodhouse
>very beautiful symplectic theories of quantization
>literally can't apply it to free boson field theories
Is synergy possible at all?

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Who's the safest bet for a 2018 Fields medal?

Is it Scholze's turn?

>>9004513
Who cares? The Fields medal is a joke. ANY prize for that matter is a joke. The only valuable prize for one's achievements is posterity. How many Fields medals did Euler win? Gauss?

How many Fields medals did Archimedes (the guy who's face is sacrilegiously sculpted on the medal) win? If your achievements are worth anything they will be cherished for aeons by the peoples to come. Everything else is pointless monkey brains craving for status in the short term.

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>>9004521
>this much edge
careful you might cut yourself

Peter has a great mane tho. I wish I weren't balding.

>>9004522
Do you even know what edgy means?

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>>9004521

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>>9004525
I'm on 4chan, of course I know what edge is.

If you're lucky someone else might take your obvious bait

>>9004533
>>9004537
4chan has rot your brains. Nothing in that post is even remotely controversial, let alone edgy.

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>>9004554
You're the odd one out here, for a reason

If you don't comprehend the stupidity of a question like
>How many Fields medals did Euler win?
then you're too far gone my friend.

>>9004557
What exactly is stupid about it? Are you so braindead you can't parse rhetorics? Here, let me hold your little hand, chimp: the point of asking that is to make you think whether Euler is a great mathematician because he won a prize or because he did great mathematics. Replace Euler with any other great mathematician throughout history. Is Grothendieck a great mathematician because he won the Fields medal?

You cling to markers of status like a pea-brained ignoramus. The whole Fields medal bullshit is parasitic, propping itself up on the prestige of some of its recipients. The Nobel is the same.

Credentialists need to kill themselves.

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>>9004513
>tfw Lurie turns 40 next year
It's now or never Jacob. Do something good for us.

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>>9004569
>What exactly is stupid about it?
That you somehow got the implication that I believed the Fields medal is some sort of binary switch that determines whether a mathematician's work is worth anything or nothing. You haven't given any actual argument against recognizing the work of young mathematicians so I'll wait for that instead of responding to any more of the bait.

>>9004455
Geometric Quantization has always seemed like a half-baked theory. I think it is going to have to evolve into something more sophisticated before it does anything significant.

>>9004614
I made no assumptions on your beliefs shitbird. I don't care what you believe. All I talked about is the value of the Fields medal. A mathematician's work doesn't need a prize attached to it to be recognised. A mathematician's work speaks for itself, as it has done for millennia before the Fields medal, the Abel prize, and all the assorted other peacock feathers that have sprung up during the last couple of decades like mushrooms after rain even existed.

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>>9004588
Lurie sounds plausible. I only recognize a handful of names on the line-up for plenary speakers (Scholze, Okounkov, Lubotzky, Pandharipande, Donaldson, Lafforgue)

https://mathoverflow.net/questions/272303/work-of-plenary-speakers-at-icm-2018

Surprised they decided on hosting it in Brazil

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>>9004631
Awards have been around forever, Euler won some too. One prize even gave Galois the opportunity to submit his work for recognition but then Fourier died and so no one looked at his work until over a decade later.

You still haven't made any argument against recognizing mathematical work, so I'm still not sure what your issue with the Fields medal is.

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>>9004620
>Geometric Quantization has always seemed like a half-baked theory
Why do you think that? Is it because it's using ideas from physics as a starting point? I think that's a perfectly valid way of searching for new mathematical theories. Just look at TQFT.
>>9004636
>Brazil
Fuck's sake. God forbid they don't fill their diversity quota.

>>9004651
>You still haven't made any argument against recognizing mathematical work
Maybe because I'm not arguing against the recognition of mathematical work but for the fact that prizes are entirely redundant. Good mathematical work is recognised with or without a prize attached to it. From the start I pointed out that the only sensible measure of merit is posterity: recognition by future generations.
>Euler won some too.
No one cares, just like no one in the future will care who will have won the Fields medal next year. Most Fields medalist will have no lasting impact on the development of mathematics, their mediocre "achievements" ignored by history.

>>9004675
>the fact that prizes are entirely redundant
And petty. Prizes ascribe fleeting recognition in the here and now. True recognition is granted sub specie aeternitatis, and posterity is its best available approximation.

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>>9004675
>Maybe because I'm not arguing against the recognition of mathematical work but for the fact that prizes are entirely redundant. Good mathematical work is recognised with or without a prize attached to it.
I've already given an example of when they're not redundant: Galois.

>From the start I pointed out that the only sensible measure of merit is posterity: recognition by future generations.
Why is this the 'only sensible measure'? Do you think good mathematical work can't be lost or forgotten?

>No one cares, just like no one in the future will care who will have won the Fields medal next year.
Frederick the Great cared about Euler's prize... and that led Euler to Berlin where he was granted academic freedom and did most of his most important work

>Most Fields medalist will have no lasting impact on the development of mathematics, their mediocre "achievements" ignored by history.
But most already have...?

