/mg/ = /math/ general

>isn't a notational quirk
Of course it is. [math] \hat{1} \neq 1 [/math].
One is a set of integers the other is a natural number. If you're just talking about the cyclic groups of 3 or 9 elements with additive notation then [math] 1 + 1 = 2 [/math] in both, uniquely.

Actually, I've seen something like that before, though more at the level of modeling superconductivity. There's actually also a nice paper about using AdS/CFT to model Navier-Stokes, there was a nice talk at perimeter about this sort thing.

>a nice paper about using AdS/CFT to model Navier-Stokes
Please share.

What's a good book on pure category theory?

Going to start learning a functional language like Haskell to apply it

Yep, it's a rather popular idea recently in the non-perturbative part of the condensed matter community. One of the main motivations for why I want to get my shit working is partly because I want a general framework for these sorts of stuff.
Some interesting readings on this:
arxiv.org/abs/0809.3402
arxiv.org/pdf/1409.1369.pdf

Better pick a non-garbage language.

Wrong choice of programming language. Haskell is too limited. You can't use it to do categorical stuff.
And you don't need to know category theory to do functional programming either. Not will you learn anything about category theory from functional programming.

>tfw an Iranian civil engineer rediscovers Fermat's elementary proof of Fermat's Last Theorem

arxiv.org/abs/1706.04186

...

But to answer your question: Awodey's "Category Theory".
For functional programming, Cutland's "Computability: An Introduction to Recursive Function Theory".

His third lemma is already wrong. Next.

>His third lemma is already wrong.
What's wrong with it?

Fear the trisector.

I had no idea; I thought it would at least make me more efficient or at least better at it, no?

The numbers 1, 2 and 11 exist only as elements of Z (or N if we're talking monoids). So if the OP's statement means anything at all, it asks to find an algebraic structure A such that the natural map f: Z->A satisfies f(2) = f(1) + f(1) = f(11). That is exactly what happens for Z_3.

>algebra 1 and calc 1
>Wasn't even able to finish the exam paper
Should I just kill myself?

Set A = 9261, B = 125, C = 105, z = 3. The lemma does not hold.
In fact, there are countably many points in [math] { \mathbb{N} }^4 [/math] for which that "lemma" does not hold.
Guess how I picked (A, B, C, z) above.

Did you study the material? Did you know what you have to do but were just slow at writing it down? Depending on the causes for your tardiness, the cure for this is more practice or alcohol in your bloodstream during the exam.

Yes to both questions.
I felt somewhat confident, but time seemed to fly a bit too fast, I only got about 3/4 of the paper done.
It doesn't really help that I had been up for over 12 hours too.

I do know how to solve them, but I'm a bit slow at thinking, I guess the only way to solve that is more practice.
Still, I'm pretty disappointed.

Also I make a lot of mistakes.

You probably focus too much on irrelevant details. Booze works wonders for that.

What would be the worst possible mathematical development in terms of falsifying a huge number of conjectures?

I was thinking probably finding a zeta function zero off the 1/2 line

I have a question. Has the function [eqn F(n) = \sum_{h=1}^{n} \frac{n}{gcd(n,k)} [/eqn] ever been studied? Does anyone have any references?

[eqn] F(n) = \sum_{h=1}^{n} \frac{n}{gcd(n,k)} [/eqn]

Why are you interested in that function? That could help us find a reference.

I'm looking for something to study in my free time and I don't want to risk re-inventing the wheel if this has already been studied.

This might help

quora.com/Given-N-what-is-the-value-of-sum_-k-1-N-frac-k-gcd-k-N

Oh, that's good. Hopefully that's all that is known about it. I'll see if I can find some asymptotic formulas.

Why do you care if it has already been studied?

Because I don't want to reinvent the wheel. If it has been studied thoroughly then I can just go back to the drawing boards and think of another function.

>it would at least make me more efficient or at least better at it, no?
No.

