/sci/ - Science & Math

Added to my replied threads.

>DO ANDROIDS PROVE THEOREMS IN THEIR
SLEEP?
https://webusers.imj-prg.fr/~michael.harris/androids.pdf

>>8988527
So i've been looking for lecture series on modern/abstract algebra

I haven't found any but a series from Harvard ... and but 1/3rd of those are seriously subpar (the prof lets some undergrad do the lecture so...)

>>8988533
:DDDDD
>>8988667
Try READING

pssh nerd

the upper bound for the moving sofa problem apparently got a nice improvement from 2sqrt(2) to 2.37 (lower bound is still 2.2195)

https://en.wikipedia.org/wiki/Moving_sofa_problem
https://arxiv.org/pdf/1706.06630.pdf

Let [math]\Sigma[/math] be a 2-surface. Poincare duality [math]H^1(\Sigma) \sim H^1(\Sigma)^*[/math] allows us to define a nondegenerate bilinear 2-form [eqn]\omega(\alpha,\beta) = -\frac{1}{8\pi^2}\int_\Sigma \operatorname{tr}(\alpha \wedge \beta)[/eqn] on the vector space [math]\mathcal{A}_{\Sigma} \ni \alpha, \beta[/math] of [math]G[/math] connections on [math]\Sigma[/math]. This makes[math]\mathcal{A}_\Sigma[/math] into a symplectic manifold.
Let the 3-manifold [math]M[/math] be such that [math]\partial M = \Sigma[/math] and let [math]\tilde{g}, A[/math] be the extension of [math]g,a[/math] to [math]M[/math] respectively, then the vector space [math]L_{\Sigma,a} \subset \operatorname{Hom}(\mathcal{G}_\Sigma,\mathbb{C})[/math] defined by the set of maps [math]f[/math] such that [math]f(eg) = c(a,g)f(e),\quad g,e \in \mathcal{G}_\Sigma[/math], where [eqn]c(a,g) = \exp\left[2\pi i \left(\frac{1}{8\pi^2}\int_\Sigma\operatorname{tr}(g^{-1}ag \wedge g^{-1}dg) - \int_M g^* \sigma\right)\right] = \exp\left[2\pi i (C_M(\tilde{g}^*A) -C_M(A)\right][/eqn] has a natural Hermitian structure.
The bundle [math]L_\Sigma = \bigcup_{a\in A_\Sigma}L_{\Sigma,a}[/math] is then a Hermitian line bundle over the symplectic space [math]\mathcal{A}_\Sigma[/math] of [math]G[/math] connections over [math]\Sigma[/math]. If [math]G[/math] is simply connected and compact, then the symplectic form [math]\omega[/math] satisfies the first integrality condition of quantization a la Dirac. This proves that the Chern-Simons theory defined by the action [eqn]C_M = \frac{1}{8\pi^2}\int_M \operatorname{tr}\left(A \wedge dA + \frac{2}{3}A\wedge A\wedge A\right)[/eqn] is a quantum field theory.

>>8988527
Physics is much too hard for physicists.

>>8988527
I'm studying calc 3 and I just learned line integrals.

When can I apply them to number theory?

>>8989012
But physicists are solving problems in math now.
https://arxiv.org/abs/1608.03679
>math is too hard for mathematicians
>physics is too hard for physicists
What does this mean?

>>8989022
Didn't that paper get torn apart on stackexchange? I can't understand the fine details but it sounds like it was very overhyped/possibly contained some errors

https://math.stackexchange.com/questions/2211278/riemann-hypothesis-is-bender-brody-m%C3%BCller-hamiltonian-a-new-line-of-attack

>In conclusion, the sloppyness of the definitions used but the authors leads to a complete mess. Nothing is correct in this paper.

>>8989028
All simply a matter of finding a more suitable Hamiltonian. Transform a math problem into a physics problem and the physicists can make up their lack of rigor with what the mathematicians lack: intuition.

>>8988875
and people say math isn't practical....

>>8989010
Thank you, I extremely like your post. If you can tell more about topics connected to physics, I will be waiting for your stories. %%It's difficult, but I'm trying hard to understand it lol.%%

>>8989068
Thanks anon. Posting here is a way for me to organize my thoughts on some interesting relationships and connections between things I've read. I'm glad people are enjoying them too.

>>8989022
>>8989028
>>8989054
Is this a joke? Have "people" not solved it yet?

>>8989019
Top kek

Looking for a recc for a subject to learn. It will be through self-study, so any texts would also be welcome.

I know the basics of algebra, analysis & topology, with a little more depth in commutative algebra and algebraic number theory.
(i.e. taken pretty much all the typical "undergrad" courses)

What should I check out next?
I'm taking an alg. geometry class next semester so I figure I may as well just wait to learn that.

Why are you making a duplicate thread?

>>8987952

I've been studying chemistry/physics for fun for quite some time now. I'm gonna become a Junior this year and try to take some actual courses.

Some statements: Should I get into organic chemistry(or is it sonething every chemist should know about)? Opinion on Biology? How can we have creativity play a bigger role in schools; Assuming only someone with high-levels of creativity could find something new in science?

Science is an important part of my life, and without it I would've ended up in a miserable situation. It just hurts that i come off as unrelatable to some others and I get less motivated to study more. How do i fix that?

Here's another extremely interesting thing.
The Chern-Simons action above can be used to define Witten's tangle operator formally via [eqn]Z_k(M) = \int_{\mathcal{A}_M/\mathcal{G}}\mathcal{D}[A]\exp\left(2\pi i k C_M(A)\right)[/eqn] as a Feynman integral. And if we define holonomy representations [math]R_i[/math] on the components [math]L_i[/math], [math]1 \leq i \leq n[/math] of the link [math]L[/math] one can define the Wilson loop operators [math]W_i(A) = \operatorname{tr}_{R_i}\operatorname{Hol}_{L_i} A[/math] such that [eqn]Z_k(M,L) = \int_{\mathcal{A}_M} \mathcal{D}[A] \prod_{i=1}^n W_i(A)\exp\left(2\pi i k C_M(A)\right).[/eqn] The exciting thing is that the quantities [eqn]\langle \psi | R_i(L_i)|\psi \rangle = \int \mathcal{D}[A] W_i(A)\exp\left(2\pi i k C_M(A)\right),[/eqn] where [math]\psi \in \mathcal{H}[/math] is a Hermitian section of [math]\Sigma[/math], actually coincides with the representation of quantum braid groups a la Nayak. In fact the moduli space [math]\mathcal{M} = \operatorname{Hom}(\pi_1(\Sigma),G)/G[/math] (obtained via the Marsden-Weinstein quotient [math]\mathcal{L} = \mathcal{A}_M // \mathcal{G}_\Sigma[/math]) can be equipped with a complex structure such that the Kahler polarization of it gives a Hermitian section [math]\mathcal{H}_\Sigma[/math] *equivalent* to the space of conformal blocks on [math]\Sigma[/math]. This indicates a connection between gauge theory and CFT that echoes my old suspicion regarding unifying strongly-coupled phenomena with category theory.
This may indicate that if I can establish an equivalence (or duality) between TQFT and CFT that I've talked about recently, then I may use this in conjunction with the equivalence of principal bundle and holonomy approaches to gauge theory to really pin down the quantum braiding found in strongly-coupled phenomena.
>>8989271
>"Intuition" is often faulty.
That's too bad for you then, sweetie.