>>9004455
>>9004663
>phyzicks
Hello... You seem to be mistaken. This isn't a thread for "Phyzicks". Please use the "catalog" function to find a more appropriate place to discuss that.

>>9004537
>>9004614
see >>9003822

>>9004696
>an example of when they're not redundant: Galois.
Except you haven't. Galois recognition as being a great mathematician has nothing to with any prizes he received. More to the point, it favours what I'm saying, what with Galois theory getting the spotlight it deserves only decades after Galois died. You're an affirmative action recipient I imagine.
>Do you think good mathematical work can't be lost or forgotten?
It is much, much less likely that good mathematical work will be lost or forgotten because mathematicians will take greater care in preserving it compared to random mediocre stuff. I'm not surprised this point is lost on your dumb nigger brain.
>Frederick the Great cared about Euler's prize...
Euler would have published regardless of Frederick the II support.
>But most already have...?
No.

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>>9004751
>Except you haven't. Galois recognition as being a great mathematician has nothing to with any prizes he received. More to the point, it favours what I'm saying, what with Galois theory getting the spotlight it deserves only decades after Galois died. You're an affirmative action recipient I imagine.
You're still strawmanning. Galois was unrecognized before his death, so how could the award be redundant?

>It is much, much less likely that good mathematical work will be lost or forgotten because mathematicians will take greater care in preserving it compared to random mediocre stuff. I'm not surprised this point is lost on your dumb nigger brain.
Why the racism? And 'less likely' is not an argument. If the work is not recognized in the first place (say by an award...) that's less incentive for that work to be preserved. Plenty of math has been 'nearly lost', Mohr's theorem was buried for 250 years, and even important work of already recognized mathematicians (i.e. Ramanujan's lost notebook)

>Euler would have published regardless of Frederick the II support.
I'm not going to speculate on alternate timelines.

>No.
I'll wait for an argument that somehow backs up this claim.

>>9004770
>Why the racism?
>>>/r/eddit/

>>9004770
>You're still strawmanning. Galois was unrecognized before his death, so how could the award be redundant?
The point is, dumb shit, that posterity is a much better judge of merit. Galois is proof of that. Not even going to bother with reading the rest of your post.

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How can other

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>>9004903
textbook covers even compete?

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>>9004903
>>9004904
Are those real textbook covers?

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>>9004910
yes

so are these

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>>9004914
Those elephants are cute.

>>9004663
Brazil is doing god-tier mathematics now, you should now. Fernando Coda is probably getting some this year.

>>9003728
>a math pic
but everything is math, anon, didn't you know?
>a link to a page about the math pic
why on earth
>what are you studying?
who gives a shit
>interesting problems, theorems, proofs, textbooks, papers?
OP already asked that
>a couple other interesting recent links
post them yourself you fucking autist

>what are you studying?
I'm learning SQL from scratch. Need a new job


Can anyone recommend a book?

>>9004914
>>9004904
>>9004903
They're fucking incredible. How to get?

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>>9004903
>>9004904
>>9004914
These are anime af

GUYS

Is there any great way to study for the GRE math subject test? All I've found is four past tests and this dude at work tells me the test content changes drastically every year so studying those doesn't help much anyway.

I've been studying calc II, calc III and a little bit of linear algebra but with how the topics that each test question addresses seem so specific it just seems like a crapshoot...

Any help/tips would be greatly appreciated!

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>>9004948
>proved Willmore conjecture
That does seem really cool. I should probably look into differential geometry deeper than just the basic differential manifolds.

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>mfw godel's theorem

>what are you studying?
I want to self-teach myself some abstract algebra. Found these Harvard lectures on youtube: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5, and I'm following along in the Mike Artin book. Anyone have better resources to recommend or does this sound like an alright path to go down?

>>9005312
>>9005336
Spivak diff geo books, they're pretty good, though I consider Fomenko to be better

>>9005396
Seems pretty good.

>>9005367
Search about IMPA, the guys there do a great job, the fields medal winner Artur Avila was made there, basically.

He was an invited speaker at ICM of 2010 in Hyderabad, and in 2014 in Seoul.
He isn't one of the speakers in 2018, so this may be a sign.

He may win the medal, who knows.

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>>9002394

>Prove that the Fourier transform F:L1(R)→C0(R) is not onto.

How about a picking a continuous, nowhere differentiable function, which decays to zero as |x| tends to infinite?
This function cannot be the image of a function in L1, since every function in L1 has almost everywhere an absolutely continuous antiderivative, and the Fourier transform behaves well with derivatives.

>>9003756
>implying calling someone a faggot is homophobia

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>>9005798
https://en.wikipedia.org/wiki/Category:Homophobic_slurs

>>9002394
>ϕ(n)=∑k=0n−1gcd(n,k)e2πikn

gcd(n,0) and exp(2πki) (=1) don't bode very well here. sure this is correct?

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>>9003756

well, you don't have to be anal about homophobia

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>>9004948

>Brazil is doing god-tier mathematics now

let's just hope this fields medal candidate doesn't run into some off duty cop.

