/mg/ = /math/ general

[Anonymous No.8974800]

Define an algebraic structure where pic related holds non-trivially. Naturally, notational quirks and concatenation are out of the question.

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

[Anonymous No.8974803]

>>8974800
The free group

[Anonymous No.8974807]

>>8974800
>Define an algebraic structure where pic related holds non-trivially.
Z/3Z and Z/9Z

[Anonymous No.8974811]

>>8974803
>The free group
>Naturally, notational quirks and concatenation are out of the question.

[Anonymous No.8974817]

>>8974807
>Z/3Z and Z/9Z
[math] \^{1} + \^{1} = \^{2} [/math]

[Anonymous No.8974819]

>>8974817
you got it :)

[Anonymous No.8974820]

>>8974807
>Z/3Z and Z/9Z
[math] \hat{1} + \hat{1} = \hat{2} [/math]

[Anonymous No.8974823]

>>8973772
There was a talk on holographic techniques in quantum Hall effect using conventional AdS/CFT (i.e. not the kind I was looking at) by this rather famous dude. In this picture you can map charged blackholes to particle excitations with fractional statistics.

[Anonymous No.8974824]

>>8974820
yep, you got it!
[math] \hat{2} = \hat{11} [/math]

[Anonymous No.8974827]

>>8974824
>Naturally, notational quirks and concatenation are out of the question.
Come on guys, try. This shouldn't be difficult.

[Anonymous No.8974838]

>>8974827
But the difference of 11 and 2 being divisible by 3 and 9 isn't a notational quirk and has nothing to do with concatenation.

Next?

[Anonymous No.8974850]

>>8974838
>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.

[Anonymous No.8974852]

>>8974823
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.

[Anonymous No.8974858]

>>8974852
>a nice paper about using AdS/CFT to model Navier-Stokes
Please share.

[Anonymous No.8974862]

>>8974800
What's a good book on pure category theory?

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

[Anonymous No.8974865]

>>8974852
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:
https://arxiv.org/abs/0809.3402
https://arxiv.org/pdf/1409.1369.pdf

[Anonymous No.8974872]

>>8974862
Better pick a non-garbage language.

[Anonymous No.8974876]

>>8974862
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.

[Anonymous No.8974887]

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

https://arxiv.org/abs/1706.04186

[Anonymous No.8974894]

>>8974887

[Anonymous No.8974920]

>>8974876
>>8974862
But to answer your question: Awodey's "Category Theory".
For functional programming, Cutland's "Computability: An Introduction to Recursive Function Theory".

[Anonymous No.8974928]

>>8974887
His third lemma is already wrong. Next.

[Anonymous No.8974930]

>>8974928
>His third lemma is already wrong.
What's wrong with it?

[Anonymous No.8974933]

>>8974887
Fear the trisector.

[Anonymous No.8974938]

>>8974920
>>8974876
>>8974872
I had no idea; I thought it would at least make me more efficient or at least better at it, no?

[Anonymous No.8974940]

>>8974850
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.

[Anonymous No.8974950]

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

[Anonymous No.8974953]

>>8974930
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.

[Anonymous No.8974986]

>>8974950
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.

[Anonymous No.8975005]

>>8974986
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.

[Anonymous No.8975011]

>>8975005
Also I make a lot of mistakes.

[Anonymous No.8975014]

>>8975005
You probably focus too much on irrelevant details. Booze works wonders for that.

[Anonymous No.8975103]

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

[Anonymous No.8975112]

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?

[Anonymous No.8975113]

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

[Anonymous No.8975115]

>>8975112
>>8975113
Why are you interested in that function? That could help us find a reference.

[Anonymous No.8975125]

>>8975115
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.

[Anonymous No.8975140]

>>8975125
This might help

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

[Anonymous No.8975145]

>>8975140
Oh, that's good. Hopefully that's all that is known about it. I'll see if I can find some asymptotic formulas.

[Anonymous No.8975147]

>>8975125
Why do you care if it has already been studied?

[Anonymous No.8975148]

>>8975147
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.

[Anonymous No.8975238]

>>8974938
>it would at least make me more efficient or at least better at it, no?
No.

[Anonymous No.8975271]

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.

[Anonymous No.8975273]

>>8975271
Best of luck.

Mention /sci/ in the preface.

[Anonymous No.8975281]

>>8974858
https://arxiv.org/abs/1006.1902
https://arxiv.org/abs/1101.2451
https://arxiv.org/abs/1104.5502
Also a pretty basic (but informative) talk about these sorts of ideas.
http://www.perimeterinstitute.ca/videos/black-holes-harmonic-oscillators-21st-century
https://www.youtube.com/watch?v=nOSm2rpz0-c
https://www.youtube.com/watch?v=Fl8vYGloaLg
>>8974865
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?

[Anonymous No.8975286]

>>8975271
"Breaking news: Animefag /sci/entist 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.

[Anonymous No.8975323]

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

[Anonymous No.8975324]

>>8975271
Why R^3?

[Anonymous No.8975334]

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?

[Anonymous No.8975345]

>>8975271
>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].

[Anonymous No.8975352]

>>8975345
>It wouldn't be the first time complex fags get cucked by [math]\mathbb{R}^3[/math].
Elaborate.

[Anonymous No.8975355]

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?

[Anonymous No.8975357]

>>8975281
>What exactly is it that you want to do?
You can read the previous 2 threads about it here.
>>/sci/thread/S8942887
>>/sci/thread/S8962385

[Anonymous No.8975387]

>>8975334
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.

[Anonymous No.8975394]

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

[Anonymous No.8975397]

>>8975324
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].

[Anonymous No.8975412]

>>8975387
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.

[Anonymous No.8975425]

>>8975412
Read Bernevig and then pic related

[Anonymous No.8975426]

>>8975387
Thanks :)

[Anonymous No.8975432]

>>8975394
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.

[Anonymous No.8975463]

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

[Anonymous No.8975504]

>>8975357
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?)

[Anonymous No.8975513]

>>8975504
>oh boo hoo information isn't spoonfed into my facehole
Don't bother, you won't understand it anyway.

[Anonymous No.8975530]

http://www.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!

[Anonymous No.8975536]

>>8975357
>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 https://www.math.uh.edu/analysis/2017conference.html
Here are some links to good articles on the subject
http://www.math.yorku.ca/~ifarah/Ftp/icm9.pdf
http://www.math.yorku.ca/~ifarah/Ftp/mt-nuc.pdf
http://www.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 anon.
There's also some nice discussions here
https://golem.ph.utexas.edu/category/2008/07/category_theory_and_model_theo.html
(Also, I'm not >>8975504, I'm >>8975281
)

[Anonymous No.8975537]

>>8975513
My time is too valuable. Who the hell are you?

[Anonymous No.8975548]

>>8975513
Sometimes you're really nice and friendly and other times you seem like a bit of an asshole. What gives?

[Anonymous No.8975554]

>>8975536
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 anon.
Also that conference looks interesting as fuck, could you film it and upload it somewhere when you get back?

[Anonymous No.8975555]

>>8975548
PMS.

[Anonymous No.8975558]

>>8975548
>>8975555

[Anonymous No.8975560]

>>8975555
What is PMS?

[Anonymous No.8975562]

>>8975555
delet dis

[Anonymous No.8975565]

>>8975554
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.

[Anonymous No.8975568]

>>8975530

[Anonymous No.8975579]

>>8975565
Fucking mint senpai, cheers.

[Anonymous No.8975603]

>>8975536
>>8975565
I don't care about the physics stuff but this is an interesting pointer. You're a certifiably cool guy.