I think I found a way to prove the Riemann hypothesis in the negative using specially defined sheaves in [math] { \mathbb{R} }^3 [/math]. If you see anything on this published in the next couple of months, know that I did it.

Best of luck.

Mention Veeky Forums in the preface.

arxiv.org/abs/1006.1902
arxiv.org/abs/1101.2451
arxiv.org/abs/1104.5502
Also a pretty basic (but informative) talk about these sorts of ideas.
perimeterinstitute.ca/videos/black-holes-harmonic-oscillators-21st-century
youtube.com/watch?v=nOSm2rpz0-c
youtube.com/watch?v=Fl8vYGloaLg
Makes sense why they'd want to, many body physics with strong correlation looks like a bitch. What exactly is it that you want to do?
Construct classes of duality relations?

"Breaking news: Animefag Veeky Forumsentist BTFOs the mathematical community"

I can dig it, but I prefer my method of putting all possible counterexamples on a list and checking all of them one by one.

I know there's some Tu fans on this board, anyone reading his new book?

Why R^3?

If/when quantum computers come out, where exactly do we see improvements in computation? For example recent computations show that there's no odd perfect numbers less than 10^1500, will quantum computers necessarily allow us to look much further than that number by some factor or does everything depend on the type of algorithm?

>Conclusion: The Riemann Hypothesis is wrong, Riemann BTFO, now on suicide watch. How will complex analysts ever recover? All this time we knew that the fucking faggots were a bunch of brainlet cucks anyways. It wouldn't be the first time complex fags get cucked by [math] \mathbb{R}^3 [/math].

>It wouldn't be the first time complex fags get cucked by [math]\mathbb{R}^3[/math].
Elaborate.

How do we fix the math education in the US for K-12? How do we stop producing high school graduates who can't do basic algebra and can't add fractions?

>What exactly is it that you want to do?
You can read the previous 2 threads about it here.
warosu.org/sci/thread/S8942887
warosu.org/sci/thread/S8962385

It is believed that quantum computing will be able to solve some NP/QP problems in P time and it will also allow us to solve problems in P that were previously thought to be unaccessible by classical means, such as calculations in critical phenomena.

Any book that explains fractions ?
Tried khan but doesn't stick somehow.

Are you asking why I am doing that or are you asking why I can do that in the first place? The answer to the former is pretty complicated and an important part of what I think will become the proof, so I don't want to give it away yet, but it suffices to point out that [math] { \mathbb{R} }^3 [/math] has a more interesting topological structure. The answer to the latter should be obvious: [math] \mathbb{C} \simeq { \mathbb{R} }^2 \hookrightarrow { \mathbb{R} }^3 [/math].

Could you give more concrete examples? I'm studying quantum computing (just the basics for now) and I'm intrigued. Also, if you could recommend a good textbook or set of notes apart from Chuang and Nielsen I'd be forever grateful.

Read Bernevig and then pic related

Thanks :)

Assume a construction of [math] \mathbb{Z} [/math] is available. Now consider the Cartesian product [math] \mathbb{Z} \times \mathbb{Z} [/math].

We now have a set that can be turned into the rational numbers. The problem we have is that it is very chaotic, in many ways. It is lacking in structure. So we shall give it to it.

One problem we have is that if we want to represent "one over two" as (1,2), it could also be represented as (2,4) so we do not have unique representations. Lets fix that.

In the last set, we say that [math] (x,y) [/math] is equivalent to [math] (a,b) [/math] if an only if [math] xa = yb [/math]. This defines a partition of our set and cuts it in a very desirable way. The proof that this is indeed a partition is left as an exercise to the reader.

Now, when referring to "one over two" we do not take (1,2) or (2,4), we now take the class in which (1,2) and (2,4) are in. That entire class is now "1 over 2".

So now we have a set and a representation. We need structure. Consider the operations defined as:

[math] (x,y) * (a,b) = (xa,by) [/math]
[math] (x,y) + (a,b) = (xb + ya,yb) [/math]

The proof that these operations give our set a field structure is left as an exercise to the reader.