>>8989215
K-theory.

>>8989195
What's so funny? Integrals helped me in number theory. But I google applications of line integrals in number theory and I find nothing.

Has there been no theorem linking line integrals with arithmetic functions?

Do we really need to spread this shit thin over 2 threads because some autist had to meme his anime pictures?

>>8987952

>>8988527
Are you saying mathematicians are the servants of physicists with that image?

>>8988527
>>8989583
Samefagging while ban evading?

>>8989583
Are they not?

>>8989553
Analytic Number Theory involves a lot of complex analysis and complex analysis involves saying shit follows from Cauchy's theorem every 5 seconds.

So yes, line/contour integrals are involved in number theory.

>>8989019
Analytic Number Theory

Basically Riemann Zeta Function

You made your bed, now sleep in it.

>>8988527
What ordinary mathematicians like to talk about? Aren't they obsessed by their theorems and proofs 24/7?

>>8990149
Yeah man I think their obsession with rigor has been holding them back for centuries desu

>>8988527
>>what are you studying?

psychohistory.

>any cool problems?

too numerous to count.

>any cool theorems or remarks?

A few, but nothing I'd like to share at the preliminary stage... analytical sociology is a a tad on the complex side.

>>8990149
Well unless they're talking about math, ordinary stuff usually: sports, current events, work, stuff like that. They're normal people, you know. They just happen to be good at math and tend to talk a lot about math.

>>8990114
>>8990136

So to apply them I need complex analysis? Complex anal is a junior course here so I guess next year.

Math noob here

What the hell is this and why is it in an introductory book to algebra?

>>8990510

To number theory, yes.

What kinds of things can we say about a matrix [math]PU[/math] where [math]P[/math] is a projection and [math]U[/math] is unitary, in particular about the eigenvalues [math]\{\lambda \}[/math] of [math]PU[/math]?

Clearly [math]\lambda\le 1[/math]. If [math]\lambda\ne 0[/math], then the corresponding eigenvector [math]v[/math] satisfies [math]Pv=v[/math]. Anything else?

>>8990731
What exactly are you confused by? It's just an exercise involving fractions, it certainly fits the criteria of elementary algebra

>>8990731
>What the hell is this
Looks like a fun exercise for noobs

> why is it in an introductory book to algebra
Idk, rational numbers are probably the most intuitive field to work in though.

This just seems like elementary proof writing if you ask me.

a) Suppose one can be simplified, lets say [math]\frac{a}{b}[/math] and lets say this maximum common factor is k. Then [math]
\frac{a}{b} = \frac{e}{f}[/math] where e=a/k and f=b/k, but are still integers. Then note that ed - fc will be an integer, but ed - fc = [math]\frac{ad - bc}{k} = \frac{1}{k} [/math] which isn't an integer.

Yeah, seems like your basic freshman level "apply your definitions" textbook. Makes sense, I guess. When we studied fields back in the day we did an in depth study of the rationals. Very quick though.

>What kinds of things can we say about a matrix
We can definitely say that you're a subhuman tripfag.

>>8990835
what's wrong with tripfriends?

>>8990841
Does it even make sense to ask "what's wrong with animals?"?

>>8990861
>Does it even make sense to ask "what's wrong with animals?"?
No, I'm still confused what your point is

>>8990864
Why are you asking me if there's something wrong with animals? Animals are just animals, they aren't really capable of thought in the way you and I are capable of it. There isn't anything "wrong" with them, they just need to know their place.

>>8989010
i saw ed witten give a talk about the chern-simons action a few years ago. i'm less interested in pure research than i used to be, is there any use of me keeping up with this sort of mathematics?

>>8990874
i'm still a student. i'm asking because i want to know if any of these maths are used in any sort of industry.

>>8990869
>Why are you asking me if there's something wrong with animals?
I'm not, that's why I said it didn't make sense to ask, you seem confused

>>8990750
Alternatively for a), suppose the fractions are neighbours and [math]\frac{a}{b}[/math] can be simplified. Then there is an integer [math]k\neq\pm1[/math] such that [math]a=ke, b=kf, e, f\in\mathbb{Z}[/math]. Then [math]\pm1=ad-bc=ked-kfc=k(ed-fc)\neq\pm1[/math], which is a contradiction. Therefore neither of them can be simplified.

>>8990882
What's the intended meaning of your posts then?

>>8990882
He means that tripsfags are subhuman.

>>8990885
>What's the intended meaning of your posts then?
To ask what's wrong with tripfriends
see:
>>8990841

>>8990892
see: >>8990869

>>8990184
Hey man, I like that picture. Do you have more math anime girls? Could you post them all? I want to save them and have math anime girls to use them when I post in the /SQT/. Maybe make it my phone's background if I can. Thank you.

>>8990731
You're still doing stuck on this exercise? (I assume you're that guy from the previous thread, right? Anon...)

1. The exercise has a typo:
Assume, to the contrary, that [math] \frac{a+b}{c+d} [/math] is a neighbour fraction to [math] \frac{a}{b} [/math].
Then [math] a(c+d) - b(a+b) = ac+ad-ab-b^2+bc-bc=\pm 1+(a+b)c-(a+b)b= \pm 1 [/math] whence [math] (a+b)(c-b) = 0 [/math]
But if either of [math] a+b [/math] and [math] c-d [/math] is 0, then [math] 1 [/math] can be written as the product of two integers (since [math] ad-bc=\pm 1 [/math]). This is absurd.

The correct neighbour fraction for (b) is [math] \frac{a+c}{b+d} [/math]. (Should be obvious why.)

2. This should have been piss easy but here's how you do (c):
Assume, to the contrary, that there exists [math]\frac{e}{f} [/math] such that (wlog) [math] \frac{a}{b} < \frac{e}{f} < \frac{c}{d} [/math].
Then [math] ( \frac{e}{f} - \frac{a}{b} ) + ( \frac{c}{d} - \frac{e}{f}) = \frac{1}{bd} [/math].
But [math] \frac{e}{f} - \frac{a}{b} = \frac{eb-fa}{fb} \geq \frac{1}{bf} [/math] and similarly [math] \frac{c}{d} - \frac{e}{f} \geq \frac{1}{df} [/math], whereby [math] \frac{1}{bd} \geq \frac{b+d}{bdf} [/math], which contradicts the fact that [math] f < b+d [/math].

>>8991362
Yep, I knew it. I fucked even this one up. Been brainfarting all day.
That should be [math] c-b=0 [/math] not [math] c-d=0 [/math]. You get no contradiction like this but the exercise is still poorly contructed since a neighbour fraction like that is less general, only holds if the numbers are of the form [math] ad = (b-1)(b+1) [/math], while [math] \frac{a+c}{b+d} [/math] is always a neighbour fraction whenever [math] \frac{a}{b} [/math] and [math] \frac{c}{d} [/math] are neighbour fractions.

>>8988527
I'm studying nonstop for my algebra qual. Dear god please save meh.

>>8991745
Algebra is pretty easy
Rule 1: a(bc) = (ab)c
Rule 2: Sometimes ab=ba

>>8991766
>he limits himself to associative algebra

>>8991770
Oh shit, he got me. I have been outed as the algebra 1 group theory kiddy I am. I must now commit sudoku.