[Anonymous No.8975614]

>>8975286
What list are you talking about?

[Anonymous No.8975877]

/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.

[Anonymous No.8975878]

>>8975323
Can you say a bit more about it? What's the audience?

[Anonymous No.8975885]

>>8975394
What exactly do you not understand? Also are you French?

[Anonymous No.8975897]

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.

[Anonymous No.8975985]

>>8975579
>>8975603
No problem anons
>>8975877
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.

[Anonymous No.8976177]

Unary arithmetic. Base 1.

[Anonymous No.8976233]

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

[Anonymous No.8976412]

>>8975425
This seems to be more about its practical implementations, but what I'm interested in is the theory behind quantum computing.

Thanks anynways.

[Anonymous No.8976515]

>>8974938
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.

[Anonymous No.8976520]

>>8976412
>practical implementations
>topological quantum computation
Not for a few decades, kid.

[Anonymous No.8976562]

>>8975897
>>8976177
This falls under "notational quirks".

[Anonymous No.8976648]

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]

>>8974800
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 ?

[Anonymous No.8976651]

>>8976648
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

[Anonymous No.8976661]

>>8974800
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.

[Anonymous No.8976662]

>>8976651
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

[Anonymous No.8976672]

>>8976661
>I generally look at the Code of Conduct first.
Kill yourself.

[Anonymous No.8976707]

>>8976672
Why is she smiling like that?

[Anonymous No.8976710]

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?

[Anonymous No.8976718]

>>8976710
>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.)

[Anonymous No.8976724]

>>8976718
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.

[Anonymous No.8976736]

>>8976724
>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 >>>/lit/ we demand rigour here. Your whole post is gibberish.

[Anonymous No.8976743]

>>8976724
What do you mean by "you only need a number to describe two points that are occupying each other"?

[Anonymous No.8976750]

>>8976661
>>8976710
Why do I have the impression that these posts are made by the same guy?

[Anonymous No.8976752]

>>8976736
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.

>>8976743
that was with regards to a point on the real number line. it'd just be the number itself.

[Anonymous No.8976760]

>>8976736
"they" being the elements that require n components to be described in their respective dimension

"component" being the elements

"extension" being the reason for which the additional component raises "they" to the next dimension

[Anonymous No.8976812]

>>8976752
>>8976760
>do you not know about n-tuple components needed to describe a vector in some vector space?
You don't need any collection of elements from a field to describe a vector. What you're talking about (if you're even talking about anything, more below) is a system of coordinates, semen slurper.
>the strictly spatial context of orthogonality
The what?
>that was with regards to a point on the real number line.
What if the point is not "on the real number line"?
>elements that require n components to be described
>"component" being the elements
Wew lad. Elements require components which are elements. Are you a dog? And are you sure you know what "dimension" means?
>"extension" being the reason for which the additional component raises "they" to the next dimension
The "reason"... components (which are elements) "raise" elements (which require components) to the "next" "dimension"...?

No, seriously, >>>/lit/

[Anonymous No.8977092]

>>8976718
>(By the way: you're retarded.)
Rude and autistic

[Anonymous No.8977111]

>>8975548
Autism.

[Anonymous No.8977112]

>>8976662
Anon's proof has an error unfortunately.
>AB-A+A-BA=A(B-I)+A(I-B)

>(and incidently, different from the identity)
That's implied by invertibility, if A is the identity then I + B = IB --> I = 0.

[Anonymous No.8977722]

Fix a strict monoidal tensor category [math]\mathscr{V}[/math] with ground ring [math]K[/math].
Define its Grothendieck algebra [math]K_0(\mathscr{V})[/math] as the [math]K[/math]-module of isomorphism classes of the objects in [math]\mathscr{V}[/math] with addition defined by [math][V] + [W] = [V\oplus W][/math] and multiplication [math][V][W] = [V \otimes W][/math]. On the other hand, define the commutative associative unital Verlinde algebra [math]\mathbb{V}[/math] of [math]\mathscr{V}[/math] as a [math]K[/math]-module generated by the basis [math]\{b_i\}_I[/math] ([math]0 \in I[/math]) such that [math]\forall v \in \mathbb{V}[/math], we have [math]v = \sum_{i\in I} k_i b_i[/math] with [math]k_i \in K[/math]. Multiplication in this module is defined by [math]b_ib_j = \sum_{r\in I} h_r^{ij}b_r[/math], where [math]h_r^{ij} = \operatorname{dim}(\operatorname{Hom}(V_r,V_i \otimes V_j))[/math].
If [math]\mathscr{V}[/math] is semisimple with a dominating family [math]\{V_i\}_I[/math] of objects such that [math]V = \bigoplus_{i\in I} V_i[/math] for every [math]V \in \mathscr{V}[/math], then the maps [math][V] \mapsto \sum_i \operatorname{dim}(\operatorname{Hom}(V_i,V))b_i[/math] and [math]b_i \mapsto [V_i][/math] are mutually inverse, and they define a canonical isomorphism between [math]K_0(\mathscr{V})[/math] and [math]\mathbb{V}[/math].

[Anonymous No.8977768]

Brainlet here.

What's a good resource to reteach myself all of High School Math?

[Anonymous No.8977779]

>>8977768
I'd say Khan Academy.

>>8977722
Why are the first two phrases exercises/questions, and the last one an affirmation ?
Also, isn''t
>[math]V = \bigoplus_{i\in I} V_i[/math] for every [math]V \in \mathscr{V}[/math]
Just a fancy way to say that you can find a base in [math]\mathscr{V}[/math] for every [math]V[/math] ? If so, why not define a single base and construct your [math]V[math] linearly ? Would be easier.

[Anonymous No.8977824]

>>8977779
>Just a fancy way to say that
No. https://en.wikipedia.org/wiki/Coproduct

[Anonymous No.8977881]

>>8977824
I thought plus sign in a O denoted direct sum. My bad.
I still don't understand why you post these.

[Anonymous No.8977883]

>Why are the first two phrases exercises/questions, and the last one an affirmation ?

[Anonymous No.8978527]

>>8977881
I am not that guy. That post outlines a pretty elementary construction. I think he was just bumping the thread.

[Anonymous No.8978534]

>>8977883
Instead of being condescending, why can't you explain ?
>inb4 muh info spoofeeding
Why go to a discussion forum if you don't want to discuss ?

[Anonymous No.8978542]

>>8978534
Because nobody in his real life gives a single fuck about his autistic ramblings. Also, he feels important here.

[Anonymous No.8978556]

>>8976562
Except that it's literally given as an example of an algebraic structure on Wikipedia you fucking brainlet

[Anonymous No.8978571]

>>8978556
Are you only pretending to be retarded? Addition is a binary operation, even in a unary numeral system.

[Anonymous No.8978579]

What's /math/'s mascot?
Why should it be Cirno, the mistress of 9s?

[Anonymous No.8978640]

supreme brainlet trying to re-understand prob and stats

I don't understand how they got to the last line from the second last line at all

[Anonymous No.8978645]

>>8978579
Cirno is dumb

[Anonymous No.8978649]

>>8978645
And she has a superiority complex.
/mg/ personified.

[Anonymous No.8978761]

Brainlet here. Also awful at trigonometry. How do i do this without substitution or parts (its supposed to be near-immediate)

[math]\int_ x cos^7(x^2) sin(x^2) dx[/math]

[Anonymous No.8978870]

>>8974800
>Naturally, notational quirks and concatenation are out of the question.

Why would concatenation "naturally" be out of the question?