>>9005810
Yeah, sorry I meant:
[eqn]\phi(n) = \sum_{k=0}^{n-1} gcd(n,k)e^{\frac{2\pi i k}{n}}[/eqn]
Also, gcd(n,0) = n whenever n is non-zero (you can take it as a convention but it's also coherent with the definition of gcd(a,b) as the positive generator of aZ+bZ)

>>9005777
Well that's not what I had in mind (though it's not completely unrelated to the solution I have) but I'm interested

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>>9005841
Let's see if my proof is what you have in mind:
Lemma: [math]\mathcal{F}[/math] is non-singular.
Proof: this follows from the completeness of [math]L^2 \supset L^1[/math] under [math]\mathcal{F}[/math] and the variational lemma: [eqn]\int_\mathbb{R}dx f(x)e^{ixp} = 0 \forall p \implies 0 = \sum_p \varepsilon(p) \int_\mathbb{R}dx f(x)e^{ixp} = \int_\mathbb{R}dx f(x)\left(\sum_p \varepsilon(p)e^{ixp}\right) = \int_\mathbb{R}dx f(x)\varepsilon(x) = 0 \implies f(x) \equiv 0.[/eqn]
Lemma: [math]\mathcal{F}[/math] is injective.
Proof: Suppose [math]f,g\in L^1[/math] such that [math]\mathcal{F}(f) = \mathcal{F}(g)[/math], then by linearity [math]\mathcal{F}(f-g) = 0[/math], which by the above lemma [math]f - g = 0[/math].
Now since [math]\mathcal{F}[/math] is injective, if it were surjective then it would be bijective, but obviously [math]L^1 \neq C_0[/math], which means that [math]\mathcal{F}[/math] cannot be surjective.

>>9005410
Fomenko is like a day at at the beach compared to Spivak. Fomenko is meant to learn from while Spivak is a refrence. Next thing you are going to say is Kobayashi is a good bedtime story. :P

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>>9006057
>dx before the integrand

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>>9006189
Do we have a problem?

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>>9006231
n-no

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happy canada day /math/

I want to learn number theory or something related. Where do I start?

>>9006394
Interuniversal Teichmuller Theory

>>9006401
It's Teichmüller! If you can't write an umlaut then just write Teichmueller.
Fucking language butchers.

>>9006405
stop being a nazi about spelling

>>9006405
How would you spell it in Arabic? German's a dying language anyway

>>9006407

[math] {{{{{hue}^{hue}}^{hue}}^{hue}}^{hue}}^{hue} [/math]

>>9006057
>[math]L^2 \supset L^1[/math]
are you sure about that ? Also, I'm not sure what you do in the proof of your lemma. What is [math]\varepsilon[/math] ? (is it a sequence ? a function ?) What do you assume about it ? (continuity ? smoothness ?) Why can you exchange the sum and the integral ?

Also, even assuming that the Fourier transform is injective, which is true anyway.
>obviously [math]L^1 \neq C_0[/math]
In what sense ? They're not equal as sets, sure, but why couldn't they be isomorphic as Banach spaces ? Also, why does this allow you to complete the proof ?

>>9006444
>>L2⊃L1
It's exactly the other way around. L1 is the largest Lp space.

>>9006496
Actually nevermind that, I didn't notice the subjacent space is R. In that case neither inclusion holds, because R has sets of both arbitrarily small and arbitrarily large measure.

>>9006444
>What is ε ?
Nothing. Assuming he was trying anything at all, I would assume he was trying to rewrite the integral in terms of some basis functions but looking at that lemma the end result is gibberish (because the subjacent space is R, and if it were some finite interval, the inclusion would be the other way around).

To prove that the Fourier transform is not surjective use the Banach–Schauder theorem, that any surjective linear and continuous operator between Banach spaces is an open map (which the Fourier transform isnt)
> why couldn't they be isomorphic as Banach spaces ?
Because C0(R) contains a subspace isomorphic to itself but L1(R) doesn't.
> Also, why does this allow you to complete the proof ?
It doesn't allow him to complete the "proof". He did not write a proof in the first place. If I were you I'd ignore posts made by the gorilaposter and the 2huposter. They're shitposts 90% of the time. Worst regulars in /mg/.

And don't forget to report them for avatarfagging.

What are some good books about Statistics and Machine Learning? Is "AI: A modern approach" by Russel and Norwig still somewhat up-to-date?

>>9006592
Ask computer science general.

>>9006629
Oh, I wasn't even aware that it existed, thanks

>>9006568
>reddit frogs
I don't want your kind around here.

Stupid question my bros but here's my thought:

How about a new number "u" to represent 1/0.

Similarly, 2u = 2 divided by zero. "u" comes from "undefined". Instead of complex numbers, I will call these "elite numbers".

