/mg/ MATH GENERAL

Last thread reached the bump limit.

>what are you studying?
>any cool problems?
>any cool theorems or remarks?
>reference suggestions?
>???

Other urls found in this thread:

vixra.org/abs/1703.0073
warosu.org/sci/image/g3R37MfrTJnr_nL6BVuNBQ
warosu.org/sci/image/QpPyH3NmVNNiB8q6YrbFhQ
warosu.org/sci/image/Gz3urX5eLH5x02UE2WfVkg
twitter.com/SFWRedditImages

I am trying to understand the solution to this exercise.

I think I get it all the way until the end. How can you guarantee that you can choose such an [math] x_0 [/math]? Can you even guarantee that the o(1) term will be smaller than b-a?

That said, if someone could explain the entire argument that would be good too because... maybe the reason I don't get that final part is because I am not truly getting the parts before.

>Can you even guarantee that the o(1) term will be smaller than b-a?
doesn't that follow immediately from the definition of little o? i.e. for all epsilon>0 the o(1) term is eventually bounded above in absolute value by epsilon, so you can just take epsilon= (b-a)/2

I didn't... realize. Little o was defined in this very chapter and I guess I haven't had time to digest.

But I think I get it. If f(x) = o(1) then that means that the limit as x approaches infinity of f(x)/1 equals 0. And therefore the limit as x approaches infinity of f(x) is 0. And that means that functions in the o(1) class get arbitrarily small.

Holy shit fuck. This was trivial. I can finally see

there is a conformal map from the unit disk, D(0,1), to the puncture unit disk, D(0,1)\{0}

you guys won't find one though

I'll puncture your unit with my disk if you keep on with that cheek

I have found something quite remarkable regarding CFT and TQFT that basically confirms my suspicion from last thread.
Will post details after I'm home.

Be honest, do you use "nootropics"?

do energy drinks count?

Does coffee count?

Does semen count?

engineer detected, lol.

xDDDDDDDDDDDD
oh wait that was really really epic and funny, let me break out my surplus supply of d's
xDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

>ywn never play chess against Urschel
>ywn never prove theorems with Urschel
>ywn never throw football around with Urschel
just kill me senpai

No, I mean stronger stuff.

What's wrong with this thing on the image?

2 energy drinks?

I know it's unknown if there is a bijection from [math]\mathbb{Q} \times \mathbb{Q}[/math] to [math]\mathbb{Q}[/math], are there other sets like this? Where we don't know if theres a bijection from its cartesian square to itself?

>I know it's unknown if there is a bijection from Q×Q to Q
wrong, both are countably infinite so there's a bijection between them.

>are there other sets like this? Where we don't know if theres a bijection from its cartesian square to itself?
i think for any infinite set that the cardinality of the square is the same as the cardinality of itself, so there's a bijection. it obviously breaks down for finite sets

Sorry, I meant where we don't know what the bijection is

it shouldn't too hard to get one from Q to Q x Q , just find one from Q to N and one from N to Q x Q and compose them

I call fake. Someone is ghostwriting his articles.

The cartesian product conserves cardinality. (For infinite sets.)

Sorry, any finite product of infinite sets has the same cardinality as the set with the highest cardinality in the product.

[math] #\mathbb{Q}^n = #\mathbb{Q} = {\aleph}_{0} [/math]
[math] #(\mathbb{C}^k \times \mathbb{R}^l \times \mathbb{Q}^m \times \mathbb{N}^n) = #\mathbb{R} = c [/math]

Yup, I can confirm this.

Im taking calc-1 and it took me a full day to understand the ε, δ concept of a limit. Am I retarded?

No. It's a difficult definition if you are not used to pure mathematics. You slowly get better at understanding things as you keep on learning. Keep at it!

Brings to mind a very interesting exercise:
>Is Q isomorphic as a group to the direct product of two non-trivial groups? Why or why not?

As far as I know there aren't any examples of sets that are known to have the same cardinality but no specific bijection is known.

Even if you have to use the Schroeder-Bernstein theorem, the proof of it is constructive so really you could describe the bijection if you wanted. It just wouldn't be a very nice description.