Are you unaware of algebraic structures that use the concatenation operator? Or are you aware of such structures, but for some personal reason you just want to arbitrarily throw them out? If so, can you site a respected author who shares your view that concatenation is not a "valid" operator to use in the study of algebraic structures?

[Anonymous No.8978888]

Ultimate retard is asking for help. Childish problem.
How to prove that if I have a/b, c/d and e/f, natural a, b, c, d, e, f and ad - bc = ±1, f < b + d, there is no such e/f: a/b < e/f < c/d ?

[Anonymous No.8978894]

>>8976710
>>8976724
>>8976760
you have zero fucking idea what you're saying, don't you

[Anonymous No.8978915]

>>8978870
>Why would concatenation "naturally" be out of the question?
It's too simple. The first thing you think of.

[Anonymous No.8978935]

>>8978870
don't be an autist anon

[Anonymous No.8978958]

>>8974800
Are there any names for algebraic structures with more than two operations on the underlying set?

I've been fucking around with commutative hyperoperations lately, wanted to know if anybody else has thought about it.

[Anonymous No.8979036]

>>8978534
Learn to read first. There were literally zero questions asked.

[Anonymous No.8979044]

If you were to train a student from scratch, what textbooks would you use? Assume they know arithmetic. So what would you use for:

>Algebra
>Geometry
>Calculus

[Anonymous No.8979051]

>>8978958
Not to my knowledge. Look into universal algebra for a broader view on what you're interested in.
This is a good textbook: https://math.berkeley.edu/~gbergman/245/3.3.pdf

[Anonymous No.8979055]

>>8978888
Reductio ad absurdum. Work the problem in reverse.

[Anonymous No.8979076]

>>8978761
Turn the product into a sum to get rid of [math] sin [/math].

[Anonymous No.8979144]

>>8979055
While trying to investigate problem, I got extra 1 which broke equality. But I'm not sure I'm right.

[Anonymous No.8979153]

>>8976710
>can two spaces not intersect?
huh?

[Anonymous No.8979161]

>>8979144
This equality [math] ad - bc = \pm 1 [/math]?

[Anonymous No.8979171]

>>8979161
No. After sex with fractions, I got cb + 1 + ef = cb + ef.
Honestly, I am totally confused with this problem.

[Anonymous No.8979197]

>>8979171
>sex with fractions

(anon, ass-ume there exist [math] e,\ f \in \mathbb{N} [/math] such that [math] \frac{a}{b} < \frac{e}{f} < \frac{c}{d} [/math]. For this to happen certain things must hold, which will contradict your given facts about the numbers.)

[Anonymous No.8979214]

Guys, I've been working on this problem for six hours a day all week with no progress, and my math professor wants me to turn it in by tomorrow.

What is 23 times 91?

pls halp

[Anonymous No.8979232]

>>8979076
cos*

[Anonymous No.8979252]

>>8979076
i fucked up, it's

[math]\int x cos^7(x^2) sin(x^2)[/math]

i don't get how you'd turn the product into a sum though

[Anonymous No.8979259]

>>8978761
This is not even close to being near immediate. Type "Indefinite Integral of (cos(x^2))^7 * sin(x^2) " in wolframalpha and you will see.

Maybe you forgot to type an x? Like x * (cos(x^2))^7 * sin(x^2) ? Cause then yeah, it is obvious.

[Anonymous No.8979264]

>>8979252
What is the derivative of ( cos(x^2) )^8 ?

[Anonymous No.8979270]

>>8979252
[math] cos\ u\ sin\ v = \frac{1}{2}(sin(u+v)-sin(u-v)) [/math]

[Anonymous No.8979278]

>>8979270
This is only for
>i don't get how you'd turn the product into a sum though
the integral is obvious >>8979264

[Anonymous No.8979282]

>>8974800
Ring of length 3 is too trivial, I assume?
1+1 mod 3 = 2
11 mod 3 = 2

[Anonymous No.8979288]

>>8979252
Set [math]f(x)=x^2, g(x)=\cos(x), h(x)=x^8[/math]. What you have now is the integral of [math]-\frac{1}{16}f'(x)h'(g(f(x)))g'(f(x))[/math], or something like that. That should help you.

[Anonymous No.8979294]

>>8979282
It's the second reply in the thread already. There are more interesting ways to approach this. I'll post my solution at 300 replies, just in case someone else comes up with similar constructions in the meantime.

[Anonymous No.8979312]

>>8979264
I don't understand how you moved the ^7 around

[Anonymous No.8979315]

>>8979197
Sorry, I don't have a clue how should I come to absurd.
What is the link between [math]ad - bc = ±1[/math] and [math]f < b + d[/math]?
Every my try to prove is dull and meaningless, I am sinking in symbols.
%%Maybe I am just epic brainlet and should stop doing math and continue play MOBA with the same retards as me.%%

[Anonymous No.8979320]

>>8975014

Are you actually suggesting taking math exams while drunk?

That seems colossally stupid

[Anonymous No.8979332]

>>8979312
Solving an "indefinite integral" means to find an antiderivative of the thing inside the "integral".

Well, what is the derivative of (cos(x^2))^8?
It is 8 * (cos(x^2))^7 * (-sin(x^2)) * (2x)
which is equal to
-16 x (cos(x^2))^7 sin(x^2)

Thus, the derivative of -1/16 (cos(x^2))^8 is equal to x (cos(x^2))^7 sin(x^2) which is the thing in the integral. Therefore it is an antiderivative. Since all the antiderivatives differ by just a constant, we have that all the antiderivatives of x (cos(x^2))^7 sin(x^2) are of the form:

-1/16 (cos(x^2))^8 +c , where c is some constant

[Anonymous No.8979382]

>>8978761
>>8979252
https://en.wikipedia.org/wiki/Integration_by_reduction_formulae

[Anonymous No.8979634]

How would you rate this textbook?
Would you replace it with something else?
What would you go onto next after this/the one you reccomended?

[Anonymous No.8979644]

>>8979634
Best introductory book I have ever read. In general, not just algebra.
It is a book which you will have to solve a lot of its exercises though, cause it has many important parts of the theory in them.
It takes a lot of time to read it, but they guy just knows how to teach.

[Anonymous No.8979646]

>>8978888
714285, this string of # shows up all the time when I divide stuff by 7 ?
7:4 this ratio on a grid makes a lot of cool shapes.

[Anonymous No.8979661]

>>8979644
thats great to hear
and my summer reading sorted

[Anonymous No.8979690]

>>8979634
A book for brainlets.

[Anonymous No.8979715]

How does one go about learning from material that doesn't have any problem sets? I'm reaching the point where I can no longer always just find an equivalent book that has them.
I feel like without working a bunch of exercises my understanding of whatever I read is extremely shaky.

[Anonymous No.8979734]

>>8979690
perfect for me then

[Anonymous No.8979738]

>>8978571
>filename
That's not Yukari you fuck.

[Anonymous No.8979739]

>>8979634
>Not reading Bourbaki algebra
Come on anon, you gotta go with the classics. Honestly though I've always felt D&F was suitable for a first course (at least the first couple of chapters) tons of exercises and examples, not abstract in the slightest, I really don't see how this couldn't be someones first foray into algebra, it doesn't assume any prerequisite knowledge (besides the ability to do proofs)

[Anonymous No.8979851]

Redpill me on the Stone-Céch compactification

[Anonymous No.8980264]

>>8979320
Tipsy, not drunk.

[Anonymous No.8980266]

>>8979851
something something universal property something

[Anonymous No.8980270]

>>8979738
Shut up nerd.