>>8988527
I'm studying a construction of the fractional Brownian motion through correlated random walks on [math]\mathbb{Z}[/math]. Does anyone have suggestions for good resources?

>>8991770
What's the point of non associative algebra? Like what the hell is left if you drop associativity?

>>8991895
lie algebras

>>8991905
That does not really count, because although Lie algebras are not associative, you still have a something similar, i.e. the Jacobi identity. It turns this is so strong in fact that you can construct for each Lie algebra its universal envelopping algebra, which is associative again.

>>8992017
>Lie algebras don't count as non-associative because you can do something to them to get a new object which is associative

>>8988527

WHO HERE /AUTISTIC NUMBER THEORY/?

[math] g(m) = \sum{f(d)}\ \ \ \ \longleftrightarrow \ \ \ f(m) = \sum{\mu(d)g(\frac{m}{d})}[/math]

>>8992056
>autistic number theory
>posts the most elementary result about arithmetic functions

????????????????????

>>8992062

It wouldn't be autistic if it was interesting now would it?

>>8992067
Autistic number theory should be the cutting edge shit that is completely detached from anything sane.

The inversion formula is even taught to kids.

>>8992073

I got it from Concrete Mathematics on the 4th chapter so eat my dick

>>8992079
kek

bump

hey /math/
what are some good resources for getting into category theory and what are the prerequisites?
thanked

I found an RNG algorithm by randomly generating assembly basically

gonna use it to calculate random numbers in an OpenGL shader

top is a bitmap made by 1 bit per iteration bottom is what it looks like with sorted pixels

>>8992429
>MacLane, Categories for the working mathematician
>ncatlab
>Abstract and concrete categories

Although you pretty much don't need to know any prequesites except maybe some basic set theory I would argue that it is very useful to know

- basic linear algebra
- basic topology up to homotopies
- some basic algebra

so one can appreciate why one should even do category theory.

>>8992469
thank you. I'll look into them.
do these books cover some motivations and uses in maths of categories?
I haven't yet read something on them, but I never understood the underlying principle behind them.
could you care to explain to me, shortly ofc, what's the fuss regarding categories all about?
thanks

>>8992429
if you have enough time and interest, study also algebraic topology along the way. the best way to learn category theory is to use category theory.

>>8992478
A good read about that is https://ncatlab.org/nlab/show/category+theory

some of it may be quite overwhelming for a complete novice. So I guess I'll try my own words:

Category theory is less an actual "theory" and more sort of a "language" or a "philosophy", which seems to inherently gives you the "right" intuition or point of view for many areas of mathematics, especially topology, algebra, algebraic geometry, etc.

The most interesting part of category theory are the definitions and less the actual theorems of category theory, because they serve to unify very different notions in different branches of mathematics back together, because from the viewpoint of category theory they behave very similarly. Thus it is makes it possible to quickly transfer knowledge in one branch to another branch. An example: Products

A product of sets or a product of topological spaces or of groups are defined quite differently, but from the viewpoint of category theory they all behave in the same way. If you know category theory and know how the product behaves, then you should be able to quickly determine what a product of top. spaces, or groups or... is, without having to memorize it for each seperate case.

>>8992544
that's truly interesting. so it's basically an abstraction on the form of some objects and how that abstraction behaves? sort of?

>>8992488
will do. just a small curiosity: does topology and algebraic geometry have any applications in theoretical CS ?

I wish more math questions were written like this.

>>8992563
> so it's basically
Plato's theory of Forms in a purely mathematical context.

>>8992563
Look into computational algebraic geometry. Also, I'm pretty sure algebraic geometry plays a role in coding theory.

>>8992563
Yes, sort of. It is sort of an abstraction of abstractions.

Category theory studies abstract structures, which is a little funny, because a category itself is an abstract structure, so you can use category theory on itself. I have not heard of any useful results obtained that way, though.

>>8992488
As that guy said, ironically category theory is best studied by working with concrete examples of categories. Some categories are very well suited for this like Topology, Algebraic Geometry and so on.

>>8992579
This guy is right. Algebraic topology is a good way to get familiar with the basic concepts, as you will encounter a lot of them. You will have functors whenever you consider homotopy or (co)homology, you will actually both covariant and contravariant functors, you will have adjoints, you will have stuff about (co)limits, you will have pushouts and pullbacks, you will have categories over objects and consider their subcategories, etc. You don't need much abstract nonsense to develop this stuff, but if you study them simultaneously, you will be like "Oh so the category of covering spaces over [math](X, x_0)[/math] is a subcategory of [math]\textbf{Top.}/X[/math] (where [math]\textbf{Top.}[/math] is the category of pointed spaces)", and so on.

You should always try to find some concrete example of those things. Does this concept make sense if the context is that of set theory, or does it make sense if the context is that of topological spaces or group theory? And one more thing, borrowed from the symmetry group of a triangle: certain objects or arrows may get extra properties when you restrict yourself to a subcategory. With the non-abelian group [math]S_3[/math], this means that all the elements commute with one another in any proper subgroup of [math]S_3[/math], and, similarly, in the full subcategory of the category of sets whose objects are all the non-empty sets, singletons are both initial and terminal objects, but in the whole category they are just terminal objects ([math]\{ x\}\to\emptyset[/math] doesn't exist for any [math]x[/math]).

>>8992566
Brainlet here... i don't really know how to approach this.

Can anyone explain to me the motivation for a coset? I'm halfway through Pinter's abstract algebra. Something to do with creating a partition? Not sure why it'd be a useful one, though.

>>8992687
1) Consider the group of integers, and suppose we want to have all the congruence classes of an integer [math]n[/math]. Consider the group [math]n\mathbb{Z}=\{ nk\ |\ k\in\mathbb{Z}\}[/math]. This is a subgroup of [math]\mathbb{Z}[/math], and its cosets are of the form [math]k+n\mathbb{Z}[/math] for [math]k=0, 1, \dots, n-1[/math]. If you look at those [math]k[/math]'s, you will see that they are all the possible remainders of an integer when divided by [math]n[/math], and hence [math]\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}[/math] is the additive group of the congruence classes of [math]n[/math].

2) For any group homomorphism [math]f\colon G\to H[/math], you have a subgroup [math]\text{ker }f=\{ g\in G\ |\ f(g)=e_H\}[/math] of [math]G[/math]. No matter if [math]G[/math] is abelian or not, that is always a normal subgroup, so you can take the quotient group [math]G/\text{ker }f[/math]. Let [math][g][/math] denote the equivalence class of an element [math]g\in G[/math] in the quotient group, and define two homomorphisms: [math]\pi\colon G\to G/text{ker }f, g\mapsto [g][/math] and [math]\overline{f}\colon G/\text{ker }f\to H, [g]\mapsto f(g)[/math]. That the first one is a well defined homomorphism of groups is easy to check, but the second one is not that obvious, so I'll do it for you. We need to have [math]g, h\in [g]\Rightarrow f(g)=f(h)[/math] for all [math]g, h\in G[/math] for the function to be well defined, so suppose [math]g, h\in [g][/math]. Then, [math]gh^{-1}\in\text{ker }f\Rightarrow f(g)f(h)^{-1}=f(g)f(h^{-1})=f(gh^{-1})=e_H \Rightarrow f(g)=f(h)[/math], and [math]\overline{f}[/math] is well defined. That it is a homomorphism follows from this: [math]\overline{f}([g][h])=\overline{f}([gh])=f(gh)=f(g)f(h)=\overline{f}([g])\overline{f}([h])[/math] for all elements in the group. Moreover, [math]\overline{f}[/math] is injective, and so, by quotienting away the kernel of a homomorphism, we can use that to construct and injective homomorphism.