Took me 3 days to understand it, so you're doing great by my books

The fact you understood it eventually means you have surpassed 50% of math undergrads. So 3 days itself is not bad. ;)

Let [math]\Sigma[/math] be a genus [math]g[/math] surface and let [math]\mathcal{M}_g[/math] be its mapping class group. We assign [math]2g[/math] level-[math]k[/math] integrable highest weights [math]\mu_1,\mu_1^*,\dots,\mu_g,\mu_g^* \in P^+(k)[/math], onto [math]2g[/math] points on the Riemann sphere and let [math] V_{\mu,\mu^*} = V_{\mu_1,\mu_1^*,\dots, \mu_g,\mu_g^*}[/math] be the space of conformal blocks of the CFT that satisfy the KZ monodromy. A TQFT functor can be defined that maps the surface [math]\Sigma[/math] to the vector space [eqn]V_\Sigma = \bigoplus_{(\mu),(\mu^*)} V_{\mu,\mu^*}.[/eqn] This shows that a TQFT can be constructed from a CFT.
Now let [math]L_1,L_2[/math] be links whose regular neighborhoods are copies [math]H_1,H_2[/math] of [math]H[/math], where [math]\partial H = \Sigma[/math], and denote by [math]M[/math] the 3-manifold obtained by gluing [math]h:\partial H_1 \rightarrow \partial H_2[/math]. Let [math]L(h)[/math] be the link that gives [math]M[/math] upon Dehn surgering [math]S^3[/math] and does not intersect [math]L_1, L_2[/math]. Take [math]T(h) = L \cup L_1 \cup L_2[/math] and let [eqn]\rho(h)_{\mu\nu} = \sqrt{S_{0\mu}}\sqrt{S_{0\nu}}C^{\sigma(L(h))}\sum_{\lambda:\{1,\dots,m\}\rightarrow P^+(k)} S_{0\lambda}J(T(h);\lambda)_{\mu\nu}[/eqn] via Witten's tangle operator [math]J(T(h);\lambda)_{\mu\nu}: V_{\mu,\mu^*} \rightarrow V_{\nu.\nu^*}[/math], where [math]S_{0\mu} = \prod_{i}S_{0\mu_i}[/math]. The map [math]\rho = \bigoplus_{\mu\nu}\rho_{\mu\nu}: \mathcal{M}_g \rightarrow GL(V_\Sigma)[/math] is a projectively linear representation that satisfies [math]\rho(fg) = \xi(f,g)\rho(f)\rho(g)[/math], where [math]\xi(f,g) = C^{\sigma(L(f)\cup L(g))-\sigma(L(f))-\sigma(L(g))} \in \mathbb{C}^*[/math]. This shows that the TQFT is unitary and not anomaly-free.
This construction gives me an idea of what sort of geometric data can be recovered from a TQFT, and how I can incorporate it into the space structure of a TQFT to describe AdS/CFT.

>Even if you have to use the Schroeder-Bernstein theorem, the proof of it is constructive
no it's not

What should I be studying if I want to prove things involving diophantine equations?

really depends on the diophantine equation, you might get something easy enough to work with using techniques from elementary number theory (i.e. pell's equation), or you might be looking at fermat's equation and need things from algebraic geometry, representation theory, modular forms, etc..

All this work and no replies, you deserve a reply, hear it is!