[Anonymous No.8980271]

>>8974800
If you're ITT and not a virgin clap your hands
>*deafening silence*

[Anonymous No.8980278]

>>8980270
I'll fucking end you kiddo. Don't test me.

[Anonymous No.8980284]

>>8977768
Lang's Basic Mathematics or Axler's precalculus.

[Anonymous No.8980288]

>>8977768
If you're serious, "Mathematics: its methods, contents and meaning"

https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

[Anonymous No.8980290]

>>8980278
It seems you're not aware what mess you're in yet.
Me and my boy Chen will fuck you up.

[Anonymous No.8980428]

>>8978571
>>8979738
>>8980270
>>8980278
>>8980290
The fuck is going on here?

[Anonymous No.8980439]

>>8974800
Does anybody know about >>8980254 ?

[Anonymous No.8980534]

>>8980428
autism

[Anonymous No.8980551]

>>8980290
>yukari
>his cats

[Anonymous No.8980589]

>>8975432
>In the last set, we say that (x,y)(x,y) is equivalent to (a,b)(a,b) if an only if xa=ybxa=yb. 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.

The partition should be defined by xb =ya NOT xa = yb

[Anonymous No.8980666]

>>8980551
I think I told you to shut up, nerd.

[Anonymous No.8980716]

As it is well known, for Gaussian processes many almost sure path regularity properties can be guaranteed by relatively easy conditions, namely conditions on the regularity of the covariance structure. The proofs rely on the "good" properties of Gaussian functions such as symmetry and being Schwartz functions.

If we weaken these properties a little bit, i.e. look at non-Gaussian processes with "sufficiently good" distributions, can we still get "easily" describable conditions for almost sure path regularity? I'd be interested in references to literature regarding this question.

[Anonymous No.8980764]

>>8979715
If you've reached that point, you should be able to understand where you've gone or could have gone wrong, check if every implication in your work is correct and not just on baseless assumptions, and feel at least somewhat sure that you have it right. If you're still insecure, then you could post a question here and on stack exchange, preferably with context and your previous working. It really helps if you don't just say solve this, but instead, where you feel like you went wrong.

[Anonymous No.8980767]

>>8980271
Tfw was a virgin until I showed I had a passion for something (maths) and girls literally started flocking to me this year. Also helps that I'm foreign, attractive and non autistic

[Anonymous No.8980803]

>>8978640
Late answer, hope it helps:
Since it's either sun or rain, we have [eqn]P_\infty(sun) + P_\infty(rain) = 1[/eqn]
which together with the second to last lines is a system of linear equations. Solving the system gives the result.

[Anonymous No.8981158]

>>8979851
A Stone-Céch compactification of a topological space [math]X[/math] is a topological space [math]\beta X[/math] such that:
1) [math]\beta X[/math] is compact and Hausdorff.
2) there is an associated continuous map [math]i\colon X\to\beta X[/math].
3) if [math]C[/math] is a compact Hausdorff space and [math]f\colon X\to C[/math] is continuous, then there is a unique continuous map [math]\beta f\colon\beta X\to C[/math] such that [math]f=\beta f\circ i[/math].

Assuming the axiom of choice, one can prove there is a Stone-Céch compactification for every topological space. You can sort of think of these like you would think about injective envelopes of modules or abelian groups: You have, for every space with any properties, a space with nice properties such as a shitload of separation axioms etc. A nice thing to note is that if [math]X[/math] is Tychonoff, then the associated map gives a homeomorphism between the space and its image, and [math]X[/math] is a dense subspace of [math]\beta X[/math]. Similarly, if [math]X[/math] is locally connected, then the associated map gives a homeomorphism between [math]X[/math] and an open subspace of [math]\beta X[/math].

[Anonymous No.8981160]

>>8981158

Moreover, this gives a functor [math]\beta\colon\textbf{Top}\to\textbf{CHaus}[/math]: For any continuous map [math]f\colon X\to Y[/math], then [math]\beta(f)\colon\beta X\to\beta Y[/math] is the unique map such that [math]j\circ f=\beta(f)\circ i[/math], where [math]j\colon Y\to\beta Y[/math] is the associated map of [math]Y[/math]. It follows that:
1) [math]\beta(1_X)\circ i=i\circ 1_X=i[/math], and, by uniqueness, [math]\beta(1_X)=1_{\beta X}[/math].
2) Let [math]f\colon X\to Y, g\colon Y\to Z[/math] be continuous. Then, if the associated maps are [math]i_X, i_Y, i_Z[/math], we have [math]i_Y\circ f=\beta(f)\circ i_X, i_Z\circ g=\beta(g)\circ i_Y[/math], and this gives us [math]i_Z\circ g\circ f=\beta(g)\circ i_Y\circ f=\beta(g)\circ\beta(f)\circ i_X[/math], and again, by uniqueness, [math]\beta(g\circ f)=\beta(g)\circ\beta(f)[/math].
Now, let [math]I\colon\textbf{CHaus}\to\textbf{Top}[/math] be the inclusion functor. Since the maps [math]\beta(f)[/math] are unique, there is a bijection [math]\text{Hom}(\beta X, C)\to\text{Hom}(X, IC)[/math], that is, [math]\beta[/math] is a left adjoint of [math]I[/math]. Adjoints preserve various things, so you should check what kind of stuff is preserved by a left adjoint.

>mfw i hit the character limit

[Anonymous No.8981188]

>>8981160
>unique map
>unique
henlo where arede broofs :DDDDD

[Anonymous No.8981211]

>>8981188
to intelegent to proofify

[Anonymous No.8981223]

>>8981211
fuuuuggggg :DDDDD

[Anonymous No.8981230]

>>8981223
:DDDDDDDDD

[Anonymous No.8981309]

>>8981158
>locally connected
Locally compact... Typos, typos, always typos.

[Anonymous No.8981489]

>>8974820
pourquoi est-ce qu'il y a des chapeaux sur ces numéros

[Anonymous No.8981495]

>>8981489
Equivalence classes.
C'est comme une partition de nombres selon l'arithmétique modulaire

[Anonymous No.8981941]

What is /mg/ reading?

[Anonymous No.8981947]

>>8981941
Spice and Wolf volume 14

[Anonymous No.8982030]

>>8981941
Spice and Wolf volume 2

[Anonymous No.8982031]

>>8981947
>>8982030

[Anonymous No.8982054]

>>8982031

[Anonymous No.8982065]

>>8977112
>Anon's proof has an error unfortunately.
>>AB-A+A-BA=A(B-I)+A(I-B)
what's the error? that's just distributivity in the ring of matrices

[Anonymous No.8982077]

There's nothing more pleasing than seeing 2nd and 3rd year undergrad Maths students realize that they'll never catch up with the geniuses at their university, let alone the geniuses in their fields, and change their major. They might get a little cocky by getting As, but even in the classes that they aced, professors realize that nothing else matters but sheer talent, and they'll be refused truly great letters of rec and research positions, which are reserved for the golden geese. But still, they'll go to office hours and try to speak with professors, before eventually being shooed away when the genetically superior student shows up to the office and says a simple "hello", immediately brightening the professor's day after being bombarded by persistent brainlets. They might even keep coping and graduate, and eventually go to a sub top-five for grad school, ultimately wasting their time.

I wonder which of you will fall.

[Anonymous No.8982114]

>>8981158
>>8979851
I have books to give away on topology through frames but I do not know what to do with hem since nobody wants them

[Anonymous No.8982124]

>>8982114
>give them away to someone on /sci/
>give to local university

[Anonymous No.8982133]

>>8978888
Help please!

[Anonymous No.8982136]

>>8982114
Does your uni have them in their library? If not, give them to those people.