>>8992728
[math]G/textker f = G/\text{ker }f[/math]
typo

>>8992687
Yes exactly, think about equivalence classes.

If you take a set, then you can "divide out" an equivalence relation, which gives you a partition of that set into equivalence classes. The useful property is then that you can consider the set of equivalence classes. This is exactly the set that you get if you start with a set and then "quotient out" an equivalence relations, which gives a this new set.

For groups the really cool thing is, that under some assumptions on the subgroup (it needs to be normal), we get that not only do the cosets form a partition (and thus give rise to an equivalence relation on the group), but also that the cosets form a new group, which we call the quotient group or factor group.

>>8992632
Oops, that example didn't quite work out LOL. If you consider the category of pointed sets, then it singletons are initial. Sorry, my bad. You do get different behaviour, though, when restricting to a suitable subcategory. If you take the category of groups, and restrict it to the full subcategory of abelian groups, you get an abelian category which the category of groups is not.

>>8992579
>I have not heard of any useful results obtained that way, though.

Studying the Category of Categories is not necessarily interesting, however there are categories of objects that built over categories which are interesting.

For instance Algebraic Stacks. When you study Schemes (which are built over sets) you pretty much always study them relative to a base. i.e. Instead of just [math]X[/math] you look at [math]X \to S[/math]

Similariy, for an Algebraic Stack you fix a base scheme [math]S[/math] and you stack is some category [math]\mathfrak{X}[/math] together with a functor [math]\mathfrak{X} \to {S_{Et}}[/math] to the big etale site of the base scheme. You require the functor to have certain properties, that sort of shape the overlying category into something useful. i.e. Fibered in Groupoids, Effective Descent, etc.
The point is categories can be used for more than just studying categories of more classical mathematical objects, but can be used to build interesting objects in their own right.

>>8992749
thanks, that makes things clearer. also, i'm really enjoying this book, it's tasty stuff.

>>8992728
just started the chapter on homomorphisms, but i'll take another look once i've read it. ta

>>8992656
claudia has bigger tits

Should I study some "standard" math such as calculus if I'm mainly interested in things related to logic and category theory?

>>8992825
What have you been doing so far? Are you in a uni? Do you need "standard" math for something? If you have no reason to restrict yourself to "standard" math, then, please, study the material you are interested in.

>>8992825
>getting memed so hard that you go around saying you're interested in category theory before you know how to take a derivative
/sci/, folks
not even once

>>8990212
Underrated post.

>>8992825
No. You will end up not understanding anything because you have absolutely no motivation for any of the definitions.

>>8992889
>What have you been doing so far?
As in my background? I've been self-studying programming language theory and now I find that math itself is more interesting to me.
>Are you in a uni?
Not yet.
>If you have no reason to restrict yourself to "standard" math, then, please, study the material you are interested in.
I guess that's what I'll do for now then. Thanks.

>>8992933
Are you okay? Do you need to talk to someone?

>>8992984
I guess that makes sense.

>>8992889

I do not see why you would start with the highest levels of abstraction possible when you would likely have difficultly with the abstractions of calculus since you don't even know what a proper derivative or limit is

Just go learn some math man, you won't understand the motivations for category theory at all

I just learned about xor swap and it changed my whole outlook on shit, mane.

math.ucr.edu/home/baez/books.html#calculus

Thoughts on this list?

>>8993071
>math.ucr.edu/home/baez/books.html#calculus

>Thoughts on this list?
looks an awful lot like a memelist

>>8989570
>>8992933
>>8989570
>meme
frrrrrrrrt
plop
plbbt, phbbbbt, prrrrp
splat

>>8993076
>>8993091

>>8993091
>>>8989570
>>>8992933
>>>8989570
>>meme
>frrrrrrrrt
>plop
>plbbt, phbbbbt, prrrrp
>splat
https://www.merriam-webster.com/dictionary/meme

>>8993107
>meme
BBBBBBRRRRRRRAAAAAAAPPPPPPPFFFFFFFFLLLLLLLLLPPPPPPPPPFFFFFF

>>8992576
>>8992579
>>8992632

thank you so very much for your patience and contribution. I'll be sure to look into each of these things and, given I have 2 months of vacation, I should be able to learn this (at least to a basic extent) alongside some other things.

>get 17k$ grant to do math research
>spend more time wondering why you got it over everyone else that applied instead of doing research

>>8993124
I spend most of my phd shitposting, then being depressed, then finishing in a hurry

Can anyone tell this brainlet what you do/learn in your final year mathematics?

>>8993220
That varies significantly.

>>8993218
That hits close to home :(

>>8993220
if you mean final year of undergrad, obviously it varies depending on your interests and your schools course offerings

i did my school's grad analysis and grad algebra courses, along with homological/commutative algebra, algebraic number theory, representation theory, catastrophe theory, differential geometry, graph theory and elementary number theory

>>8993007
Just make sure you pass the possible entrance exams! Good luck!

>>8993017
That's why I asked for extra info. It turns out we have a guy with a proper reason to look into that stuff. If that was some hs kid wanting to be cool, then it would be a lot different.

Guys, take a cute koala gif.

What about falling for Edmund Landau meme analysis? It looks very nice despite its age.

I'm a medical student in Europe so I don't have to deal with any math at all outside of common logarithms. I want to get some grasp of it though, at least at a basic college level but I feel like khan academy is unbearably slow and simplistic. Any good youtube channels that go through things faster?

Heh, i wish i could do maths.

> For ever positive integer n there's a prime p such that n < p <= 2n.
> this is essentially Bertrand's postulate which Chebyshev proved for all n in 1850.
> Use Bertrand's postulate to prove that there's a constant b ~= 1.25 such that the numbers
> [math] \lfloor 2^b \rfloor, \lfloor 2^{2^b} \rfloor, \lfloor 2^{2^{2^b}} \rfloor , ...[/math]
> are all prime.

So they calculate the constant by figuring the value of p_n and then taking the log_2 of that?

damn pham

>>8990874
>>8990879
Chern-Simons TQFT can be used to describe topological defects and the quantum Hall effect in condensed matter systems. You won't need much more than Chern numbers though.
>>8990899
Here's another one I made.

Math grad school is for those who have mommy and daddy backing them off financially

feel free to prove me wrong

>>8997362
>not getting your grad school funded by the department
you shouldn't be doing grad school

>>8993278
what country are you in?

after you said "along" , are those courses or just topics you briefly touched upon?

>>8997366
I am not talking about that idiot. I am talking about what happens afterward. I guess you would be pretty stupid to think you will always get fellowships and a tenure. Are you really that blindly confident?