Given that we are able to obtain a TQFT from a CFT by constructing a modular functor from the conformal blocks, perhaps we can reverse engineer the construction to obtain a CFT from a TQFT.
My idea is to equip the space structure of an unitary decorated TQFT with a set of points [math]p_1,\dots,p_{2g} \in \mathbb{C}\cup \{\infty\}[/math] where [math]g[/math] is the genus of the surfaces [math]\Sigma \in \mathscr{A}[/math], such that the TQFT functor maps these set of points to the "KZ structure" on the conformal block [math]V_{\Sigma}[/math], i.e. it would ensure that the covariant tensors [math]\Phi \in \operatorname{Hom}(V_{\mu_1}\otimes V_{\mu_1^*} \otimes \dots \otimes V_{\mu_g}\otimes V_{\mu_g^*},\mathbb{C})[/math] satisfy the KZ equation [eqn]d\Phi = \omega \Phi[/eqn], where the KZ form
is given by [math]\omega = \frac{1}{k-2}\sum_{i < j}\Omega^{(i,j)}d \ln(z_i - z_j)[/math], and fibres over to a linear bundle [math]\varEpsilon(p_1,\dots,p_{2g})[/math] with a flat connection [math]\nabla = d - \omega[/math]. The TQFT functor maps [math](\Sigma,\{p_i\})[/math] to the space of conformal blocks [math]V_{\nu,\mu^*}[/math] in the sense of CFT, and then I will make sure that everything else also works as expected and that the operator invariant for this TQFT correspond to Witten's invariant. This will give me a pretty promising framework to base the AdS/CFT correspondence on.
Thanks user.

JUST
U
S
T
It should be [math]\mathcal{E}(p_1,\dots,p_{2g})[/math] and [math]V_{\mu,\mu^*}[/math].

Someone needs to shoop that picture with some more appropriate Veeky Forums reading.

so you are saying if there is a bijection QxQ->Q, the cardinality of QxQ is the cardinality of Q?

I basically have to find the smallest natural number n for [math] a^n\equiv 1 (mod 100) [/math].
I get a solution (n=40) with Euler's theorem, but how do I proof that 40 is the smallest solution or if it is not the smallest solution how do I get to the smallest one?

>if there is a bijection QxQ->Q, the cardinality of QxQ is the cardinality of Q?
of course, there's a bijection between two sets if and only if the two sets have the same cardinality

you need more details here, specifically conditions on this 'a'

for example if a=0 then a^40 is not 1 mod 100

Well, you could just use a cas, e.g. maple, to brute-force check for [math]1\leq n \leq 39[/math], but this i probably not what you want. Also, if this is not only for some specific [math]a[/math], need what already said.

Oops, forgot to add that a is any integer with gcd(a,100)=1 .

you have phi(100)=40 but there's no primitive root mod 100 so (Z/100Z)^x is a non-cyclic group of order 40

so by the fundamental theorem of finitely generated abelian groups (Z/100Z)^x is either Z/2Z x Z/2Z x Z/2Z x Z/5Z or Z/4Z x Z/2Z x Z/5Z, which tells you the smallest n is either 10 or 20

Ok, seems like the smallest n is 20. But is there a way to get that answer with just Euclid's theorem and some other really basic number theory stuff? Like "Euclid gives 40, but because of xyz we can cut it down to 20"

No primitive roots or groups, that's already too advanced. This is for students 2 months into number theory.

Bump. The problem is:

Find the smallest number [math] n \in \mathbb{N} [/math] that satisfies [math] a^n \equiv 1(mod100) [/math] for all [math] a \in \mathbb{Z} [/math] with [math] gcd(a,100)=1 [/math] .

primitive roots are pretty basic, what other theorems has your class covered?

Euclid, Euler, [math] \phi (n), \tau (n), \sigma (n) [/math] , multiplicative functions, Wilson, some basic stuff about primes, modulo (including the Chinese Remainder Theorem). Things like that.

anyone able to give a broad rundown on the current state of motives?

i get that there's all these different pure motives, mixed motives, chow motives, tate motives, artin-tate motives, and that some are basically supposed to be the ideal building blocks to work with in algebraic geometry, but the 'right' category of motives hasn't been found yet? i've seen tate motives used in vast generalizations of birch swinnerton-dyer conjecture but I've never really seen some kind of introductory book on motives, just scattered research papers

>I've never really seen some kind of introductory book on motives, just scattered research papers
also on this point is it fair to compare motives to something like L-functions, where the definition depends on the setting but all L-functions generally have the same flavor? and so we wouldn't expect a book on 'general' motives like we wouldn't expect a book on 'general' L-functions

Thanks friendo

@8963627
@8963629

I have interests in the theory of dynamical systems, the theory of functionals, real analysis, complex analysis, tensor analysis, and numerical analysis. In numerical analysis, I have an interest in developing algorithms whose error terms can be correlated with decoherence and/or violation of conservation of information in quantum theory. I have also been working on non-coordinate bases for general relativity and I think number theory is relevant to my interests in this regard.

