/sci/ - Science & Math

>>11769296
nice picture

>>11769325
Yeah, took me a while to find it.

>kenken edition
>no kenken in the OP
fucking anime retards

Anyone have any recommendations for Representation Theory?

>>11769341
representation theory of what

>>1176934
Not that anon, but tell me about finite groups.

>>11769367
Anon that links to a 2010 thread

>>11769408
Only ten years too late, well, better late than never.

Im learning pre calc from youtube lectures but Im a bit lost in terms of what I should be exercising.

What exactly are the problems that will put all my understanding up to test? I just want to sit down and start practicing over and over to get good.

>>11769296
is there a collection of pictures with anime girls holding math books like /g/ has one for programming books?

>>11769449
>like /g/ has one for programming books?
Where is that one?

>>11769467
I believe it's this one https://github.com/laynH/Anime-Girls-Holding-Programming-Books

I am torn between mathematics and computer science. Is computer science still a meme degree even at a top 50 univeristy? Do you have any advice? Double major?

>>11769645
Mathematics will take you into a whole new world where things you can't even begin to imagine are possible, it's a world of endless possibilities, trials, fun and by the end of it, you'll be left with a sense of accomplishment in life that won't feel in no other area.

Computer science will take you inside a room and make you write basic and repetitive shit until the end of your life.

Does Kenken actually train any math related skill?
I never bothered because I thought it was just complicated sudoku.

>>11769871
C'mon... You already know the answer to that, my man

Does anyone know a good resource for either Hypergraphs, Hamiltonian cycles or toroidal graphs.

I need to do a graph theory project and I wouldn't mind doing either of these three topics, but don't know where to start for good resources as most graph theory books don't touch these topics

I feel pretty messed up that the schooling system made me believe I was bad at math. Math is beautiful, its pure.
Instead you had to take this thing and turn it into something that was a step-based process.

>>11769645
Math is a very different beast to actual science subjects. Though there's always applied mathematics which is underrated.

Id say double major which is what I did with physics/math. You can always drop one later or reduce one to a minor, which you'll probably want to do so you can do more papers of whatever interests you more (while I got 2 majors, the draw back was missing out on interesting math/physics electives)

>>11769943
I think the step based process is good as it helps break down very long complex methods. The problem is schools get a little too fixated on that so you can pass test problems, rather than actually fostering an intuitive understanding of math concepts.

>>11769944
Computer science isn't science and has nothing to do with science.

>>11769958
have you heard of artificial intelligence? calabi-yau spaces? advances in physics that require advance computation? "science" is literally in the name

>>11769948
https://www.youtube.com/watch?v=VW6LYuli7VU
watch this by feynman. explains much better than I can

>>11769960
>artificial intelligence
doesn't exist
>calabi-yau spaces
math, not CS. Structures and ideas invented by mathematicians.
>advances in physics that require advance computation?
You mean like advances in physics that require particle accelerators and electrical grids? Engineering must be physics now!
>science is in the name
you are fucking stupid

>>11769986
>doesn't exist
If a program is told that it feels hurt, does it actually feel hurt?
Motivate your answer by drawing comparisons to humans :^)

>>11769963
He doesn't really explain much. What does he mean by solving 2x+7=15 "algebraically" ?

>>11769960
All of those things you mentioned are closer to mathematics than they are to "actual science" as distinguished in the op. The person you're replying to was correct.

can anyone help explain inner products?

using formula pic related
take example (1 1 0)^T the norm is root 2,

but isn't it supposedly 2?

Am i tripping or did i get the dot product and inner product mixed up

It is known that, in general topological spaces, the (Minkowski) sum of two closed sets [math]A + B = \{ a + b : a \in A, b \in B \}[/math] need not be closed. A sufficient condition for the sum to be closed is that one of the sets is compact. Now, I doubt that compactness is a necessary condition, but I haven't really seen much discussion about this anywhere, and I'm not sure if closedness can be satisfied without compactness of one of the sets in some special cases, in particular when adding cones (which cannot be compact).