Numbers have an elite part if the u component is non-zero. Elite complex numbers can be plotted as a point on a 3D graph. And you also get interesting math like this:

u * 0 = 1

2u * 0 = 2

Multiplying a real number by zero will always yield zero. But if you multiply an elite number by zero, you will get a real number. You can think of it like a rotation around the origin in the 2d real/elite plane. You should have an intuitive sense of how these numbers work now.

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>>9006639
>I don't want your kind around here.
Too bad

>>9006649
>reddit frogs
I said I don't want your kind around here.

>>9006656
You're an affirmative action recipient I imagine. I'm not surprised this point is lost on your dumb nigger brain.

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>>9006693
>Take your English classes, dumb fucking mongoloid.
English is my second language so I was hoping for some examples.

>They are an insult to life itself.
Do you have a frog phobia or something?

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>math general

>>9006726
You don't have to do this.

>>9006724
this is a math thread. please post math instead of low quality racism and arguments interested to the topic.

clarifying: don't give a shit of you're a racist, you just shouldn't talk about it here.

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>>9006761
>We made a vow. Your kind needs to die.

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>>9006763
Frogs are made to be abused.

Just hide this post >>9006556 and it will hide most of the aboves retarded argument

>>9006786
The irony here is that post is spot on.

>>9006788
see >>9006801

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>tfw 99% of the posters in /math/general cannot prove this: if [math] m,n \in \mathbb{N} [/math] with [math] (m,n)=d [/math] then [math] {\mathbb{Z}}_m {\otimes}_{\mathbb{Z}} {\mathbb{Z}}_n \simeq {\mathbb{Z}}_d [/math].

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>mfw Mochi's ABC proof finally clicks

>>9006788
>>9006796
>>9006829
>>9006834
Who are you quoting?

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>>9006829
We are the 99%.

>>9006829
The math general is not for jerking off or being elitist. We pride ourselves in being significantly more loving and caring than the rest of /sci/ so please do away with your elitism.

>>9006945
This, makes me sad that 1/10 of this thread was a flame war
Are all generals doomed to this fate?

>>9006960
No. Earlier ones were much better. Depends if avatar fags present themselves because then 90% of the thread is them doing avatar fag stuff and people complaining about it.

The last math general was pretty good because an avatar fag made a second one after the good one was made, so he was contained in his own thread and then the rest of us did a few problems.

But definitely in the past they were better and much kinder.

>>9006829
Let m=1 and n be arbitrary. Then 0 \otimes Z_n is isomorphic to Z_n, not 0.

>>9006967
not quite, you are thinking more of the direct product, this is the tensor product.

>>9006301
Thanks anon. Didn't know Coxeter was Canadian.

>>9006829
Let me do a wild guess (not a rigorous proof by any means)
First: Any simple tensor verifies [math]l \otimes k = lk 1 \otimes 1[/math], so they are all multiples of [math] 1 \otimes 1 [/math]. Thus, not only simple tensors, but also all tensor are.
Second: By Bezout, [math] d 1 \otimes 1 = (ns+mt)1 \otimes 1 = ns \otimes 1 + 1 \otimes mt = 0 [/math]
Third: You should proof [math]d[/math] is the minimum number with that property. This is the tricky part, maybe I'll dedicate a sec with paper and pen later.

>>9006984
Just realized, third is easy as well. If we have [math] k 1 \otimes 1 = 0 = ns \otimes 1 = 1 \otimes mt [/math], so [math] n|k, \quad m|k[/math], so [math] d|k [/math].

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>>9006444
>are you sure about that?
Pretty sure all [math]L^1[/math] functions are [math]L^2[/math] on [math]\sigma[/math]-finite measures senpai.
>What is [math]ε[/math] ?
Arbitrary continuous function. Read about the variational lemma.
>Why can you exchange the sum and the integral ?
Lol don't worry about that dude.
>They're not equal as sets, sure, but why couldn't they be isomorphic as Banach spaces ?
How do you have an isomorphism without it being a bijection? Sorry I thought that was one of the requirements.

>>9006945
>>9006964
The math general is for discussing math. Sorry that me steering the fucking thread back on topic rubs you the wrong way.
>>9006984
You're on the right track.
Welcome to the 1%.

>>9006994
>Pretty sure all L1L1 functions are L2L2 on σσ-finite measures senpai.
This is not true. You need some remedial functional analysis lessons.

>>9006998
The topic is discussing math, what you did is "be a cunt about math", hence you didn't steer the thread back on topic
There is a difference between "Heres a problem" and "If you cant solve this problem youre retarded"

>>9007006
Oh nevermind that's true for finite not [math]\sigma[/math]-finite measures because of Cauchy-Schwarz. I just remembered the basic example of [math]\frac{1}{x}[/math] being in [math]L^2[/math] but not [math]L^1[/math]. Haven't touched this in 3 years.

>>9007013
Something tells me you're one of the shitposters above.

Question:

I'm reading Lie Groups; Lie Algebras by Hausner & Schwartz

in the first section on functions of operators, they call on circles to help define some of the conditions in their lemmas.