>>8997370
>"you need mom and dad to support you for gradschool"
>you don't
>"OBVIOUSLY IM TALKING ABOUT SOMETHING ELSE"

there's a reason you won't get anywhere near math

>>8997370
>I am talking about what happens afterward. I guess you would be pretty stupid to think you will always get fellowships and a tenure.
I'm an actuary, so grad school has already been worth it for me

>Are you really that blindly confident?
Do you really think being a professor is the only thing a math degree is used for?

>>8997367
>what country are you in?
Canada

>after you said "along" , are those courses or just topics you briefly touched upon?
each of those were semester long courses

>>8997362
>>8997370
Why do engineers even try to troll math threads? This is just pathetic.

>>8997374
You know it's true.
Bet mommy and daddy will gladly take care of you until you get your tenure , which would be never

>>8997380
I literally asked every professor that I only want enough money to survive and they said not to pursue grad school

Really made me think

>>8997389
How does it feel you won't be financial independent until your 50's?

>>8997393
>I literally asked every professor that I only want enough money to survive and they said not to pursue grad school
It sounds like they don't have much faith in you

>>8997384
I can only take up to 5 courses and i need deans permission to work.
How do you even deal with all the HWs and tests and finals?

>>8997397
>I can only take up to 5 courses and i need deans permission to work.
I listed 9, it was 5 math classes the first semester, then 4 maths the second semester along with a language course

>How do you even deal with all the HWs and tests and finals?
The same way I dealt with them for the first 3 years, work hard, study with classmates, a few sleepless nights

>>8997394
I got A's on their classes moron and some of them i met during talks

How are you even in grad school and make the dumbest statements?

>>8997400
ah dude i thought u took all of them in one semester. I was pretty jealous.

>>8997401
>I got A's on their classes moron and some of them i met during talks
It's not that hard to get A's in math.

>How are you even in grad school and make the dumbest statements?
I already said I finished grad school.

It sounds like you're part-way through a bachelor's in math. You don't have to go to grad school, but it doesn't sound like you have a plan B for whatever reason.

Is it possible to solve a right triangle given only it's area and three angles? I know you cannot solve an AAA triangle because there is no way to determine it's size, but if the area is know, can I determine the size?

>>8997404
Maybe in your community college.
> it doesn't sound like you have a plan B for whatever reason.
If i go to grad school, am i gonna be a moron like you?
Of course i have a plan B. Why are you so stupid to think just because there is a plan B it works like a golden parachute? Let me guess, you were a spoiled lil brat werent you? faggot.

>>8997413
>Of course i have a plan B.
So what's the problem with you not going to grad school then?

>Why are you so stupid to think just because there is a plan B it works like a golden parachute?
Where did you get that implication? That's why you also have a plan C etc...

>Let me guess, you were a spoiled lil brat werent you?
What's with all your resentment? I grew up in a shitty duplex and paid for my rent throughout university with money from the government

>faggot.
Why the homophobia?

>>8997421
>Why the homophobia?
>4chan
>homophobic

faggot just means you are a faggot

>>8997434
>faggot just means you are a faggot
You're not very personable my friend.

Anyway, you shouldn't go to grad school unless you want to, so don't bother, you'd just be taking up valuable resources that someone who actually wants to be there could use.

>>8997421
faggot

>>8997421
>reddit spacing
>complains about "homophobia"
Fuck off faggot. You don't belong here.

>>8997448
>You don't belong here.
Sorry but I think I'll stay. It'd be hard to leave after being here for 10 years anyway

>>8997435
are you stupid enough u can convince someone to not go to grad school with your incoherent argument? Like seriously, how can someone this stupid have gone through grad school?

>>8997453
you are so fucking full of yourself.

>>8997519
>are you stupid enough u can convince someone to not go to grad school with your incoherent argument?
What's incoherent about it? Grad school's not particularly fun, easy or well-paying compared to "real-world" alternatives, especially if you don't want to be there. There's plenty of people who dropout, and going there without wanting to go greatly increases your chance of being one of them.

>>8997520
>you are so fucking full of yourself.
I didn't mean to come off that way, there's a reason everyone memes that no one ever really leaves this website

>>8997406
Suppose you know [math]A, \alpha, \beta, \gamma[/math], and let [math]a, b, c[/math] be the opposite sides of the angles, respectively. If now [math]\gamma = 90^\circ[/math], and so [math]c[/math] is the hypotenuse. This gives [math]a=c\sin(\alpha), b=\sin(\beta)[/math] and, for the area, [math]A=\frac{1}{2}ab=\frac{1}{2}c^2\sin(\alpha)\sin(\beta)\Leftrightarrow c^2=\frac{2A}{\sin(\alpha)\sin(\beta)}[/math]. This gives us [math]c=\sqrt{\frac{2A}{\sin(\alpha)\sin(\beta)}}\Rightarrow a=\sqrt{\frac{2A\sin(\alpha)}{\sin(\beta)}}\land b=\sqrt{\frac{2A\sin(\beta)}{\sin(\alpha)}}[/math].

>>8997533
yeah now you sound more like a person. Thanks for the warning but i am gonna steal some aspiring math kid money and waste it on myself :^)

Good luck getting your tenure fag

>>8988667
Group Theory
https://www.youtube.com/watch?v=O4plQ5ppg9c&list=PLAvgI3H-gclb_Xy7eTIXkkKt3KlV6gk9_
Vector Spaces
https://www.youtube.com/watch?v=KMlzcvmhaLA&list=PLAvgI3H-gclbucHp-i_p1OxzeFxadw8b3
Ring Theory
https://www.youtube.com/watch?v=qFBDbpjtjcM&list=PLAvgI3H-gclZ-DYOVyyTkJBUXZ_R1JEZP
Field Theory
https://www.youtube.com/watch?v=HPNSE0KppSI&list=PLAvgI3H-gclbhFtavB7BFUvHKm6QMGDXR
Galois Theory
https://www.youtube.com/watch?v=8H8vkR9Q_24&list=PLAvgI3H-gcla8xWr-UuLPxwhia2up0W8j

Make sure to watch these videos at at least 1.5x speed for your own sanity

>>8998068
>replying to a post 9 hours later knowing they won't see it

>that edginess

>>8992656
Claudia so THICC she has her own gravitational pull

>>8998084
>robbing someone who might be more talented and motivated is edgy
what

>>8998149
>what
I meant the
>Good luck getting your tenure fag
part

>>8998161
he is a lil obnoxious conceded faggot. he doesnt deserve

>>8991770
FRENCH FAG DETECTED

>>8998352
i-i'm canadian

>>8998183
>conceded
you're not fit for grad school m8

>>8998368
>get autistical for a spell check mistake
you arent fit to live with humans
I ll get that tenure before you, double nigger

>Math fags think they can determine who can go to grad school by looking at someones posts
Math fags are deplorable

>>8998366
Oh yeah I forgot to mention I emailed Bryant a few days ago and he told me that he doesn't know much to give any substantial advice.
>J
>U
>S
>T

>>8998084
>>8998149
>>8998398
>>8998407
>>8998410
Who are you quoting?

>>8998366
It must have been painful, right?

>>8998420
>i was pretending to be retarded

>>8998435
Who said this?

>>8998420
>mass quoting with no purpose
see u in 3 dayz

>>8998398
>I ll get that tenure before you, double nigger
best of luck, i'll be waiting

>>8998410
That's too bad

>>8998454
you ll be waiting unemployed?