I'm currently struggling with the following example. The space of interest is [math]B(\mathcal{H})[/math], bounded operators on some Hilbert space. Let [math]S[/math] be some closed and bounded set in this space, and let [math]P[/math] denote the closed cone of positive operators. Is [math]S + P[/math] closed? If not, can you find an example of a closed and bounded [math]S[/math] for which [math]S + P[/math] is not closed?

>>11769296
Noob here.
Is memorising derivation rules the worst way to learn differentiation?
Should I be going about this differently?

>>11770066
Your pic is the norm induced by the standard inner product on R^n. That is the Euclidean norm, I.e. the standard notion of distance from high school.
Now why the fuck would it be 2? What is the length of the hypotenuse of a triangle with two sides of length 1?

What does /mg/ think of Lee's book on Topological Manifolds? It started out slow and a bit too verbose, but I really liked his introduction of simplicial complexes.

>>11770066
The norm is the square root of the inner product, my definition.
[math]
||x||^2 = (x, x)
[/math]
This is the case for each and every inner product, on each and every Hilbert space.

This allows you to clearly see, for instance, Pythagoras theorem (assuming you are using a real inner product), as :
[math]
(x - y, x -y) = (x, x) + (y, y) - (x, y) - (y, x)
[/math]
But in a "square rectangle", [math]x[/math] and [math]y[/math] are orthogonal, so [math](x, y) = (y, x) = 0[/math].

Then :
[eqn]
(x - y, x -y) = (x, x) + (y, y) \\
||x - y||^2 = ||x||^2 + ||y||^2 \\
||x - y|| = (||x||^2+||y||^2)^{\frac{1}{2}} \\
[/eqn]

As expected.

>>11770093
>Learn differentiation
Yikes. What would you even need to memorize? There's literally two rules. Just apply chain rule & product rule mechanically....

>>11770093
You just need to know a few rules and then practice, practice, until it becomes almost automatic. It's a boring part of math, for sure, but it trains you to not be spooked by tons of symbols and focus on the deeper meaning of books rather than calculation steps.

/gmmg/

>>11769335
Anime posters should be b&.

>>11769341
Check these out:
https://www.elsevier.com/books/representation-theory-of-finite-groups/burrow/978-0-12-146356-4
http://math.mit.edu/~etingof/replect.pdf

>>11769367
They are nice.

>>11769408
Maybe the anon misses times when we were less cold and bitter.

>>11769871
If you consider reasoning in general a related skill, then yes.

>>11769943
It is a sadomasochistic endeavour leading to certain pain and possible pleasure.

>>11770093
Don't memorise them. Derive them.

>>11770119
There's basic derivation rules and rules for the following functions; exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic and inverse hyperbolic.
I have already come across applications of the derivation of exponential functions, and I was wondering where a^x = a^x ln a came from.

Where do all these derivation rules actually come from?

>>11770132
The only thing you'll ever need to learn is that [math]f(g(x))' = f(g(x))'*g'(x)[/math], which you can prove just by using Newton's formula.

https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx

Look at this and then try your hand at other functions.

gmmg

>>11769334
Why is the book reflecting? Very funky.
Is there a real book just called mathematics?

Also, yesterday I bought Converse of that color.

>>11770093
There's really only the chain rule.

>>11770129
Thanks for the book links.

>1st ed. 2019 Edition
What's your relation to those works?

This is a nice book:
http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf

Also,
https://youtu.be/3-XxmQv9gPY

>>11770134
Mistake, I meant
[math]f(g(x))' = f'(g(x))*g'(x)[/math]

>>11770135
>What's your relation to those works?
I've looked at both in the past for some facts and they seemed good. Maybe they are completely useless for learning, but at least they they were recommended online. I don't know. Nice shoes.

>>11770137
\cdot, you meant.

>>11770137
So I can get to all derivations through the chain rule?

>>11770132
>Where do all these derivation rules actually come from?
They come from solving the definition of a derivitive for each function.
Try this video:
>https://m.youtube.com/watch?v=WUvTyaaNkzM
and the play list it corresponds to. You just have to memorise the rules through practice but this video will give you an more visual understand of how those rules were derived

>>11770152
Chain rule and product rule for 90% of cases. The remaining 10% are for arcsin and other things like that.