You now understand fractions. Congratulations.

>he/she's not working on at least one (1) mathematical prize problem
What's your excuse Veeky Forumsentists?

Bitch, do you seriously expect people to comb through two whole threads for your relevant posts? Tex isn't even working on warosu. (Or is there something wrong on my end?)

>oh boo hoo information isn't spoonfed into my facehole
Don't bother, you won't understand it anyway.

fractalforums.com/index.php?action=gallery;sa=view;id=20351

>Every pixel is either white or black.

>This is what the algorithm is, for each pixel's integer coordinates (in binary):

>0 - y -> a
>a & x -> a
>a - x -> a
>bitwise_population(a) -> a
>if(a is odd){the pixel is black}

>How does this generate patterns that look like Koch snowflakes? I don't know!

>All of the latex is in code form and not in equations
Had to put it into sharelatex before I could read, seems you're doing work in the same vein as lurie and hopkins, at least at the basic level of applying category theory to tqfts, I was actually able to follow some of it thanks to lurie's 4 videos, my C*-algebra background, and just googling a few things (Analysis and some C*-algebra stuff is what I do, so most of the physics and geometric aspects are out of my field). This reminds of me of something that might interest you given some of your comments in the last two threads, namely the subject matter of a conference I'll be attending math.uh.edu/analysis/2017conference.html
Here are some links to good articles on the subject
math.yorku.ca/~ifarah/Ftp/icm9.pdf
math.yorku.ca/~ifarah/Ftp/mt-nuc.pdf
ma.huji.ac.il/~sustretov/notes/hse.pdf
Since category theory and model theory are "complementary" fields in a sense and model theory has been used in classification problems of C*-algebras it might provide you with some nice tools user.
There's also some nice discussions here
golem.ph.utexas.edu/category/2008/07/category_theory_and_model_theo.html
(Also, I'm not , I'm )

My time is too valuable. Who the hell are you?

Sometimes you're really nice and friendly and other times you seem like a bit of an asshole. What gives?

This actually looks valuable. I'd never expect mathematical logic to be useful in operator algebras. Even though my aim is towards a more "specializing" direction in going from TQFT to CFT I think these higher level perspectives may as well give me some inspiration. Thanks user.
Also that conference looks interesting as fuck, could you film it and upload it somewhere when you get back?

PMS.

...

What is PMS?

delet dis

I'll email the conference organizers to see if they plan on recording the lectures themselves, if not I'll just record them and upload em to a google drive or maybe youtube.

...

Fucking mint senpai, cheers.

I don't care about the physics stuff but this is an interesting pointer. You're a certifiably cool guy.

What list are you talking about?

/mg/, while it is not mandatory, I want to write an undergrad thesis for graduation. Are there good reads to find inspiration. Maybe a book about open problems? I am particularly interested in analysis and number theory, and while I'd gladly do something in either, I would prefer doing something in their intersection.

Google searches on this topic are not satisfactory.

Can you say a bit more about it? What's the audience?

What exactly do you not understand? Also are you French?

That algebraic structure you're looking for OP is base 1
It's literally elementary:
>I+I=II, II+I=III, etc
Roman numerals also works.

No problem anons
When writing an undergrad thesis you typically work with a professor, they should be the first person you ask, especially if you aren't already well versed in either of the fields. Next if you want a field that's at the intersection of analysis and number theory there is of course analytic number theory, you could write up an exposition on aspects of the field like recent major results, a summary of current open problems with progress made towards them, maybe even some notes on a special topic within the field, at the very least it'd show an understanding of the basics and current landscape of the field. Another field that uses techniques from analysis to solve problems in number theory is ergodic theory, specifically ergodic theory applied to the theory of diophantine approximations, a major open conjectures in the field where people have been successfully applying these techniques are the littlewood conjecture and variants/generalizations of said conjecture. A recent fields medal was given for work of this kind (lindenstrauss 2010) a decent thesis might be on the littlewood conjecture and generalizations, essentially you'd start with the first chapter being a historical expose on the problem with motivation, then an intro to the require number theory, measure/ergodic theory, then the basics of lie groups and lattices with a little bit about manifolds, this'll cover most of what you'll need, then you can go about explaining the results towards the conjectures, generalizations of the conjecture, and variants of the conjecture that have been solved. All in all that'd make for a decent thesis. If you already have good research (whether or not publishable) you can always wrap that up and call it your thesis.