These circles are actually indexed and defined by sets

I don't know latex but for example a circle of index i is defined as:

C_i = {a | |a - a_i| < eps_i} where a is the eigenvalue for some arbitrary operator A. eps_i > 0.

I just don't see what defining a circle in a lemma for operators achieves.

>>9007015
Hölder's inequality actually, [math] {\| fg \|}_1 \leq {\|f\|}_p {\|g\|}_q [/math]. You get the Cauchy-Schwarz inequality from it for p=q=2.

>>9006994
Nay, take an integrable function that makes sufficiently big spikes, square it and you can imagine that it won't be integrable (more precisely, you can check that f(x) = n^4(1/n^3 - |x-n|) on [n-1/n^3, n+1/n^3] for each n >0 and 0 elsewhere is L^1 and not L^2)

This post >>9006556 already has the answer.

>>9007015
>that's true for finite measures
This is not true either.

>>9007069
https://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion

>>9007071
That thread says exactly what I said. If you have a finite measure not all L^1 functions are L^2.

>>9006945
It is the mentally ill avatarfags fault.

>>9006788
>>9006829
>>9006941
>>9006994
Subhuman trash is not welcome here.

>>9002320
How many of you did a double major in math and something else? I was considering doing a computer science and math double major. Is it doable or is it pointless? I'd plan on going to grad school for comp sci. I was also thinking of doing a comp sci and Econ double major.

>>9007176
>Reminder: /sci/ is for discussing topics pertaining to science and mathematics, not for helping you with your homework or helping you figure out your career path.
>If you want advice regarding college/university or your career path, go to >>>/adv/ - Advice.

>>9007176
Fucking retard...

>>9007176
>>>/g/

>>9007176
I know someone who did double major in math an philosophy. Honestly seems like the way to go.

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>>9007409
[math] \forall \epsilon > 0, \quad \frac{1}{\epsilon} < u \implies u > x \quad \forall x \in \mathbb{R}[/math]
[math]\pm u = \lim_{x \rightarrow 0^\pm}\frac{1}{x} = \frac{1}{\lim_{x \rightarrow 0}x} = u \implies u = -u[/math]
[math]\mathbb{R}\cup \{u\} \cong S^1[/math]

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>>9002320 >>9007252
The Singer Katy Perry asked to Neil deGrasse Tyson: Is Math related to Science?
http://youtu.be/9nsaNXNsg28

Pew Die Pie Reaction
http://youtu.be/OB4znOVsfnA

Katy Perry &Neil deGrasse Tyson Full Video
http://youtu.be/3ujWVbjKBCo

My question is Why severely retarded people are allowed to become rich?

I make simulations of discrete quantum mechanical systems

https://www.youtube.com/watch?v=sso5GutGTkc

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>>9007172
>subhuman because of anime
>not subhuman because not capable to prove the claim

>>9007480
>subhuman because of anime
No. Subhuman because of being an attention whore.

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>>9007482
Ok.

Random Question: I got a BS in Physics, and want to go to grad school, but my GPA isn't very good (3.2). What do?

How feasible is going back to school for a different undergrad vs studying for the GRE and trying to get in with a low GPA?

>>9007488
You're fucked.
>>>/x/

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>>9007486
She's on the spectrum for sure.

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>>9007492
>she
Who?

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>>9007498
Everyone in this thread is a girl anon.

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>>9007501
That is precisely why I asked for clarification. If you take pic on a street in Hong Kong and talk about the guy who looks like -_-, you are not giving enough information. Take any topological space and prove all points have disjoint nbds, take any group and prove all its subgroups are normal, take any topos and prove it is étale.

>>9007486
>>9007492
>>9007498
>>9007501
>>9007507

pls ban this avatarfag

>>9007450
Wow, you guys really think he took the time to latex that even though no one's going to read it?

I just finished taking intro to real analysis, should i take Real analysis I and topology at the same time? This is my last year in college but could stay ne more year to make myself a bit more competitive.
Any advice appreciated.

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happy canada day again /math/

moar challenge problems, if anyone has some

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>>9007676
I posted this one in the last thread and it got no replies

You do need to know some elementary number theory though (at least for the solution I know)

>>9007676
Prove that you should be allowed to live

>>9007676
Prove that you shouldn't be allowed to live

The only time i have felt more alive is when I am studying math.

>>9007621
competitive for what? grad school in math?

>>9007700
yeah, i need more advanced coursework for the next 2 years

>>9007684

too wordy.

>>9007676
Let [math]X[/math] be a calabi-yau variety/manifold. Prove there exists a calabi-yau [math]Y[/math] s.t. [math]{\operatorname{D} ^b}\operatorname{Coh} \left( X \right){ \cong _{{A_\infty }}}{\operatorname{D} ^\pi }\operatorname{Fuk} \left( Y \right)[/math] .

>>9007732
>calabi-yau variety/manifold

fuck if i know what that is

>>9007621
>>9007708
The amount of overlap you'll get between analysis and topology is unlikely to extend past the amount of topology you learned in intro analysis
you won't have any overlap problems unless your instructor is a complete fucking weirdo

>>9007738
How much do you know ?