>>11770152
He's talking about rules like the quotient rule, integrating by parts, substitution... They come from the chain rule, but he's being an arse. You'd still want to use those rules when applicable rather than the basic chain rule.

He's also misunderstanding your question, you want to understand where the basic derivations come from. Like I said here >>11770154
They come from solving
dy/dx = (f(x+h) - f(x)) /h
For each function (eg, x^a, sinx, srt(x) etc), some of them are quite involved.

reminder to take the langpill

>>11770166
So looking at some proofs would show that the rules work and where they come from?

>>11770177
Yes. For example, the rule for the product follows easily if you notice that f(x+h)g(x+h) - f(x)g(x) = f(x+h)g(x+h) - f(x)g(x+h) + f(x)g(x+h) - f(x)g(x) and then split the limit of that into the sum of the limits of f(x+h)g(x+h) - f(x)g(x+h) and f(x)g(x+h) - f(x)g(x). When h approaches 0, you end up getting f'(x)g(x)+f(x)g'(x). Check the details out as an exercise.

>>11770186
hmm. Okay I will look tomorrow.
First I will watch the 3blue1brown calculus series I think as that seems better for some intuition.

>>11769342
representation theory of dubs

>>11770069
The answer to your example is NO.
Choose an orthonormal basis [math](e_i)_{i \in I}[/math]. I will denote by [math]v \otimes w[/math] the unique operator [math]A \in B(\mathcal{H})[/math] given by the formula [math]Ax = \langle x, w \rangle v[/math].
Then, let
[math]A_k = (-1 + \frac{1}{k})e_1 \otimes e_1 - e_k \otimes e_k[/math],
[math]S = \{A_k : k \geq 1 \}[/math],
Then [math]T = -e_1 \otimes e_1[/math] does not belong to [math]S+P[/math], but it belongs to the closure.

>>11770198
>tomorrow
Anon... It's half past ten, you are not taking a day off, are you?!

>>11770199
Nice. Too bad they weren't the irreducible dubs 11.

>>11770209
Not here anon.
by the time I finish the videos it will be bed time

>>11770199
Checked.

>>11770215
Grab your teddy bear, go to bed and watch those videos together. Try not to wake up from your phone falling on your face. I got a nice bruise around my eye when that happened.

>>11770225
It's too early yet...
But I'm having some /comfy/ drinks while watching them.
So I might watch the last of it later in bed.

>>11770228
Yeah, it's a bit hard to guess what the time is in East Mongolia or some other country around that like Australia maybe. However, I recommend you do these tomorrow:
>prove the sum formula (f+g)' = f'+g'
>prove the product formula (fg)' = f'g + fg'
>use the product formula to get the formula (af)' = af'
(where a is a constant)
>use the previous formula and the sum formula to get the formula for difference (f-g)' = f' - g'
>prove the division formula (f/g)' = (f'g - fg')/g^2
>check how the chain rule is derived
Those will help you when you need to differentiate stuff like [math]e^{x^2 + x}[/math]. Then you just have to go look for ways to obtain the derivatives of some concrete functions like x^n, e^x, sin(x), etc. The first one is easy to get directly from the definition, but the rest will require some trickery, so maybe check the details and try to understand what is going on.

Did you know Feynman was a fan of the Tuvan republic? Well now you know. Also, reindeer are cute https://www.youtube.com/watch?v=O5DCxaGy3wY

Is animeposter a femboy?

>>11770200
S needs to be closed

>>11770249
The animeposter irl. Essentially middle aged and east euro so by necessity Kyriakos Grizzly himself

>>11770253
it is

>>11770253
Sneed?

How do prime ideals behave in a tensor product? If [math]B\otimes_A C[/math] for rings [math]A,B,C[/math] and algebras [math]A\to B,C[/math], is there a neat way to see how the primes relate to the primes in either of the rings [math]A,B,C[/math]?