Is calculus critical for studying Probability Theory?

>>8998458
>you ll be waiting unemployed?
I already said I'm an actuary, so unless I get fired or bored then no, probably not.

The next /math/ general should have a no academia discussion rule.

>>8998461
have i been arguing with an actuarial brainlet all this time? wow really brings things into perspective.

>>8998460
Yes. You should learn measure theory too.

>>8998465
>have i been arguing with an actuarial brainlet all this time?
Yes.

Feel free to prove me wrong by graduating grad school, I'd be very impressed with you.

>>8998464
>THINGS I HAVE NO RESPONSE TO BECAUSE I AM A BRAINLET ARE NOT ALLOWED
t. brainlet

>>8998470
does mr sheckelstein pay you to shitpost instead of sucking his dick? go be a mind slave somewhere else

Why would any non brainlet actuarial aspiring math person would ever take advice from a faggot like you

>>8998476
>Why would any non brainlet actuarial aspiring math person would ever take advice from a faggot like you
Because I've gone through grad school, and you're an undergrad who seems confused about the prospects post-grad school.

>>8998474
Can't we just do math without an academic dick waving contest?

>>8998485
You do actuarial dude. You shouldnt even be here. Maybe if i show you this page of proofs with symbols ur mind cant comprehend will make you go away

>>8998493
>You shouldnt even be here.
What do you think I did while studying math in grad school? You've literally been studying math for less than 4 years and most likely haven't contributed anything to the field

>Maybe if i show you this page of proofs with symbols ur mind cant comprehend will make you go away
You know you don't magically lose your ability to understand math after finishing school right?

Is [math]k\left( x \right){ \otimes _k}k\left( y \right) \cong k\left( {x,y} \right)[/math] ?

>>8998574
Maybe if [math]k[/math] is algebraically complete? [math]p(x) \otimes q(y) \mapsto p(x)q(y)[/math].

Anyone who knows number theory and quadratic congruences give me a quick hand here.

I am looking at the problem in pic related and I have a problem with pic related. You see, they refer to theorem 6 which says:

If p divides [math] x^2 - Dy^2 [/math] and gcd(x,y) = 1 then D is a quadratic residue mod p

The problem is that they are applying the theorem, but I see no guarantee of x and y being relatively prime. The problem says that x and y could be any numbers, not necessarily relatively prime. So how come they can just assume x and y are relatively prime and apply theorem 6 like its nothing? What am I not seeing.

>>8998448
>>8998474
Who said this though?

>>8998591
what even forces 4kxy-1 to be prime?

Consider an extreme where the bus to Bridget departure 1 minute after the bus to Claudia. If the buses departure every hour then obviously it is more probable that he takes the bus to Claudia, since he will only visit Bridget in one out of sixty days. I made various assumptions here but the general idea of how to approach the problem should be clear.

>>8993278
>single semester courses on any of those subjects
>algebraic nt before elementary nt
uhhh, i think you got memed lad

>>8992566
>>8992656
See >>8998621

>>8998624
I didn't take them in the order they're listed, obviously elementary was taken before algebraic

and what's the alternative to a single semester course?

>>8998627
a full year course? you've only scratched the surface and learnt definitions for a one semester course

>>8998614
Nothing, but remember that what theorem 6 says applies now to every prime that divides 4kxy - 1

Therefore, -1 is a quadratic residue mod every prime that divides 4kxy - 1

Then by the complete multiplicativity of Legendre's symbol, this applies to all the prime powers. And by the chinese remainder theorem, this applies to 4kxy - 1

>>8998629
my school only offers full year courses in calculus and linear algebra for first years, and if you can't get past definitions in a 12 week course something's wrong with your class

Any reqs for discrete math that isnt a meme applied book? I tried reading Rosen but it was just a million examples and exercises that are all the same with different numbers and too many application sections. Hopefully one that goes to the point, and has important exercises and problems

>>8998634
stanley - enumerative combinatorics

>>8998637
is it beginner?

>>8998638
the first exercise is adding one and one together to get two

>>8998076
This is awesome

>>8998639
ok just dled it and skimmed fir bit
something a bit more brainletier pls

>>8998630
factor out the gcd of x and y and the proof still works out

>>8998667
That is what I was thinking but how would you accomplish this? It is not detailed in the paper.

>>8998669
might mess up some details but something like this:

let d=(x,y), without loss of generality assume n>=m

then
x^m+y^n
= d^m (x/d)^m+(d^n)(y/d)^n
= d^m[ (x/d)^m + d^(m-n) (y/d)^n]

so by the assumption (x^m,y^n,4kxy-1)=1, if 4kxy-1 divide x^m+y^n then 4kxy-1 divides (x/d)^m + d^(m-n) (y/d)^n

since
(x/d)^m + d^(m-n) (y/d)^n
= [(x/d)^(m')]^2+d^(m-n)[(y/d)^n']^2

this implies that (jacobi symbol here):
(-d^(m-n)/ [4kxy-1])=1

but then by multiplicativity
1= (-1/[4kxy-1]) (d^(m-n)/ [4kxy-1])

and since m and n are even in this case of the proof, d^(m-n) is a square so the jacobi symbol on the right is 1, so
1 = (-1/[4kxy-1])

>>8998679
>and since m and n are even in this case of the proof, d^(m-n) is a square so the jacobi symbol on the right is 1, so

But in the proof this is used even when m - n wouldn't be even.

>>8998695
>But in the proof this is used even when m - n wouldn't be even.
can you point out where?

>>8998701
In case 2, it is taken that m is even but n is odd.

>>8998705
but where is
>and since m and n are even in this case of the proof, d^(m-n) is a square so the jacobi symbol on the right is 1
used? the situation is different there because of the extra y so the author breaks it down into two cases

>>8998710
>used? the situation is different there because of the extra y so the author breaks it down into two cases

But notice that he does that after he applies theorem 6, with D now instead of being -1 is -y.

>>8998710
Oh wait no, I am retarded. You are a genius. I think I got it now. Thanks man.

TURN ON CNN

[math] e\pi [/math] is irrational
>[math] e\pi [/math] is irrational
[math] e\pi [/math] is irrational

https://arxiv.org/pdf/1706.08394.pdf

>This note shows that the product eπ of the natural base e and the circle number π is an irrational number.

>>8998824
Is that paper serious? I read "the circle number" and think it is either some troll or some other powerful black woman who don't need no man, but does need a sum of super thin rectangles.

>>8998830
The central result is presented on the first page dude lfmao

>>8998651
Check out

Concrete mathematics

and

A course in combinatorics

if you haven't already. They're more on the "beginner" side, although they go into more advanced topics.

>>8998824
lol this guy also proved the Riemann Hypothesis in 2008

>>8998824
I mean, what do you expect when you make it easier to put a paper out there? Guy proved the obvious in a way he thought was interesting.

What math jobs allow me to work remotely / set my own hours?

>>8998833
More papers should work that way. Think of all the time we could save!

>>8998891
I disagree, actually. Imagine if someone put the main result as the first part of the paper and in the proof just quoted a bunch of other theorems that will be proven later. You read the proof and it all looks swell. You think that the result is now solid truth.