>>9007758

fairly little geometry, although i hope to learn more shortly.

>>9007758
Enough to disprove the Riemann hypothesis.

>>9007765
I mean about math in general, have you taken any classes so far ?

>>9007771

not a math major, so not many. a couple of grad-level math classes, but that's about it. i have a bunch of math books that i've been meaning to go over, but never seem to have the motivation

>>9007772
>not a math major
What is it with you people that just study math without any real goal? Do you really think you are smarter than the average proletarian because you think you are "le smart hobbyist"?

Fucking kill yourself

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>>9007789
>What is it with you people that just study math without any real goal?
Why is that a bad thing? You have some problems if you think fame and money are the only reasons to do stuff. You can just do things for fun, and math isn't harmful for you or people close to you, so why not?

>>9007803
>fame and money
>not simply to go to grad school and see what happens afterwards
Pretty sure nobody here has unrealistic goals and if they do they are in denial
Anyway, if i wasnt a math major, i would be trying to enjoy more life. You arent proving anybody that you are smarter. You just have too much time in your hands and you ll never be smarter than a freshman in a math major
Why u even try?

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>>9007806
You shouldn't try, that's certain. Stop wasting your time trying to learn stuff you don't enjoy.

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>>9007676

>>9007025
The lemma in question essentially states the following:

Pick an operator [math]A[/math] with eigenvalues [math]\lambda_1,\dotsc,\lambda_n[/math]. We define [math]C_i[/math] as a disjoint open ball around the eigenvalue [math]\lambda_i[/math] which is disjoint from [math]C_j[/math] for all [math]j\neq i[/math].
If this is the case, we can find a [math]\delta>0[/math] such that whenever we have another operator [math]B[/math] for which [math]|B-A|<\delta[/math] (that is, B is a perturbation of A within the possible error of [math]\pm\delta[/math]) then the eigenvalues of [math]B[/math] lie in the [math]C_i[/math].

In otherwords: if we can find disjoint open balls (your circles, or also called "neighbourhoods") around eigenvalues of some operator A, then we can find some positive [math]\delta[/math] such that if B perturbs A within error [math]\pm\delta[/math], then the eigenvalues of B lie in the neighbourhoods of the eigenvalues of A.

Even more dumbed down: if we can find disjoint neighbourhoods around the eigenvalues of A, then we can guarantee with some certainty that small perturbations of A preserves these neighbourhoods!

Do not be too worried about these lemmas at the start, he's just doing some technical lemmas which he will refer to later so he doesn't have to do it there. You will see it used later.

>>9007676
What level of problem do you want? What topic? I can give you some good ones if you tell me what you know or are interested in.

>>9007789

it was a recommended minor, i had to take the courses.

>>9007836

linear algebra or maybe some easier analysis or easier abstract algebra.

>>9007772
>>9007772
Here are some challenges:
. Let [math]M \in \mathcal{M}_n(\mathbb R)[/math]. What is, in terms of M, the minimal number of coefficients you need to modify to make it invertible ?
. What is the average number of cycles in the decomposition of a permutation of {1,...,n} ?
. Let K be a field and let [math]F \in \mathbb K(T)[/math]. Prove that F is transcendental over K and prove that [math]K(T)/K(F)[/math] is an algebraic extension. What is its degree ?
. Let [math](a_n),(b_n),(c_n)[/math] three real sequences such that [math]a_n + b_n + c_n \to 0[/math] and [math]e^{a_n} + e^{b_n} + e^{c_n} \to 3[/math]. Prove that the three sequences converge to 0.
. Give a necessary and sufficient condition on a sequence [math](u_n)[/math] for there to exist a permutation [math]\sigma \in \mathfrak S(\mathbb N)[/math] such that [math](u_{\sigma_n})[/math] is nondecreasing.

>>9007816

if i didn't enjoy it why would i be asking for problems?

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>mfw "people" ITT judging others for the reason they do math
your insecurity is showing

>>9007856
>let [math]F \in \mathbb K(T)[/math]
I meant [math]F \in K(T) \setminus K[/math] (ie. a nonconstant fraction)

>>9007856
>. Let M∈Mn(R). What is, in terms of M, the minimal number of coefficients you need to modify to make it invertible ?

>coefficients
do you mean how many entries in a rank-deficient matrix M need to be modified to make it invertible? n-rank(M).

you have n-rank(M) vectors that can be written as a linear combination of the remaining rank(M) vectors, and each vector only needs to be modified in one position to make it linearly independent from the rest.

>>9002320
Can any american here point to a good resource to how grad school starts in the United States for pure mathematics? I am not in the US but I aspire to go there for my masters and then my PhD but I want to know at what level americans start their graduate studies so that I can compare and maybe fill in any gaps I may have.

Also any general tips to go to a famous school for grad school after being a nobody in a nobody school for undergrad?