>>11770245
Definitely East Mongolia.
But yes I will give a few of these ago tomorrow.

>>11769296
is it classified how many prime ideals a ring can have? for example, is there a ring that has exactly 2 prime ideals? (note that prime ideals have to be proper ideals)

>>11770245
Mongolian music is very based.

>>11770368
That's interesting.
Unrelated to your question.
But can mathematics have a security classification?

>>11770364
Great!

>>11770370
Indeed it is, but I like the Tuvan style more for some reason (for example, Huun-Huur-Tu and Alash are from Tuva but Khusugtun is from Mongolia). It could be the influence of Slavic music as Tuva is one of the many states of the Russian federation. Dunno. This guy is one of the first ones I heard, and I still love his voice https://www.youtube.com/watch?v=HwANedEkqaY

>>11770368
For the first question: no idea. For the second one, I think you could take a field [math]k[/math], and then [math]R = k[x, y]/ \langle x^2, y^2\rangle[/math] should have as its prime ideals [math]kx, ky[/math], but I may be mistaken.

>>11770249
Writing fanfic, eh? You can insert me in your story as anything. I can be a big bulky guy like Kyriakos, I can be a feeble femboy, a princess, a platypus, a Lovecraftian being without a form describable by human languages. The world is yours to create!

>>11770373
>But can mathematics have a security classification?
The NSA has probably backdoors to some cryptographic systems, these inevitably will rely on mathematics unknown to the rest of the world and most definitely under very strict security measures.

>>11770397
It seems to be the best gym music.
I actually think this would be fantastic mixed with a chello or something.

>>11770368
No, there are many rings with the same amount of ideals that are non-isomorphic. For example, [math]\mathbb Z/9\mathbb Z[/math] only has prime ideal [math]3\mathbb Z/9\mathbb Z[/math]. Similarly, [math]\mathbb Z/p^2\mathbb Z[/math] only has prime ideal [math]p\mathbb Z/p^2\mathbb Z[/math] for every prime. None of these can be isomorphic or even have homomorphisms between them for obvious reasons.

More interesting examples can occur if the ring is infinite. For example, we know that the intersection of all primes is the nilradical, so any non-reduced ring [math]R[/math] has exactly the same primes as [math]R/\sqrt R[/math], and in fact, it is also so for every ideal [math]0\subset I\subset \sqrt R[/math] that [math]R/I[/math] has the same primes. So for example, in some non-Noetherian ring, it can happen that there are infinitely many such [math]I[/math], thus giving a family of rings with the same spectrum. For this reason, [math]\text{Spec}[/math] alone is not enough to give an equivalence of categories between commutative rings and affine schemes, and thus you also need a sheaf of rings over each spectrum to achieve the equivalence.

>>11770423
>NSA has probably backdoors
That's a big if.
Maybe?
But you're better of relying on implementation flaws then mathematical cryptographic flaws.
I would say the NSA relies on implementation flaws, but who really knows.

>>11770175
Dated. You had better not read him about advanced topics. I like his algebra book and his analysis textbooks (not listed here: Analysis I and Analysis II) but his books about algebraic geometry (eg. Abelian Varieties) use Weil's language and are virtually unreadable.

>>11770427
Cello and morin khuur/igil are the same instrument (up to homotopy), so it could very well work. Here you have a violin, although it is under some effects (and its player seems to be under the influence of something as well) https://www.youtube.com/watch?v=IVyqMj5Zhy4
and here accordion https://www.youtube.com/watch?v=QzAI1ISxoUg
Thanks for reminding me of all the nice walks to the uni and back while listening to stuff like this. There was a nice park along the way with a bridge, and ducks underneath it. Sometimes I'd buy some cucumber on my way home would feed its chunks to the ducks while explaining different concepts and proofs to them. Good times. But yeah, getting a bit off topic and soon some angry janny will punish us.