[Anonymous No.8982165]

>>8982077
>projecting on /sci/

[Anonymous No.8982193]

https://en.wikipedia.org/wiki/Guess_2/3_of_the_average

>In game theory, "guess 2/3 of the average" is a game where several people guess what 2/3 of the average of their guesses will be

> if choices are restricted to, for example, the integers between 0 and 100... it becomes advantageous to select 0 if one expects that at least 1/4 of all players will do so, and 1 otherwise.

[Anonymous No.8982212]

>>8982124
Universities must die.

[Anonymous No.8982214]

>>8982193
>my ban expired

[Anonymous No.8982222]

>>8982214
t'was fun, I watched War and Peace

[Anonymous No.8982259]

hey /mg/. currently waiting my graduation ceremony to start, just finished bachelor's in math. starting master's in applied math this fall. recommended summer reading? i was thinking about doing my master's thesis in functional analysis.

[Anonymous No.8982285]

Hey mathfags. Sorry to shit up this thread. The furthest I've gotten in math is differential equations and linear algebra "for engineers."

Is "real" math actually as cool as it seems? I think geometry might be kinda interesting, but... a lot of textbooks in other topics seem dry. I get bored trying to learn real analysis, for example.

[Anonymous No.8982311]

>>8982285
Patience. The beginning is dry and technical, always. Eventually, the boring stuff turns into cool stuff. Although, there are pretty dank proof tricks on the first pages, occassionally.

[Anonymous No.8982519]

>>8981941

[Anonymous No.8982532]

>>8980428
Children playing around with their cartoon images

[Anonymous No.8982536]

Can someone explain to a dunce how you represent rotations with a quaternion and why it's so efficient on conputers?

[Anonymous No.8982553]

>>8982536
Rotations SO(3) have a universal cover of SU(2), which means that they both have the same Lie algebra and therefore the same linear (read: matrix) representation. However the algebra of su(2) is isomorphic to the Clifford algebra [math]C_{0,1}^+[/math] generated by its positive definite basis, and [math]C_{0,1}[/math] is isomorphic to the quarternions.

[Anonymous No.8982569]

>>8982553
I was hoping for an explanation without abstract algebra. I want to understand why they're numerically efficient.

[Anonymous No.8982582]

>>8982569
Either you define the algebra and then do rotations with quaternions in [math]\mathcal{O}(N)[/math] or define su(2) basis and do rotations with matrices in [math]\mathcal{O}(3^N)[/math].

[Anonymous No.8982626]

So, can you do it?
>>8982618
>prove 2 is prime without appealing to the principle of the excluded middle

[Anonymous No.8982686]

>>8982626
The only positive integers less than or equal to 2 are 1 and 2, both of which divide 2. Moreover, if [math]n>2[/math] is an integer, there is no integer [math]k[/math] such that [math]2=kn[/math], so 1 and 2 are the only positive integers dividing 2.

No LEM, only that [math]a>c, b >0 \Rightarrow c<ab[/math] for all integers.

[Anonymous No.8982689]

>>8982686
That proves that 2 is irreducible.
Prove it is prime.

[Anonymous No.8982692]

>>8982689
>doesn't know the definition of prime
?? what board is this

[Anonymous No.8982700]

>>8982553
>>8982582
>Can someone explain to a dunce
Right. This all will make perfect sense to a dunce.

Mathematicians are autistic.

[Anonymous No.8982701]

>what are you studying?
Number Theory, Real Analysis I
>any cool problems?
The Collatz Conjecture is my obsession right now.
>any cool theorems or remarks?
The creation of the real set using infinite sequencies of cauchy is very sophisticated.
>reference suggestions?
No.

[Anonymous No.8982703]

>>8982689
How do you define a prime number? An integer [math]p>1[/math] is prime if it is divisible only by [math]\pm1, \pm p[/math], no?

[Anonymous No.8982705]

>>8982692
Yeah, I'm surprised too. 99.99% of /sci/ doesn't know that an irreducible element is not the same thing as a prime element.

[Anonymous No.8982712]

>>8982705
Do you not know prime and irreducible are the same for integers...?

[Anonymous No.8982717]

>>8982703
Let [math] p\in\mathbb{Z}\setminus\{0, \pm 1\} [/math]. We say that [math] p [/math] is prime [math] iff [/math] whenever [math] p [/math] divides a product [math] ab [/math] it divides at least one of [math] a [/math] and [math] b [/math].

>>8982712
Yes, the two sets (primes and irreducible integers) coincide. And?

[Anonymous No.8982729]

>>8982701
>The Collatz Conjecture is my obsession right now.
Everyone who wanted to prove it already proved it and moved on.

[Anonymous No.8982747]

>>8982569
>I want to understand why they're numerically efficient.
Because 4 < 9.

[Anonymous No.8982749]

>>8982717
>Yes, the two sets (primes and irreducible integers) coincide. And?
It means an integer is irreducible iff it is a prime. What you gave is the definition of a general prime element, and in the case of integers all irreducible elements are prime elements.

[Anonymous No.8982763]

>>8982749
So you don't know how to prove that 2 is prime without using the principle of the excluded middle. Got it.

[Anonymous No.8982781]

>>8982763
If that makes you happy.

[Anonymous No.8982790]

>>8982781
I'm not happy, I'm disappointed.

[Anonymous No.8982797]

>>8982790
So am I. At you.

[Anonymous No.8982799]

>>8982797
Why is that?

[Anonymous No.8982805]

>>8982799
That is left as an exercise.

[Anonymous No.8982816]

>>8982700
>retard rationalizing his retardedness
If you understand the second post you shouldn't be on this board at all.

[Anonymous No.8982820]

>>8982797
Why, I only want what's best for you. It's not enough to know that in [math] \mathbb{Z} [/math] every irreducible element is prime. Any brainlet can memorise trivia like that.

[Anonymous No.8982834]

>>8982816
>If you understand the second post you shouldn't be on this board at all.
why are you here then?

[Anonymous No.8982836]

>>8982834
Oops it should be "don't understand". Thanks m8

[Anonymous No.8982845]

>>8982820
Then you could probably tell me why I should make a distinction between irreducible and prime elements in [math]\mathbb{Z}[/math]. If we pick some arbitrary ring, then they may be two different sets, but not in this case. So, please do tell me why, and accompany your explanation with proof that you are not this >>8982689 poster. Otherwise I will waste not a single minute more on you.

[namefag No.8982847]

>>8982536
>>8982569
>>8982700
Firstly, I will assume you know about the quaternions, though if you do not, we write quaternions in the form of [math]q = a + bi + cj + dk[/math] where the a,b,c,d are real numbers, and i,j,k follow the rule that [math]i^2 = j^2 = ijk = -1[/math]. I will write the conjugate of q as [math]\bar q = a-bi-cj-dk[/math], and we define a "norm" (or length) of q as [math]\|q\|=\sqrt{q\bar q}[/math].

Consider a vector (x,y,z) in 3D space: we will write this as [math]xi+yj+zk[/math], something you might have seen in highschool or university already. That is, we view a quaternion as a real number and a vector in R^3---this is the perfect number of information to consider a rotation of an angle around an axis in 3D space! But how exactly do we carry out this calculation with the quaternions?

The process is simple. Pick an axis in R^3 to rotate around, call it [math]u[/math] and make sure it is a unit vector. Pick an angle [math]\theta[/math] as the angle we rotate everything around u. Define a quaternion [math]r = \cos(\theta/2) + u\sin(\theta/2)[/math]. To rotate a vector v, around the rotational axis u by [math]\theta[/math], we calculate the vector [math]rvr^{-1}[/math] where $r^{-1} = \frac{\bar r}{\|r\|^2}$. This is the rotation of v about u by the angle [math]\theta[/math]!