Unary arithmetic. Base 1.

[math]\left( \mathbf Z / 9\mathbf Z,\, +\right)[/math]

This seems to be more about its practical implementations, but what I'm interested in is the theory behind quantum computing.

Thanks anynways.

Category theory underpins a lot of the stuff to do with types in Haskell. It doesn't use anything deep from category theory, nor does it use it beyond an organisational scheme to ensure consistency. Haskell has an excellent type system, possibly the tightest one going around.

>practical implementations
>topological quantum computation
Not for a few decades, kid.

This falls under "notational quirks".

Here is a funny one for you.

Let [math]A, B[/math] be two square, invertible matrices of the same dimension.
Show that [math]A + B = AB \implies AB = BA[/math]

A natural ensemble in which 2, 3, ... and 10 have been removed. As such, the successor of 1 would be 11. Or does that falls into trivial group ?

Does only one need to be invertible (say A)?
A+B = AB
=> I+A^-1 B = B
=> B-I = A^-1B
=> AB-BA = AB-A+A-BA=A(B-I)+A(I-B)=AA^-1B-AA^-1B=0

I'd really like to learn math this year and to do that I usually try and get involved in a part of a community. Before I try and take part in a community, I generally look at the Code of Conduct first.

I noted that the Code of Conduct mentions 'gender' but that doesn't really feel like it includes non-binary or agender folk.

Well done, the proof in my textbook is a bit less contrived but require both to be invertible (and incidently, different from the identity)

A + B = AB => (A - I)(B - I) = I

So (A - I) = (B - I)^-1

And then (B - I)(A - I) = I
As a result A + B = BA, and so AB = BA

>I generally look at the Code of Conduct first.
Kill yourself.

Why is she smiling like that?

two points that don't overlap occupy the same line.

two lines that don't intersect can occupy the same plane

two planes that don't intersect can occupy the same space

can two spaces not intersect?

>two points that don't overlap occupy the same line.
They can "occupy" the same line even if they "overlap".
>two lines that don't intersect can occupy the same plane
They can "occupy" the same plane even if they intersect. Lines that don't intersect can "occupy" different planes.
>two planes that don't intersect can occupy the same space
Two planes "occupy" the same space even if they intersect. Planes that don't intersect can "occupy" different spaces.
>can two spaces not intersect?
Yes.
(By the way: you're retarded.)

i meant that they can be oriented in a way that requires an extension of the components needed to describe them. for two points that aren't occupying each other you NEED a line to describe them. otherwise you just need a number. same with the rest but with spaces i don't see how a fourth spatial component can occur as with the line and plane, they extend infinitely, so an infinite tesselation would lead to an intersection as a limiting case.

>i meant that they can be oriented in a way that requires an extension of the components needed to describe them.
"They" being? "Component" being? "Extension" meaning? "Oriented" meaning?
Fuck off back to we demand rigour here. Your whole post is gibberish.

What do you mean by "you only need a number to describe two points that are occupying each other"?

Why do I have the impression that these posts are made by the same guy?

what are you complaining about? do you not know about n-tuple components needed to describe a vector in some vector space? i'm just asking how that applies in the strictly spatial context of orthogonality. if you don't know algebra why even reply? i'm honestly baffled here.

that was with regards to a point on the real number line. it'd just be the number itself.