>>11770368
Usually, they will have infinitely many, but you can play with localizations, products and quotients to get whatever finite number of prime ideals you want.
For example, it is easy to find rings with only one prime ideal: Take any field [math]k[math] Then, taking the product of it with itself will yield a ring with exactly two prime ideals [math]k \times \{0\}[/math] and [math]\{0\}\times k[/math].
Otherwise, you can take an integral domain of dimension 1 and localize at a maximal ideal: For example [math]k[t]_{(t)}[/math] has exactly two primes: [math](0)[/math] and [math](t)[/math].

>>11770437
>But you're better of relying on implementation flaws then mathematical cryptographic flaws.
But the NSA does *not* control the implementation and if the implementation is free software it becomes really hard to hide something.
At any given point some person might look closely enough and you are fucked.

>But you're better of relying on implementation flaws then mathematical cryptographic flaws.
This seems just totally false.

>That's a big if.
I would say that it is a near certainty.

For some cryptographic systems the NSA is the one who published "random" numbers which it wanted to be used in implementations.
Obviously nothing suspicious about those numbers has been found, but this doesn't mean they are actually secure.

>>11770459
Music can be explained with mathematics so I would argue it's very related.
When you look at Music and the complexity that's required to come up with a unique piece it's actually quite interesting that people can come up with complex music without completely understanding those principles.

A while ago I tried making a music player and editor, but it turned out I was lacking an understanding of the underlying physical properties of waves to make a decent one that would also interface with instruments, one of the reasons I'm here now.

Fuck jannies

If I have a space X and a continuous map f:Xx[0,1] -> R^n such that f(x,0) is an embedding, is it true that X is homotopy equivalent to the image of f at t=1? If not, is there a similar statement that's true?

>>11770468
>This seems just totally false.
Mathematical cryptographic flaws are sure to undo any cryptographic algorithm, but are far less common and the amount of people that have that knowledge to do this is far less than finding flaws in an implementation.
So it comes down to a risk analysis. Impact times likelihood.
It's far more likely that someone has botched an implementation of asymmetric crypto then that asymmetric crypto is flawed. So you devote your avenue of attack towards finding implementation flaws over finding mathematical flaws, given that finding mathematical flaws has already been tried and your trusted person has said it can't be done, P != NP.
But finding a flaw in the protocol is far more valuable.
Sure the NSA could have done that, but that's only true of people, companies, organisations etc etc, that use that library of the implementation.
That is an implementation based exploit, as they know the number from which the private keys are generated and potentially able to spy on people.
The idea of that cryptographic system is still secure, even if the NSA release numbers that libraries are based on.

>>11770499
>Sure the NSA could have done that
I mean they could have generated the specific numbers to use.

>>11770475
>Music can be explained with mathematics so I would argue it's very related.
It can, indeed. There is actually a book connecting music to topos theory, but I can't say I'd understand toposes enough or music theory at all, so I don't know if that is good on any level.
>A while ago I tried making a music player and editor, but it turned out I was lacking an understanding of the underlying physical properties of waves to make a decent one that would also interface with instruments, one of the reasons I'm here now.
We are talking about a software, right? Care to explain to a dumbo how the physical properties are reflected by bits and bytes? I mean, sure. I can think of it like an algebraic topologist would and say that all the structure in this mp3/flac/your favourite audio file type comes from the physical instruments that are making noise, but at the same time I could just use a computer to construct the same file without the need of anyone actually playing anything. In the latter case, I would not have the limitations given by the physical world, so would I end up getting the similar kind of resonance and interference I would get if I would tell a hundred people to play the same thing with a flute and record that? Not even baiting, I am literally a potato when it comes to things like this.

>>11770497
Let X be the interval [-1, 1] with 0 removed, let f(-, 0) be the inclusion, and let f(x, t) = (1-t)f(x, 0). What is the image of f(-, 1) then? What are the path components of the images of f(-, 0) and f(-, 1)? Something to think about!

>>11770506
This obviously for n=1.

Just discovered a new proof of a^p = a (mod p):
1^p = 1
(a+b)^p = a^p + b^p (mod p) because all binomial coefficients are divisible by p.
So a^p = (1+...+1)^p = 1^p + ... + 1^p = 1+..+ 1 = a (mod p).