Why does this work? Well if you want a proof, I can do that, but really it just amounts to showing that this really is the rotation you know in a different sense. Just ask if you want a proof.

Why is this more efficient than other methods? Well the "other methods" usually is using rotation matrices. Keep in mind that these are 3x3 matrices, meaning that each rotation matrix is up to 9 values. Recall quaternions are 4 always. This is the bare minimum really, and so we see that quaternions can be more efficient than matrices in this sense. Also the operations in matrix multiplication exceed that in quaternion conjugation.

[namefag No.8982851]

>>8982847
Sorry, I forgot to use a math tag:
[math]r^{-1} = \frac{\bar r}{\|r\|^2}[/math].

Again, feel free to ask for more detail if you need it!

[Anonymous No.8982878]

>>8982717
To be fair, >>8982703 is right.
A prime number is an integer p such that p>1 and d|p means d=1 or d=p. This is a perfectly fine definition. You provided the definition of a prime element in Z, where these two definitions coincide.

[Anonymous No.8982884]

>>8982878
correction, this is nonnegative integers p and d.

[Anonymous No.8982886]

>>8979171
If you assumed that there exists natural numbers e and f such that f < b+d and a/b<e/f<c/d when |ad - bc| = 1 then got the equality cb + 1 + ef = cb + ef which implies that 1=0 then you're done since you found your contradiction.

[Anonymous No.8982900]

>>8982878

[Anonymous No.8982910]

>>8982900
Nice hat

[Anonymous No.8982917]

>>8982910
You too, monkey boy.

[Anonymous No.8982924]

>>8982626
>>8982686
>>8982689
>>8982692
>>8982703
>>8982705
>>8982712
>>8982717
>>8982749
>>8982763
>>8982781
>>8982790
>>8982797
>>8982799
>>8982805
>>8982820
>>8982845
>>8982878
>>8982884
>>8982900
>>8982910
>>8982917
>this is the state of /math/

[Anonymous No.8982943]

>>8982924
Why is she so sad?

[Anonymous No.8982948]

>>8982845
Bitch, I didn't link that post because it asked something about prime numbers. I linked it because it asked for an interesting direct proof that 2 is prime without using the principle of the excluded middle. Making it a question about irreducible numbers is just sidestepping the problem entirely.

[Anonymous No.8982966]

>>8982948
If someone asks a proof that 2 is prime without an explicit reference to rings, it is reasonable to assume that person is talking about prime numbers.

[Anonymous No.8982995]

>>8982966
The primality of 2 is not the point. Neither is the fact that [math] \mathbb{Z} [/math] is a greatest common divisor domain.

[Anonymous No.8983192]

>>8982995
... still no takers? Here's a variant:

Let [math] a,\ b \in {\mathbb{Z}}* [/math] and suppose [math] 2 \mid ab [/math].
Suppose [math] 2 \nmid a [/math]. Then, either [math] a=\pm 1 [/math] (in which case we are done) or, by Bézout's identity (working Euclid's algorithm backwards) there [math] \exists x,\ y\in \mathbb{Z} [/math] such that [math] 1 = 2x + ay [/math], whence [math] b = 2bx + bay [/math], baby.
Similarly, if [math] 2 \nmid b [/math] then [math] 2 \mid a [/math], necessarily.
Therefore 2 is prime.

This brings the count to 2 intuitionist proofs that 2 is prime >>8982644
Who is nerd enough to be the third loser to post one? (Irreducible numbers brainlets need not apply.)

[Anonymous No.8983224]

Hey /mg/. I just was gifted some money. I'm looking to spend some of it on math texts. Recommendations?
Interested in numerics, functional analysis, abstract & linear algebra. Maybe some topology too?

[Anonymous No.8983232]

>>8983192
>intuitionist proofs
constructive proofs*

[Anonymous No.8983234]

>>8983224
download books first to see if they're worth buying

[Anonymous No.8983239]

>>8983234
sure, but I am still looking for recommendations.

[Anonymous No.8983263]

>>8983224
Conway, Rudin, Reed-Simon, Simon are all good for functional analysis
D&F, Bourbaki, Lang, Artin, Jacobson, Knapp, Rowen for algebra
Munkres, Fomenko, Janich, Brendon for topology
At the end of the day it comes down to taste and what you'll be applying these to but all of these books will get the job done

[Anonymous No.8983272]

How do I draw marks on the circumference of a circle in 5 degree increments. Diameter is 20 inches. Haven't done much geometry in 5 years so am rusty.

[Anonymous No.8983292]

>>8983272
1) Find a way to draw a 72 sided polygon inscribed in a circle. (Maybe difficult.)
2) Done.

[Anonymous No.8983300]

>>8982763
>principle of the excluded middle
Where did he use it? Or are you saying he can't use constructively provable special cases of it?

[Anonymous No.8983314]

>>8983192
There was already a constructive proof here: >>8982686

[Anonymous No.8983342]

>>8983272
I would need to find b correct? Then I could just slide a square up and down the a radius until it intersects with the perimeter of the circle and mark it accordingly. Then straight edge that point to the center and do the same thing all the way around the circle.
But what is it that I must do to find b? I figure it has something having to do with sin but I am drawing a blank.
Would it be sin(5)= b/10?

[Anonymous No.8983347]

>>8983342
tried me sin of 5 = and obviously isn't it.

[Anonymous No.8983350]

>>8983347
Make sure you're doing 5 degrees and not 5 radians.

[Anonymous No.8983367]

>>8982729

Show me a proof then, I'm interested :^)

[Anonymous No.8983379]

>>8983367
>:^)
Why would I be helping someone of your ilk?

[Anonymous No.8983384]

>>8983350
So was I correct on my equation? I did have my calculator set to radians.

The radius is 10square root of 2". So sin(5)=.087 so .087x10 square root of 2= 1.23" correct? Mark it from radius a and straight edge that to the center , mark off of that and repeat around the circle?

[Anonymous No.8983871]

>>8983300
>>8983314
That's a proof that 2 is an irreducible number.
>muh all irreducible elements are prime in GCD domains

[Anonymous No.8984113]

>>8983192
>Who is nerd enough to be the third loser to post one?
this >>8983502 is by far the most elegant proof yet.

[Anonymous No.8984579]

/mg/, I have a question. Recently I have been studying a function that I created, by taking a process used to define a really famous function but generalizing it to see what happened. Naturally there were a lot of trivial things. This generalized function kept some properties of the original function and lost some others. Nothing surprising there. But then I started using this function and constructing functions from it, and I found a function that has a really interesting property. And I say interesting not because it is important or maybe useful, I say it because I have never seen functions with that property before. And when I was googling to see what was out there, I could not find anywhere detailing functions with that property.

So I may, MAY have found something new. And I want to study it and maybe publish something (I am not a grad student but I am willing to do it on my free time). But I have a question to the people in this thread who have published before.

What is the minimum you would consider publishable? And by that I mean, what should I first accomplish before knowing that I have enough content to start writing and then publishing. By the way, I am not talking fancy here. I just want to post it on arxiv for fun, then shill it here for some reviews, and then show it to my professors if you people tell me it is not complete shit.

This is what I have at this moment. A first example of a function with this property, with a method to construct it (which is fun because it uses a famous function).

And I can explore other variants of my method and prove that the same method with other constraints also yields functions with the same property, or if that property is unique to that constraint.

I can do more, but how deep should I go before writing a paper?