>>9007877
>Reminder: /sci/ is for discussing topics pertaining to science and mathematics, not for helping you with your homework or helping you figure out your career path.
>If you want advice regarding college/university or your career path, go to /adv/ - Advice.

>>9007881
Stop pretending anyone outside this little circle knows fuckall about mathematics and the structure of mathematics education.

Seriously dude, do you need a surgeon to get that old stick out of your ass or what?

>>9007883
>Stop pretending anyone outside this little circle knows fuckall about mathematics and the structure of mathematics education.
It doesn't matter, it's not what the board is for

You could try asking on mathoverflow too, where there's hundreds of actual professional mathematicians, and they'd point you somewhere else too, because it's not what that website is meant for either

Is that too confusing for you?

>>9007893
I've gotten good responses for questions like that before. /sci/ can give good advice if you stay out of the shit threads. And this is the main website I use so it is only natural to ask here.

I have a question for all you /adv/posters. Have you ever actually gone to /adv/? Do you know what that place even is? Do you also know that the sticky is 4 years old and is completely outdated and only remains there because no one in 4chan administration cares about /sci/... right?

You know at least that, right?

>>9007903
>I have a question for all you /adv/posters. Have you ever actually gone to /adv/?
No, why would I ask for academic advice from 4chan? Most people here are undergraduates who think their couple years of studying gives them the capability to give guidance

>>9007914
>No
Great, so you are uninformed swine. I did not expect any better.

>>9007854
Algebra:
>Let [math]G[/math] be a finite group of order [math]n[/math] and let [math]m[/math] be the number of elements of [math]g\in G[/math] with order exactly 2. Prove that [math]n-m[/math]. What do we know if [math]n[/math] is even?
>Is [math]\mathbb Q\cong A\times B[/math] as groups where [math]A,B\not\cong \{\text{id}\}[/math]?
>Prove that a positive integer [math]p[/math] is prime if and only if [math](p-1)!\equiv -1 \pmod{p}[/math].

>>9007877
At the end of your undergrad, the universal bar is pretty much:
Algebra: know dummit &foote material front to back.
Analysis: know the material in Baby Rudin & Rudin's functional analysis book.
Geometry: know the material in Munkres, and maybe some of Hatcher's algebraic topology book. And maybe some of Pressley's differential geometry (not a lot of schools actually force this in undergrad).

That's pretty much all you will need to know.

>>9007960
Topology is not Geometry

>>9007960
Thank you. The only thing my curriculum misses from that is algebraic topology, as we only do general topology and then differential geometry. And I'll use your recommendations of books to compare when I get to those classes.

By the way, is there any difference from starting at a masters or a PhD? And by that I mean, if I go directly to a PhD program, will I just start with masters level classes until I climb up, or am I expected to be more advanced if I want to jump directly into a PhD program?

Here everyone says that going for a masters is absolutely necessary and I want to know how that holds up in the US?

>>9007962
>Topology is not Geometry
it's close enough

>>9007854
Linear:
>Prove [math]\mathbb R[/math] and [math]\mathbb C[/math] are isomorphic as vector spaces over [math]\mathbb Q[/math].
>Suppose [math]V[/math] is a vector space over [math]k[/math] and let [math]R=\text{End}(V)[/math] be its ring of endomorphisms. Show that [math]\text{End}(V\oplus V)\cong R^4[/math] as an [math]R[/math]-module.
>Let [math]V[/math] be a finite-dimensional vector space and [math]\varphi\colon V'to V[/math] be a linear map. Prove that there is an [math]n\in\mathbb Z[/math] such that [math]\ker\varphi^{n+1} = \ker\varphi^n[/math] and [math]\im\varphi^{n+1} = \im\varphi^n[/math]. Does this hold if [math]V[/math] is infinite-dimensional?

>>9007962
https://en.wikipedia.org/wiki/Geometry_and_topology

>>9007974
>>9007981
Topology includes a whole bunch of shit that looks nothing like geometry.

>>9007974
REEEEE, way to piss both the Topologists and the Geometers off.

Topology is mostly concerned with qualitative statements about the space, while geometry is more quantitative.

>>9007854
Analysis:
>Does there exist a function [math]f\colon\mathbb R\to\mathbb R[/math] such that it is discontinuous everywhere except a point where it is differentiable?
>Define the functions [math]\text{max}[/math] and [math]\text{min}[/math] in the obvious ways. Show that if [math]f,g\colon\mathbb R\to\mathbb R[/math] are continuous, then [math]\text{max}(f,g)[/math] and [math]\text{min}(f,g)[/math] are continuous functions.
>Are countable unions of compact sets compact?
>Let [math](X,d)[/math] be a metric space and [math]x\in X[/math]. Show that [math]d(x,\cdot)\colonX\to\mathbb R[/math] is continuous. Is it uniformly continuous?

>>9007989
>>9007985
wow guys, even topologists and geometers in accademia are okay with the sorting of topology and geometry into the same category. way 2 b.