>>11770499
Your post seems entirely incoherent, I really do not get what you are trying to say.
Do you have any actual argument?
I mean, there could be reasonable arguments people could make, like that the NSA also has an interest in creating secure communications, or that surely any such flaw would have been found by other researchers.

>So you devote your avenue of attack towards finding implementation flaws over finding mathematical flaws, given that finding mathematical flaws has already been tried and your trusted person has said it can't be done, P != NP.
If you are an extremely well funded organization you obviously do both.
What even is this argument.
The NSA isn't a single person who has to decide whether to spend it's time on finding an exploit in some implementation or to investigate the mathematics of the algorithm in search for possible exploits.

>and your trusted person has said it can't be done
What?
No sane person has said "it can't be done". Do you have any idea of how this works?
Nobody knows whether there is an actual flaw, there is no theorem to exclude that.

>P != NP.
What the fuck am I reading?

>But finding a flaw in the protocol is far more valuable.
Implementation exploits are nearly useless for the goals of the NSA.
They only apply to the people who actually use the implementation and they might very well be gone tomorrow.

>Sure the NSA could have done that, but that's only true of people, companies, organisations etc etc, that use that library of the implementation.
This sentence is incoherent.

>That is an implementation based exploit
What is?

>The idea of that cryptographic system is still secure, even if the NSA release numbers that libraries are based on.
That is pure speculation on your part.

>>11770500
>I mean they could have generated the specific numbers to use.
What?
They *did* generate the numbers and told people to use them.
This is not a hypothetical, they are the ones who are creating cryptographic standards.

>>11770513
nice one

>>11770513
>Just discovered a new proof of a^p = a (mod p)
a=2, p=2
2^2 = 4
2 (mod 2) = 0

:^)

>>11770436
perhaps I should have phrased my question a bit better. I meant to ask if we know which numbers can occur as the number of prime ideals of a ring

>>11770533
Z/(p1p2...pk) has k prime ideals

>>11770506
>software
yes.
>physical properties are reflected by bits and bytes
This is interesting because reading and then converting it into binary is for music is very similar to how internet connections work on a computer.
You have an analog instrument making string vibrations, the magic bit of hardware that converts those vibrations into electrical pulses. Those pulses range from low to high voltage at a particular frequency. Then the amp then converts those voltages into sounds. I'm probably over simplifying but it doesn't matter.

It's pretty much the same for a computer. You're trying to get those analog signals on some sort of input, convert those high low voltages to a 1 or 0 format, and then identify key things with that ""packet"" of analog signal, to make it do what you need it to do (I don't really understand how to do that).
If you know how a computer works, you will know that electrical binary signals are high or low voltage, which is pretty much what an electric instrument is doing anyway, high or low vibrations converted into high or low voltage.
As for mp3, flac etc, they are just encodings, so they are away to arrange the bits to make a sound. So if I recorded the pure bits from a guitar and then ran an encoding or compression algorithm on it for say a mp3 format, it would do some operations to convert it into the approximate sound in a smaller format (by rearranging the bits), with some loss in what it is supposed to be. Flac attempts to do the same thing, but without a loss in quality, but usually results in larger file sizes.
So the answer to your question is that if I could arrange the bits in a manner that follows some musical piece, encode the frequency that it is supposed to play at, then encode the sound it is supposed to make (woody, etc) and then write software to interpret the "packet", then you could digitally play the guitar. But as you notice with music software, it's really hard to encode an accurate sound of what it sounds like.

>>11770467
thanks, that answered my question

>>11770542
yes, I see

>>11770517
>"it can't be done"
I'm not sure if you're aware, but asymmetric cryptography is based on prime factorisation.
https://en.wikipedia.org/wiki/RSA_Factoring_Challenge
Using specific numbers is implementation based. The idea of it being secure is that it becomes (possibly) impossible the compute those numbers the higher your go.
P != NP. This is why RSA Is supposed to be secure... but it might not be as you mentioned.
>Sure the NSA could have done that, but that's only true of people, companies, organisations etc etc, that use that library of the implementation.
Using specific numbers is an implementation of that cryptographic system.
If I was to write my own library using different numbers, it would be a different implementation.
If the NSA have provided numbers or an algorithm that people use, it is an implementation. The NSA didn't invent RSA, but they might have made an implementation in the form of a library that individuals use.