Note that this will be a toy paper. This function and if there are others, this class of function, are probably completely useless because the property is very specific.

[Anonymous No.8984658]

>>8983224
Why don't you try one subject first before you attempt any other? They way you've written it sounds like you just heard cool things and want to try it.

If you are mathematically mature (if not, book of proof or some other like it, then some advanced calculus textbook, maybe Spivak Calculus), then for Linear Algebra: Shilov if you wanna cheap out (Dover), Strang for more computational side or Axler if you want to learn the theory foremost(latter is recommended). For algebra, try Dummit and Foote or Artin, or if you want a light and interesting read with lots of exercises (with a view towards codes), then Pinter, but the latter is not enough for rigorous Algebra.

After that try Munkres for Topology, then maybe Rudin or Tao for Analysis

[Anonymous No.8984691]

>>8983224
Algebra: Aluffi's Algebra. Chapter 0
Topology: Frames and Locales: Topology without points by Picado, Paltr
If you like good memes

[Anonymous No.8984718]

So I'm going to university soon and I'm somewhat gifted in mathematics (not by /sci/'s standards but by normal people's), so I've been thinking about how to nourish it properly so I can maybe do stuff with it in the future.

In particular, while I've got a pretty good conceptual basis (I just finished a multivariable calculus class in my high school and I seem to have a more solid understanding of it than most of the people in that class), I think I still need to develop a more quick and clever approach to doing math, like how people who do math olympiads have it. Where have you guys found a good place to get problems that aren't complicated conceptually, but mostly just require really clever thinking?

[Anonymous No.8984808]

>>8979214
2093.
Speaking of which, I found a really simple shortcut for multiplying two-digit numbers. You round both of them to the closest multiple of ten, add each error multiplied by the other rounded number, then add the product of the errors. For example, [math] 23 + 91 = (20 + 3)(90 + 1) = 1800 + 270 + 20 + 3 = 2093 [/math]. It's just distribution but it's way different then how I usually multiply and way simpler for mental math.

Speaking of which, what arithmetic shortcuts do you guys do to be able to do arithmetically complicated things in your head?

[Anonymous No.8984814]

>>8974894
divi...

[Anonymous No.8984818]

>>8984718
If you do not feel comfortable with higher level math yet then do Olympiad problems. Remember, there is no shame even if they are for high school kids. There are many problems not even PhDs can solve. Olympiad problems are general and beautiful.

If you feel comfortable with higher level math then find undergraduate textbooks from authors who put top tier problems in their books. A good example is Tom Apostol, He is a motherfucker who will make you depressed and reconsider doing math, but that's the point. Obviously I am talking about his analysis textbooks here. I haven't read his calculus books but calculus is high school so it doesn't matter.

[Anonymous No.8984875]

>>8984818
>olympiad problems are beautiful
Out of curiosity, what does it mean that a problem is beautiful?

[Anonymous No.8984891]

>>8984875
I don't know if you've written proofs or not but when you get into uni you will and you will see there are two types of proofs

Type 1: Dull proofs. In dull proofs you just apply some definitions to prove really obvious theorems. Easy example: Prove that the 0-element of a group is unique. You just apply the definition. You do not need any kind of discovery. You just need to apply definitions and be good at juggling logic. Of course, juggling logic is important in mathematics, but logic juggling is not the only thing there is.

Type 2: Discovery proofs. In discovery proofs it is not enough to apply definition and previous theorems. In a discovery proofs you will have to do some kind of discovery, which will usually come about because of a unique and creative construction that makes a certain property shine.

Beautiful problems are those that need a big discovery. And I get it also helps that the problem is not tedious (olympiad problems about inequalities are sometimes very fucking tedious, not beautiful). Proofs that come about because of a clever idea or construction are the best. A good example is Gauss' proof for the value of the Gaussian integral. That shit is fucking majestic. Like holy fuck, amazing. 10/10 math right there. And really simple too. If you did multival calc you probably already saw a proof or something close to an argument for the Gaussian integral. Majestic isn't it?

[Anonymous No.8984926]

Lads, am I truly, honestly fucked when it comes to getting into top 20 grad schools for Maths?

I've got awesome letters of rec, done research since late freshman year, and have a 3.97 GPA. My general GRE is V: 167, M: 170, and I'm gonna be taking the Math Subject GRE, and feeling pretty good.

Here's the kicker, though. I go to fucking UC Riverside. I know I'm screwed when it comes to schools like Berkeley and UCLA, but do I have decent chances for some mid-tier UCs, like Irvine, and some similarly ranked schools? Will adcoms cringe when they see my undergrad institution?

[Anonymous No.8984955]

>>8984818
Thanks, bro!
>>8984875
For me, one element of it is, does that problem have a very easy, elegant solution that it is very tricky to think of, an requires a lot of cleverness? For example, I remember I did this one problem: if you have three numbers a, b, c, randomly chosen between 0 and 1, what is the probability that they form the side lengths of a triangle?

This problem isn't very difficult, and there are a few solutions, but I remember reading one that completely blew me away. WLOG, assume c is the largest side. Divide each side by c and you three side lengths a/c, b/c, and 1. Assign x = a/c and y = b/c and graph the set of all possible x and y (shitty diagram related). It forms a square, since x and y can't be larger than 1 (since a and b < c). Finally, since the inequality for side lengths is a + b < c, just draw the line x + y = 1. Everything below satisfies the inequality and everything above doesn't, so you've split the set equally into can and can't form a triangle, therefore, the probability is 0.5

That solution is so marvelously simple and extremely easy, yet it's not obvious, which is what makes it beautiful.
This guy >>8984891 raises some good points too.

[Anonymous No.8984964]

1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = i * i = -1

Debate this

[Anonymous No.8984995]

>>8984891
I see, I see. Thanks for an inspiring answer. I can see your love for math in your text, in your word choices, and in the investment to write all this.

Another example of a type 1 proof would be the uniqueness, say, products up to isomorphism, and stuff like that. One of my all time favourites is how the Hopf fibration induces the long exact sequence
[math]\cdots\to\pi_{n+1}(S^2)\to\pi_n(S^1)\to\pi_n(S^3)\to\pi_n(S^2)\to\cdots[/math].

>>8984955
Nice one. I'd probably not even know how to begin in the first place.

[Anonymous No.8985403]

>>8984964
sqrt(a^2)=+-a

[Anonymous No.8985786]

>>8984579
>What is the minimum you would consider publishable?
No such thing.

[Anonymous No.8985791]

>>8984818
Olympiad problems are awful. Olympiads suck.
Those sort of problems are usually a calculator grind fest with a couple "non-obvious" steps.

[Anonymous No.8986092]

Would it be worth the time for a mech eng student to learn analysis?

[Anonymous No.8986112]

>>8986092
What counts as "worth the time"? If you like learning cool things, then yeah. If you dislike learning things you will probably not utilize frequently in your job, probably not.

[Anonymous No.8986294]

>>8984818
Sorry for the late response, but this just came to me this morning. When you say Olympiad problems, do you mean stuff like the AMC, which a large portion of high schoolers take and do well on, or do you mean the higher level high school olympiad problems, like those given at Olympiads on the international level, like IMO and stuff? Obviously I'm going to search out the one that's closest to my level, but I'd like to know which was in your head when you wrote this post.

[Anonymous No.8986387]

>>8985791
>Those sort of problems are usually a calculator grind fest with a couple "non-obvious" steps.

You don't even need a calculator for olympiad problems.

>>8986294
I do mean things like IMO. But less famous stuff too like the national olympiads from countries like america and Russia, and if you look at continental competitions from all around the world you will also find good problems.