But then 10 years later someone decides to check the rest of the paper where the bunch of lemmas are proven and they find that right in lemma #1 the author fucked up and thus the original proof is wrong, and now we gotta roll back an entire field of mathematics.

Wouldn't be the first time it happens.

>>8998988
He's being facetious.

>>8998368
Grad school is special? Since when?

>>8998890

>>8998625
>>8992566

There's also the approach where one bus arrives at regular intervals, and the other one arrives as a poisson process, with the same rate as the first bus. If you do the math, it turns out that you'd catch the first bus 2x as much as you'd catch the 2nd bus.

>>8999051
yes, how?

>>8999090
Be a trust fund baby.

>>8992566
>from whence

>>8998824
Where did he go wrong?

>>8999092
too late for that

>>8998824
>https://arxiv.org/pdf/1706.08394.pdf
AYOO
HOL UP
SO YOU BE SAYING
THAT [math]\pi \times \frac{2}{\pi}[/math] WUZ IRRATIONAL AND SHEET

What kinds of things does number theory skillz enable a person to do?

>>8999188
>whenceforth

>>8999798
Cryptography

bump

>>9000029
Based feelsposter.

>>8990731
SL(2,R)

Reminder that Cirno is actually good at math
>>9000000

>>9000156
You mean [math]SL(2,\mathbb{Z})[/math]? It's the subgroup of the conformal group that fixes fractional quantum Hall states and dual quantum Hall states are related by exactly [math]ad - bc = 1 \mod (2\pi)[/math].

>>9000269
w-what's the difference between base 2 and binary

>>9000204
you have any waifu tattoos yet?

>>9000278

base -2

it's better because you can represent positive and negative integers without needing a sign bit

-3=1101
-2=10
-1=11
0=0
1=1
2=110
3=111

>>9000278
oh nevermind now i see it's base -2 instead of base-2

>>9000286
Maybe next year when I get that PhD money.

https://www.youtube.com/watch?v=AmUI2qf9uyo

>tfw will NEVER be this smart
>tfw my children will NEVER be this smart
>tfw my children's children will NEVER be this smart

>>9000473
I may not be smart, but atleast I won't be having kids haha!

>>9000473
>Witten's voice
I want him to read me his textbook on string theory to sleep.

>>9000485

>not wanting to teach your son or daughter the beauties of mathematics so they carry on in your footsteps

maybe when your IQ is higher in your later years you'll realize the true beauty of it

>>9000493
>the beauties of mathematics
Nice meme. Besides, I can feel my IQ decline slowly.

>>9000503

>Nice meme

debatable

>Besides, I can feel my IQ decline slowly.

more the reason to spend your last remaining days as a 200 IQ master transferring your knowledge to your children, then spend your brainlet years in luxury. Essentially you transfer your intelligence and live forever.

>>8989553
If you just learned them, you have quite a way to go.

>>9000493
Sorry anon, but "beauty" is purely a property of theories in physics. Anyone can think up random crazy fantasies out of the blue; it's only beautiful if you can make a correspondence with something concrete.

>>9000516
Why? I'd get nothing out of that.

>>9000528

to be expected from a brainlet anime posting fag to not know the values of family

>>9000529
Wasn't I supposed to be a 200 IQ genius?

>>9000534

I was mistaken apparently.

>>9000536
It is highly advisable not to make mistakes.

>>9000538

Mistakes are fine. Not acknowledging and learning from them isn't.

>>9000542
Keep that in mind when you are defusing a bomb and cut the wrong wire.

>>9000516
>transferring your knowledge to your children
kek, these 'smart gene' memes, fucking skip the entire raising children part so you can actually spend your time in your field, and just transfer the knowledge to your younger colleagues

>>9000550

last I checked this is a math thread not a bomb defusing thread

>>9000554

>he can't raise a kid and develop his field

mark of a true brainlet

>>9000556
What if you make a mistake in the bank you work at's cryptographic algorithm and lose all your clients' money?

>>9000556
>this is a math thread not a bomb defusing thread
Yeah, but you could do something useful for once. It's more likely that a randomly picked person has to defuse a bomb than that a student of pure mathematics is useful to anyone.

>>9000529
>gets cucked and bound to a succubus harlot and your crotchfruits that does nothing except suck your wallet dry for the rest of your life
>intelligent
The best (only) option is to marry a male colleague and adopt a generically edited baby.

did somebody say something?

must be an echo in here

>>9000557
sorry to inform you that it doesn't take a genius to change diapers and raise a mini autist into a grown autist, dumbass

>>9000563

>being so blind to reality due to constant shilling and internet memes

embarrassing

>>9000566

would expect this kind of autism in a math thread

>>9000567
>thinks alimony is just an ""internet meme""
You ever leave the house?

>>9000567
You keep being the good goy you are and I'll cultivate my own army of superchildren.

>>9000573

>marrying a shit woman

dug your own grave. only low IQ faggots can't find the perfect woman

>>9000575

sure thing. have fun when you're 40+ with nothing but anime pics and sub par math scribbles to your name

>>8997393
>they said not to pursue grad school
Well duh, probably for the same reasons we're telling you not to pursue grad school. Because you're a fucking moron.

>>9000586
>only shit women get alimony
What is this meme?

>>9000521
Imagine having this much shit taste.
I bet you don't believe in the form of beauty either.

>>9000612
t. Plato.
Get lost before I throw my shit at you.

>nothing interesting on the arxiv tonight

>>9000622
Back to your barrel, Diogenes.

>>9000734
Reminder that conforming to any school of philosophy of mathematics that isn't formalism is a sign of assbergers.

>>9000743
Reminder that conforming to any school of philosophy of mathematics is a sign of assbergers.

anyone go into tech or engineering with a math degree? if so, what did/do you do?

>>9000758
Reminder that conforming to any school of philosophy is a sign of assbergers.

>>9000770
Reminder that school is a sign of assbergers.

>>9000780
Reminder assbergers.

which field of math is the comfiest?

>>8988683
>Implying i'm not also reading the textbook
>Not wanting to also hear lectures
found an autistic brainlet

>>9000762
I'm an ECE student. I'm planning to double major in Math.

I'm just a beginner & studying Analysis by myself by books like Rudin.
Also downloaded the Barry Simon Analysis books, & started studying by it.

I plan to do the required Math courses (as Real Analysis, Abstract Algebra, PDE, Differential Geometry, Topology, Complex Analysis)

>>9001011
Differential topology/smooth manifolds. Everything is soft as fuck.

>>9001011
Algebraic Geometry

which field of math is the least comfy?

>analytic number theory

>>9000521
Stop posting already.

http://www.ams.org/notices/200210/comm-morin.pdf

>The World of Blind Mathematicians

>>9001067
>Stop posting already.
>>>/reddit/

>>9001069
Pontryagin is so fucking based.

>>9001060
That's not how you spell combinatorics.

bros

why did he have to go so young

:(

>>9000521
Physics should be banned though. We are already working on it.

Suppose you stumble upon a (seemingly random looking) combination of letters/numbers/symbols.

What kind of structure/details do you look for that would suggest there's information deliberately encoded in the combination?

>>9001188
On the other hand what suggests it's actually just random nonsense?

>>8990932
Get a load of this brainlet

>>9000521
Only brainlet """mathematicians""" and brainlet """physicists""" think there is some true difference.

>>8988527
The substitution axiom - occultism!!