The specific "new" thing I have been trying to break into is the analysis of hyperreal numbers. Classical field theory is extended into quantum field theory by extending the real numbers into the complex numbers and I want to study how to extend that to "hypercomplex" field theory by applying the concepts of hyperreal analysis to the real and imaginary number lines in the domain of QFT field variables. The goal in this regard is to explore the solutions to the Hamiltonian action principle that are maxima of the action. Usually the minimum action is selected because the maximum action is almost always infinity but I think I am developing a good workaround based on hyperreal numbers. I think this could equally well be a problem in dynamics or analysis.

I have a lot of well-developed applications in mathematical physics, but recently I came up with a purely mathematical application of the principles I have been struggling to make rigorous (and failing to do so because I am not a proper mathematician). In the link below I hope you find a thoughtful and well-reasoned argument against the Riemann hypothesis (that hopefully also has an application to the axial current anomaly in QCD). This is not what I consider one of my best results but it is one that can be understood without having to go into the nested references of my other papers.

>On The Riemann Zeta Function
>vixra.org/abs/1703.0073

Sophomore here. I want to focus on something related to topology next year, but I don't know exactly what. I thought I'd go for K-theory, since I'm arleady kinda good with basic algebraic topology, although homological algebra is a viscious bitch when it is detached from topology.

Anyways, if you were a young student aiming to get to a good grade school eventually and knows some very basic topology, what would you go for?

Go back to Tumblr

warosu.org/sci/image/g3R37MfrTJnr_nL6BVuNBQ
warosu.org/sci/image/QpPyH3NmVNNiB8q6YrbFhQ
who were you quoting in all of these posts?

the guy you're responding to isn't me

oh, sorry
warosu.org/sci/image/g3R37MfrTJnr_nL6BVuNBQ
warosu.org/sci/image/QpPyH3NmVNNiB8q6YrbFhQ
warosu.org/sci/image/Gz3urX5eLH5x02UE2WfVkg
who were you quoting in all of these posts?

>who were you quoting in all of these posts?
what do you mean? you can see the post number i quoted in each, i assume they're mostly anonymous

wayq?

you lost me m8

>>>/global/rules/13

reaction images aren't avatars m8

Unfortunately you got baited into admitting you're an avatarfag, so see yah I guess lol

>Unfortunately you got baited into admitting you're an avatarfag
which post was that in?

13) Do not use avatars or attach signatures to your posts.

t. rulefag

When will we have AI strong enough to be able to search the math literature to see if combinations of theorems can prove new ones?

I'm desperately trying to understand why the divisor [math](\pi^*(A) - dE)[/math] defines a morphisms that becomes [math]/phi_{\bar{k}}[/math] when you base extend to [math]\bar{k}[/math]. I've been feeling like a brainlet since I started working on that proof.

That [math]/phi_{\bar{k}}[/math] should be a [math]\psi_{\bar{k}}[/math]

Homological Algebra is really abstract subject, but once you get the hang of it it is actually pretty easy.

>grade school
Symplectic geometry. The geometric quantization people desperately need new talents.

why not master topology

From the sounds of it, you don't know a whole lot.

Personally, I would go algebraic geometry if I were able to start over from highschool.

Math student here. How do I git gud at triple integrals?

And I don't mean how to compute antiderivatives, change variables, limits of integration, etc. I know all that.

I mean about when you FEEL the integral. How do I FEEL the triple integral? I remember back when doing normal integrals I could FEEL mistakes, and I could also FEEL when my solution was right. What I did was that after integrating I would then mentally calculate an estimate of the area under the curve and see how close my result is to my estimate.

But how can I estimate what I can't FEEL? If I am doing triple integrals for volume then I can't really estimate 3d volumes as well as I could estimate areas under curves mentally. And if the integral is literally 4D then my brain can not even try to feel it.