Maybe the NSA are behind RSA, I don't know.

>>11770115
Didn't even know Lee had a book on that, intredasting. Will check it out. His two other GTMs, Introduction to Smooth manifolds & Riemannian Manifolds are great, but not optimal to start out with. Not because it's like nothing is explained or motivated, think Hartshorne style, but because the first book (actually the second, from 2003) might make a "synthetic" impression on someone who doesn't already know differential geometry. By synthetic I mean that the goal, namely the geometry, obfusciated by roughly 600 pages of definitions and abstract vector bundle stuff.

>>11770568
>I'm not sure if you're aware, but asymmetric cryptography is based on prime factorisation.
That is just false.
There are many different schemes, e.g. ECC is becoming quite popular as it gives you basically the same security for vastly shorter key lengths.
Literally learn a tiny bit about cryptography before talking about it.

>Using specific numbers is implementation based.
That is false.
The NSA has standards for Cryptography, this standards gives guidelines on the implementations.
The numbers are part of the standard, they are totally implementation independent.
But even if they were SO FUCKING WHAT?
WHAT THE FUCK WOULD IT MATTER?
WHO GIVES A FUCK?

>The idea of it being secure is that it becomes (possibly) impossible the compute those numbers the higher your go.
Thanks for teaching me cryptography 101 GEEZ I really had no Idea!!!!
You mean to tell me that is how cryptography works?????????

>P != NP.
You have no clue of how computing works.
P != NP is:
- Just a claim of yours
- Even if it were true it means LITERALLY NOTHING about any particular algorithm
You do not understand a tiny bit about computation spouting this nonsense.

>If I was to write my own library using different numbers, it would be a different implementation.
Are you on drugs?

>The NSA didn't invent RSA, but they might have made an implementation in the form of a library that individuals use.
THEY CREATED A STANDARD. THEY SAID "THIS IS HOW YOUR IMPLEMENTATION SHOULD LOOK LIKE".
This debate you are having in your head is fully retarded and missing the fucking point so fucking hard.

>Maybe the NSA are behind RSA, I don't know.
YOU ARE AN IDIOT.
RSA STANDS FOR Rivest Shamir Adleman. THE NAME LITERALLY TELLS YOU WHO DEVELOPED IT.


Why am I even talking to you? You clearly know nothing about cryptography.

>>11770599
big seethe

>>11770545
I see, thanks for the explanation. Have you made any progress? I know the signal analysts like Fourier, but you know a lot more about this than I do, which means that you already knew that. Does it help you in any way here? It would actually be pretty cool if you could somehow separate different instruments from the track, but that probably requires a bit more than a Fourier transformation... Suppose you could characterise sound sufficiently well using for example frequency, volume and for example direction if you had a recording sophisticated enough. Then you would get vectors of the form (f, v, d) which would give you a bit more wiggle room and maybe allow distinguishing between different overlapping sounds. However, as mentioned before, I'm a potato when it comes to this, so this idea can be already tried, completely useless or already tried and found completely useless. The volume parameter probably would be replaced by something else due to it being a bit questionable.
>>11770545
>But as you notice with music software, it's really hard to encode an accurate sound of what it sounds like.
Yep, and because of this for example the Hauptwerk organs use recorded sounds from different place. When you specify that you want to be in Toulouse, the sound you get is not synthesised but actually individual sounds recorded in the Toulouse cathedral. It works with big instruments like these, but when you want to go full jazz fusion without actually touching a guitar and have to either use similar recordings and be a prisoner of their sound, or use the horribly terrible synthesised guitars which could allow you to make shoegaze if you just pretend it sounds so off because you were using so many effect pedals.

