MATH GENERAL

Math general - cute duckie edition
>what are you studying?
>any cool problems?
>any cool theorems or remarks?
>reference suggestions?
>???

Other urls found in this thread:

youtube.com/watch?v=m6rfpQXzXu0
ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf
link.springer.com.ololo.sci-hub.io/article/10.1007/BF03024397
arxiv.org/pdf/1609.08458.pdf
www3.nd.edu/~andyp/notes/Isoperimetric.pdf
web.stanford.edu/~danlass/esslli2011stus/petrovic.pdf
corelab.ntua.gr/studygroup/Tarski_ElGeom.pdf
stacks.math.columbia.edu/browse
twitter.com/SFWRedditImages

>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.

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)

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.

>[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.

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

>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/

>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

>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

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.

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

Mathematics is much too hard for mathematicians.

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.

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)

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).

>linking to numberphile unironically
Kill yourself.

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.

Has anyone here tried studying magic?

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

Yes and how do you prove that inequality ?

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

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

>he's not studying outside

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

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?

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)

>implying calculus isn't high school math

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: link.springer.com.ololo.sci-hub.io/article/10.1007/BF03024397

Another elementary proof: arxiv.org/pdf/1609.08458.pdf

another www3.nd.edu/~andyp/notes/Isoperimetric.pdf

Heh, that's really neat. Thanks !

Kein Problem. Geometry is underrated.

a common feel, friend
i wasn't blaming math. just expressing the well known feel of going down a rabbit hole of sources.
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.

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

Magic, math, what's the difference?

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

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.

You should be able to solve this.

web.stanford.edu/~danlass/esslli2011stus/petrovic.pdf

>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

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

You have much to learn.
corelab.ntua.gr/studygroup/Tarski_ElGeom.pdf

Back off! Patchouli is mine.

That doesn't solve it.

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

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

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

You're free to fuck off anytime.

No, you faggot.

...

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

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

Blow me anyway you incurious fag.

>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.

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.

>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...

>something must be complicated for it to be elegant

what a cancerous opinion

Who are you quoting?

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

Physics should be banned and all physicists should be hanged.

Amen.

Wow talk about math anytime.

>s___ _p

>reddit frogs
Subhuman garbage belongs in some other thread.

>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.

The last word is "up".

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

It might be Ip.

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

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

Hello... Please see the attached post below:

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

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

Hey... Please see the attached message:

Reminder to report all spammers.

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

I think reading that gave me autism. Send help.

stacks.math.columbia.edu/browse

Oooops... Seems like you made a mistake! This is not a thread for "Phyzicks".
see the attached message below:
hello... it appears you are using the wrong thread.
please use the so-called "catalog" function on this website to find a "Phyzicks" thread.

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).

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

Can't help those afflicted with brainletism.

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.

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.

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

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

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

see

see

Anime site.

see

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

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!

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

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

>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.

>I don't know which structure exactly.
I mean the "internal" structure that makes a category internal.
>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?

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].