So how do geniuses do it? I need to feel triple integrals. I have never been able to do math that I cannot feel and I worry that I may get something less than an A for the first time in my life because I can't really feel the integrals.

Rewrite your post in a less retarded way if you want an answer from me.

I usually feel integrals by printing out a 3d model of the equation and turning it around in my hands for a few hours.

If that doesn't work, I sometimes laser etch the equation onto my butt plug. It helps to feel it from the inside

>muh feels
All of my what.

How do I feel triple integrals?

If you don't can't feel mathematics then you are not going to make it my brother. Better drop out and change majors before it hurts.

What is she doing....?

>I usually feel integrals by printing out a 3d model of the equation and turning it around in my hands for a few hours.

I wish I could do this but first I can't really bring a 3D printer during a test (and while I love calculus, my current primary concern is being able to pass this final round with an A) and second I have no 3D printer.

Therefore, as has often been the case, I have to feel it. Mentally. I need to in some way feel connected to integrals like I feel connected to ideals, groups, rings, vector spaces, single integrals, double integrals, topologies, etc.

Category theory already does that.
>find duality/equivalence between categories
>formalize theorems in them in terms of categorical language
>see how it behaves under the duality/equivalence
>???
>profit
You can probably do this with Coq or other Turing-complete language that has a closed compact monoidal category of types.

I have two conundrums. The first one is that I can try and have another go at an exam I did badly in and maybe get an honorable grade (or whatever it's called, it's the thing they give you when you have the best grade), but it also substitutes my original result, so if I fuck up then I'm fucked for good. If I don't do anything I'll have an 8. What should I do?

The second one is that I'm interested to do my Master's in Cambridge, but I'm starting to realise that there's no fucking way I'm getting in (I know of two people going there next year, and their average grade is ~9.7). I went and asked my advisor about it and it looked like he simply didn't want to tell me that I'm beyond fucked. So I want to know what I'll be missing and how much I should care (stupid, I know).

Also I'm a worthless piece of trash.

>it also substitutes my original result
Are you sure?
>I'm interested to do my Master's in Cambridge
What for? Do you want to do maths or do you want to acquire social prestige?

>the thing they give you when you have the best grade
The word is "distinction".

Do more research and get a paper published, and try to sound passionate during interviews. If you don't have the grades at least have the research skills and drive to back it up.
Also I hope your grades increased throughout undergrad, or you're fucked.

>What for? Do you want to do maths or do you want to acquire social prestige?
I want to do good maths, to feel I deserve the respect people have for me and for more people to respect me.
They did (probably not by enough, though), but I'm already convinced that I'm beyond fucked, and no way I'm getting anything published and beating the competition for the few places they have. I just want to know what I'll be missing out, and if there's any way to at least be as good a mathematician as someone getting their master's there.

Do masters elsewhere mate lmfao

>what I'll be missing out
Nothing.
>if there's any way to at least be as good a mathematician as someone getting their master's there.
Where you're getting your degree is irrelevant. If you have the knack for math you got it, if you don't have it, you don't have it. Institutions are a filter, nothing more. They don't impart anything on you.
>to feel I deserve the respect people have for me and for more people to respect me.
So you just want prestige. Try to switch majors. I recommend marketing.

Amen.

>Where you're getting your degree is irrelevant

EPIC reddit meme LOL!!!!! You are making a very FUNNY comment chain!!!!! GOLD!!!!

>Institutions are a filter, nothing more.
>They don't impart anything on you.
is this how you cope with going to a no name school? GOLD!!!

That wasn't me, frogposting retard.
>GOLD!!!
hahahahaha LOL!!! Epic meme breh!!!

Is this how _you_ cope? It's readily apparent you are trying to compensate for something.

I can't bear to see this retardation.

>It's readily apparent you are trying to compensate for something.
oh please elaborate, I'd love to hear what you think I could be compensating for by not deluding myself into living in a bubble where reputation and prestige don't affect reality

you're not serious are you?

Fuck off idiot.

>Veeky Forums poster
>completely unfamiliar with the research
Pick both. As far as your ability to do maths is concerned, which is what I was talking about, yes, it's irrelevant. The best universities are those which select for the best students.