Axiom says this. Let φ be a correctly constructed formula of the language of set theory of the first order, in which the letter B does not enter freely.
Let ρ be its parameter string. Then for any set A and any set of parameters ρ if φ is functional with respect to A, then there exists a set B consisting precisely of elements that are φ-images of the elements of A.

This axiom is esoteric because it makes it easy to prove the existence of sets, of which nothing in principle can be said about the emptiness or non-emptiness.

For example, a correctly constructed formula is the formula "the cardinality x is strictly greater than alef-0, but strictly less than the continuum". Applying this formula to some cardinal greater than the continuum, we obtain a set of cardinals M whose emptiness is equivalent to the continuum hypothesis. The set M exists, but its elements can not be studied by any means. Absurdity somehow.

But f is **ANY** well-formed formula! Which, in addition, can depend on any number of parameters. This, if you think about it, allows you to literally prove the existence of Cthulhu.

Indeed, the proof is reduced to an exact set-theoretic description of what we mean by Cthulhu. If C (x) is an assertion that the set x is Cthulhu, then a formula f (x, y) can be constructed that is functional in x and assigns Cthulhu to each of the x. Applying this formula to any one-element set, we prove that there exists a set whose only element is Cthulhu.

Moreover, if we consider that we have the right to renumber the Great Ancient by ordinals, then we can prove the existence of any preassigned number of copies of Cthulhu. Even an alefa with a transfinite numbering of different Cthulhu.

Proof me wrong...

>>9001188
Why do you keep posting pictures of black people?

>>9001468
I'm black.

>>9001321
>an exact set-theoretic description of what we mean by Cthulhu
Not possible by Tarski's undefinability theorem.

>>9001479
So you're Sarafa Iyaniwura. http://www.math.ubc.ca/~iyaniwura/
You know what to do anons.

>>9001490
Nothing to do with that pal.

>>9001479
Yes, I can tell by your use of frogs.

>>9001188
If I understand it then it's perfectly valid and trivial, if I don't then it's gibberish.

Ok you faggots, time for more elementary algebra exercises. Prove that any non-abelian group with 8 elements is isomorphic to [math] Q_8 [/math] (group of quaternions) or [math] D_4 [/math] (dihedral group).

http://www.amazon.com/gp/richpub/syltguides/fullview/R1GE1P236K3YSV

This link I was given for a good math series of books is just redirecting to amazon's home page

Anyone got a working link?

It's from the sticky

>>9001707
Do you know what it's about?

>>9001710
From here
https://sites.google.com/site/scienceandmathguide/subjects/mathematics

>Advanced Mathematics Book List HIGHLY recommended Amazon book list with reviews. Assumes high-school level knowledge of math.

>>9001490
Is the proposition "traps are gay" undecidable in ZFC+V=L by Gödel's Incompleteness Theorem?

>>9001188
>>9001189
if you had access to a random generator that produced this you could potentially try to fit a distribution (uniform more often than not) to see if the characters were chosen randomly. So if you see each letter appearing about as often or perhaps 1/3rd of the time its a letter/number/symbol or perhaps # of x's / # of total characters appeared of the time its an x then that would suggest a random uniform process of generation.

I have absolutely 0 experience in the art of decoding whatever it's called, but I've heard every language has grammatical patterns, in this case the language is just a string of characters. But you would look for these patterns intensively, see what repeats, keep track of what repeats and how often and then you would know that there is potentially information hidden. As to what it is, though, that would be a more difficult task unless you can convert it to a language we already know about (which considering a human would have produced this generator, and had it had information produced, the information had to have been converted from a human language to a string of characters).

Need some help. Was checking out previous years' admission tests for a university I'm planning on applying to, and in the math section, pic related came up.
I'm supposed to find out the missing digit, a, and that I can do (some more information regarding the relation between A, B, and C is provided), but I don't understand what do the line over the top and the smaller digit at the bottom right mean. Not asking for a solution to the problem, just asking what do the symbols mean so I can get to study that.

>>8988527
Will I become famous if I can prove that ZF is inconsistent? I already know that it is, now I'm just thinking if I should even bother writing down the proof.

>>9001768
>if I should even bother writing down the proof.
Don't. It's wrong anyway.

>>9001827
It's correct though. I know it for a fact.
The question is should I actually bother writing it down? What do you think?

>>9001841
>I know it for a fact.
Lol ok write it down and put it on vixra please.

>>9001844
>Lol ok write it down
Will I become famous?
>put it on vixra please.
I'll post it in this thread instead.

>>9001851
>wants to become famous
>posts results in this thread

>>9001861
I think that would be pretty funny.

>>9001732
The smaller digit at the bottom is the number base. The bar over the top could either mean that the order of the symbols is as given (that they appear in the order they are written), or that they repeat, but the former is most likely.

>>9001891
Ah, so base 5 meaning the number sequence goes 1 2 3 4 10 11 12 13 14 20 and so forth, right? thanks m8.

>>9001872
Post it in the new thread so more people could laugh at it.

>>9001651
Let G be nonabelian with order 8. Then, we cannot have [math]\forall g \in G g^2 = 1[/math]. Hence, one element in G has order at least 4. Again, since G is nonabelian, there cannot be any element of order 8, hence there is an element x of order 4.
Now, let's count. Assume that x and x^{-1} are the only elements of order 4. Since x doesn't commute with the whole group (otherwise, Z(G) would have index at most 2, hence G would be commutative), there is an element y that doesn't commute with x. y must have order 2 and yxy is an element of order 4 different from x, ie. x^{-1}. Besides, x and y generate G since <x> has index 2, and is thus maximal. Hence, G is generated by an element y of order 2 and an element x of order 4 such that yxy = x^{-1}, ie. G = D_4.
Assume that there is an element y of order 4 outside of <x>. Then <x> and <y> have an intersection of order 1 or 2. If their intersection were trivial, then they would generate a group of order 16, which is absurd. Hence, x^2 = y^2 <> 1. Now, z = xy is neither in <x> nor in <y>. Let's see that it also has order 4, which will allow us to conclude.
Assume that z has order 2. Then we have xyxy = 1, hence xyx = y^{-1} = y^3 = x^2y, hence x and y commute. Since x and y generate G (same argument as before), G is abelian, which is a contradiction.
Finally, G = {1, x^2 = y^2 = z^2, x, x^3, y, y^3, z, z^3}, which is isomorphic to Q_8 via x -> i, y -> j

Is it just me or is this general the most consistently on-topic general on /sci/?

>>9002009
Beautiful palindromic number.

>>9002009
It just became less on-topic because of your stupid post.

>>9002051
Thread's bumplocked. Suck my cock.

>>9001490
>**This** statement is false
How you going to use logic system, where can easily proved existence of ctulhu, dagon and even allah?

>>9002114
You can not define Allah in ZFC.

Damn, how to apply mathematics to real life?
I`m spended almost three hours thinking how to make a box containing 60 liters of sand with a minimum consumption of material.
I'm somehow defective or is it a normal?

>>9002146
>>>/sqt/

>>9002146
It takes many years of training to get to the point where you'll be comfortable enough with math to use it effectively in real life. You'll get better so long as you keep working hard.

New thread soon.

>>9002320

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

>>9002146
the theory of minimal surfaces is an active area of research