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>>9007676
>>9007676
Let [math]G[/math] be a group such that, for any elements [math]g, h\in G[/math], there is a positive integer [math]n[/math] such that [math](gh)^k=g^kh^k[/math] for [math]k=n-1, n, n+1[/math]. Prove that [math]G[/math] is abelian.

>>9007859
Your post made it look like you did math just to make yourself look intelligent. Pro tip, that only works for redditor tier nerds, as most people you could think you are fooling just think you solve hard equations.

>>9008013
In my experience if Topology and Geometry are grouped together then it really only means Topology and Differential Geometry. Algebraic Geometry gets grouped with Algebra.

>>9008024
ya ag is better put in algebra, but the origina guy was only referring to differential geometry and topology, (see >>9007960)

Why is this thread so full of obnoxious weeaboos and their pathetic anime?

>>9008030
It might have something to do with this being a website inspired by a popular anime-related website in Japan.

>>9008039
Thats what /a/ is for. It is just obnoxious on /sci/, especially the need to do it on every single post they make.

>>9008043
>Thats what /a/ is for.
Actually the entire website is anime-inspired. You might want to search for other sites. I'm sure you will find something.

>>9008043
I find the stackexchange chats are better than /sci/.

>>9008019
>Your post made it look like you did math just to make yourself look intelligent.

you got me, i was doing it for anonymous 4chan cred.

challenge problems are good exercises, thanks to everyone who posted. i'll do some of them tomorrow when i'm less inebriated.

>>9008043
>>9008052
You don't know what you're talking about. This website WAS about anime... This is the science board not the faggot board so fuck off back to a with your stupid anime waifus and general loser hobbies.

>>9008075
This. Saying anime belongs everywhere because the first board is an anime board is like saying the USA belongs to the natives just because they got here first. Like Europeans and the natives, non weeaboos found somewhere fertile and turned it into something great rather than the primitive things of the weeaboos.

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>>9008062
>you got me, i was doing it for anonymous 4chan cred.
Yeah, my point exactly.

>>9008075
>>9008079
No... You seem to be confused. This is strictly an anime website. Please use some other place if you dislike that inherent property of this site.

>>9008079
Do you even know how boards were created? These boards were made to contain off-topic (read: not anime-related) discussions, so by your analogy every other board is the natives.
The first board is /b/ anime-random btw, so you newfags shouldn't really even be talking about this in the first place.

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Here is a Japanese image of a smug girl to piss someone off.

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>>9008075
>>9008079
Just posting some math related pictures.

I won't argue with subhuman weeaboos.
Your containment board is this way: >>>/a/.

>>9008082

it's easier to do problems when you can ask questions and get feedback.

OMG THIS IS SO KAWAII KAWAII KAWAII UGUU UGUU
STICKING BLACK DILDOS IN MY ASS IS NOT GAY, IT IS KAWAII

>>9007833

dude thank you that was a great explanation. definitely helped me a lot. I wish I could interpret the text that well.

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>>9008090
>>9008092
>>9008093

When I was younger I used to like some anime like DBZ and Naruto but yeah after you're 18 it's pretty much a loser thing, specially the waifu pillow thing. Specially considering you're studying math and posting on 4chan I'd not be surprised if you are pretty much losers by all standards (virgins, live with their parents, don't party or do drugs etc).

>>9008100
SO KAWAII
I WANT THIS KAWAII WAIFU'S PENIS IN MY ANUS
KAWAII KAWAII KAWAII!

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>>9008101
Please calm down and conduct yourself.

>>9008105
Nice try, PSYOP disinfo shill.

>>9008101
>don't party or do drugs etc
What an embarrassment.

>>9008107
I am calm. In just saying the obvious. You're a loser with no friends or future. You're a virgin who doesn't go out and probably have some kind of meant disorder like anxiety or depression or even symptoms of autism. Am I not right?

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>>9008101
>Specially considering you're studying math and posting on 4chan I'd not be surprised if you are pretty much losers by all standards (virgins, live with their parents, don't party or do drugs etc).
how does anyone go to college without losing their virginity and doing drugs?

>>9008107
THIS ANIME IS SO KAWAII
UGGUUUUU UGGUUUUU
SITTING ON COCK IS NOT GAY - ITS KAWAII

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>>9008092
It sure is. That doesn't mean doing math without being a student would be to seem smart. You can download FL Studio and make bangers without trying to seem like the next Bach, you can draw dicks without trying to be seen as the next Tom of Finland, you can collect rocks without trying to seem like you were building a monument worthy enough to challenge the pyramids of Giza. You probably get my point.

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>>9002320
>what are you studying?
I'm trying to establish an equivalence between anime and math.

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>>9008114
Not being normalfag scum like you.

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>>9008120
If you think about it, that allows you to literally prove the existence of Cthulhu.

>>9008118

not even sure what point you're trying to make anymore. your posts are rambling

>>9008120
The correlation is non existent because anime belongs on the autism/aspergers containment board (it's called /a/ for a reason).

>>9008129
He is an obnoxious anime poster, what else did you expect? Just leave him be.

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>>9008130
Really? OK that was silly.