[Anonymous No.8986699]

Which book covering needed knowledge in proofs is the best for mere brainlet?

[Anonymous No.8986701]

>>8986699

[Anonymous No.8986752]

>>8986699
http://www.sciencedirect.com/science/book/9780444508133

[Anonymous No.8986886]

Does anyone know of a chain rule generalized to linear operators other than differentiation/the Laplacian? It seems like the Cole-Hopf transformation used on the Burgers equations could be generalized to analyze other nonlinear equations.

[Anonymous No.8986905]

You will probably call me shit but how do I solve this one? I can't think of any solution for nearly an hour

[Anonymous No.8986924]

>>8986905
I looked at that and in two seconds know that you just split the denominator up into something like A/x + (bx^2+cx+d)/x(x^2+2)

Or whatever, it's been years since calc II but you will have a 1/x which integrates to lnx then a 1/x(x^2+2) which is a u substitution.

[Anonymous No.8986928]

>>8986905
[math]\frac{1}{x^2(x^2+2)}=\frac{1}{2x^2}-\frac{1}{2(x^2+2)}[/math]

[Anonymous No.8986947]

>>8986924
if this were true then you would have Ax^2 + 2A - bx^2 - cx -d= 1/x, which is impossible

[Anonymous No.8986949]

>>8986947
>something like
just use partial fractions, it's not hard, look up the theorem

[Anonymous No.8986953]

>>8986949
i agree that partial fractions was the answer, as shown by >>8986928. the thing to learn from this is that if you want to do partial fractions, then the degree of the numerator minus the degree of the denominator must match between each term in order for the decomposition to make sense algebraically.

[Anonymous No.8987565]

Bump.

[Anonymous No.8987586]

>>8984926

I don't understand

All the UCs are fine schools except for maybe Santa Cruz

Why would they cringe at a pretty good app? Did you write a good thesis or do any undergrad research? Did you take grad classes?

[Anonymous No.8987587]

>>8987565
Can you describe what is on you beautiful picture, please? I want to google it, but I cant.

[Anonymous No.8987604]

>>8987587
>I want to google it, but I cant.
What prevents you from googling it?

[Anonymous No.8987616]

What do you lads think about alcohol and math?

Sometimes I get really nervous during a real analysis exam and it makes me tense up so much that I have trouble recalling relevant information in order to prove something.

Saw somewhere else on /sci/ to try a little alcohol during exams since it lowers inhibitions, but I am not too sure. It seems like a good way to fuck up when you're logic juggling during an exam question.

Does anyone have any anecdotes?

[Anonymous No.8987647]

>>8984818
>There are many problems not even PhDs can solve.
This isn't because they're hard questions, or even good questions.
It's because olympiad questions require a highly specialized skillset very distinct from mainstream math, and you're fucked trying to do much without that training no matter how bright you are.

The number of "general and beautiful" olympiad problems are outnumbered 20/1 by the "how can I rearrange this page-wide fucker into one of the 17 forms of AM-GM I practiced?" problems.

[Anonymous No.8987659]

>>8987616
You are not trying to get wasted, but you are trying to get the warm and relaxed feeling you get after a beer or two. Basically, just drink a bottle or two of some proper beer (that is, from Europe or Asia) before entering the exam hall. I haven't done exactly this, but I have studied like that, and it felt a bit better and easier than without beer.

[Anonymous No.8987781]

>>8987616
>>8987659
Depending on your body size, a bottle of beer or two would be too much (how much is a bottle anyway? 500ml? 350ml?). Eat a couple of those chocolates filled with whiskey before the exam. It also has the benefit of a lot of easy calories from the chocolate.

[Anonymous No.8987785]

>>8986387
By calculator grind fest I mean that most olympiad problems actually require repeated applications of pretty standard heuristics, not that they require a lot of number crunching.

[Anonymous No.8987809]

There was this group of mathematicians (I forget their names, Russians) who used to gather at bars and sketch theorem proofs over vodka and fill out the details sober. Also, my PDEs professor in 3rd year was visibly drunk at almost every other lecture and seminar. He still strolled through the material like it was nothing. Although, to be honest. he probably drank because he was bored out of his mind. He was an old guy, teaching since the 1970s.

Also, there's stuff like this: http://www.sciencedirect.com/science/article/pii/S1053810012000037
Take it with a grain of salt though, the study has a low number of participants in the experiment. I've found a couple other similar studies, but they also have a low n count.

[Anonymous No.8987831]

>>8987809
Russians liking vodka is a stereotype for a reason, I studied there for a semester and in the school's fridge next to the coffee cream sat an open, half-finished 1.75 litre bottle

[Anonymous No.8987880]

>>8987831
Russian liking beer is absolutely truth, I say as a russian. Vodka is much lesser fav, but also widely used.

[Anonymous No.8987882]

>>8987880
>tfw can't buy cheap Бaлтикa anymore

[Anonymous No.8987884]

Should I buy Mathematica lads?
Is it useful for higer maths?

[Anonymous No.8987888]

>>8987884
>paying for digital things in the year 2000+10+7

[Anonymous No.8987891]

>>8987882
Loool, do you like it? In my location, It's almost most cheap beer in a shop. Have you ever tried Бaлтикa 9 lol?

[Anonymous No.8987896]

>>8987888
I couldn't find any crack to linux version

[Anonymous No.8987898]

>>8979252

Substitute x^2 to u. du = 2x dx

Substitute Cos(u) to t; dt = -sin(u) du

[Anonymous No.8987901]

>>8987891
>Loool, do you like it?
very much

>Have you ever tried Бaлтикa 9 lol?
Looking at the cans I can only remember 3 and 7 for certain, I'd just buy anything off the shelf that wasn't #0

[Anonymous No.8987908]

>>8987901
9 is very hard, it is 9%, only the strongest can handle it. How much It costs near you? I can buy "7" in a nearest shop for about 0.6-0.7$.
Sorry for offtoping, but It is realy interesting for me to know.

[Anonymous No.8987917]

>>8987908
I've only seen 500ml bottles of #7 here and they go for 1.80-2.11$ depending on which province I'm in

[Anonymous No.8987918]

>>8987917
and that's in USD btw, in canadian loonies that's 2.39-2.80$

[Anonymous No.8987921]

>>8987917
Thank you a lot for a respone, mate. Have a good day.

[Anonymous No.8987943]

You faggots actually like beer? I'd drink wine exclusively but it's too expensive and spoils after you open the bottle. I want to buy a vineyard and a winery someday. Or start one myself. It's on my bucket list for the next 10 years.

Btw gorrillaposter, thank you for the confirmation. My search has narrowed down significantly. If worse comes to worst, I will just order pizzas for the whole math department at U of British Columbia. Including Brian Marcus.

[Anonymous No.8987975]

>>8987781
0.333 l x2 isn't much.

[Anonymous No.8988027]

>>8987917
>7% alcohol beer
>less than 2.5 euros
>mfw even Canada has cheaper beer than Romania
I do not understand how this could happen. Beer used to be cheap as piss only 7 years ago. If this trend continues we'l have price parity with Denmark in 5 years.

[Anonymous No.8988029]

>>8987616
>sometimes i get really nervous during a real analysis exam
just how many real analysis exams are you doing anon

[Anonymous No.8988093]

>>8987943
>You faggots actually like beer?
You've never had a good beer? Obviously there's plenty of bad beer but you don't have to look too far to find something enjoyable

[Anonymous No.8988284]

>>8987587
https://ncatlab.org/nlab/show/Chern+class

[Anonymous No.8988529]

NEW THREAD
>>8988527