>>11770568
And I mean. Again. Do you have any actual argument here?

The claim is "the NSA has secret mathematical knowledge about cryptographic algorithms".
You seem to heavily disagree with that but yet you have not made a coherent argument against it.

>>11770605
Oops double reply.

>>11770601
>big seethe
I am indeed seething.

If you talk to somebody who makes very big claims but literally has not even a "I read the first paragraph on Wikipedia" tier knowledge about the subject, then you have a good reason to be upset.

>>11770506
>book connecting music to topos theory
Which book?

>>11770599
>That is just false.
Soz I meant RSA specifically.
>guidelines
Exactly. Guidelines are an implementation (lol).

>P != NP
It's the principle, but I get that it's not really how it works out.

>Are you on drugs?
Alcohol is technically a drug.

>THEY CREATED A STANDARD
>IMPLEMENTATION SHOULD LOOK LIKE
Exactly. So you agree with me that it doesn't have to be done the way the NSA specify. Hence, making a library using the NSA specification is an implementation.

>Rivest Shamir Adleman
Who was from MIT?

You're really salty over nothing.
Maybe you should talk to someone about it.

>>11770618
I can't say anything about its contents, but it exists https://www.amazon.co.uk/Topos-Music-Geometric-Concepts-Performance/dp/3764357312

>>11770607
I'm not arguing with you, not initially anyway.
Consider that the NSA might not be able to hack your Asymmetric cryptographic systems.
Symmetric is different as that is just a matter of finding someone's password (sort of).

NSA's standards are not dependent on RSA or any cryptographic system being secure. That is my new argument seeing as you want to accuse me of being an idiot.

>>11770621
>Guidelines are an implementation (lol).
No.
An implementation is a specific program.
But this LITERALLY does not matter.

>It's the principle
NO. IT IS NOT A PRINCIPLE. IT IS A CLAIM ABOUT COMPUTATIONAL COMPLEXITY BETWEEN DIFFERENT CLASSES OF ALGORITHMS.
IT LITERALLY IS IRRELEVANT TO THIS DISCUSSION.

>Alcohol is technically a drug.
It isn't "technically a drug" it literally is a prime example of a psychoactive substance.
It also explains a lot why you are unable to form any coherent thought.

>So you agree with me that it doesn't have to be done the way the NSA specify.
NOTHING HAS TO BE DONE THE WAY THE NSA SPECIFIES IT.
You could literally use Casear encryption if you wanted to.
But again, this couldn't possibly be less relevant.

>>11770631
>Consider that the NSA might not be able to hack your Asymmetric cryptographic systems.
>Symmetric is different as that is just a matter of finding someone's password (sort of).
You literally know nothing about cryptography, right?

>NSA's standards are not dependent on RSA or any cryptographic system being secure.
And?
Yes, the NSA saying "you should do X" does not guarantee that X is secure, what of it?

test

>>11770640
low

>>11770605
>made any progress
It wasn't a major goal of mine, so I did the research and took some notes and decided my time could be better spent.
But I might have another go once my mathematics and physics knowledge is better.
It's been done plenty of times I'm sure, but in programming you find that solving for a particular problem, has applications in other areas. Like the comparison I drew between internet communications and analog signals.

>>11770628
>£159.99
Keeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeek

Btw, you know any other books that links math to music?

>>11770635
>Guidelines
Guidelines tell you how you should do something. You should use these numbers, for example.
I don't have to do it that way.
I would agree that most people do it following the guidelines, but there are implementations that don't.

>A CLAIM ABOUT COMPUTATIONAL COMPLEXITY BETWEEN DIFFERENT CLASSES OF ALGORITHMS
Seeing as we're both using wikipedia.
>It asks whether every problem whose solution can be quickly verified can also be solved quickly
So factoring large primes is a "P != NP?????" problem. You're just being obtuse.

>literally is a prime example of a psychoactive substance
You're a prime example of autism

>HAS TO BE DONE THE WAY THE NSA SPECIFIES IT
No it doesn't.

>the NSA saying "you should do X" does not guarantee that X is secure
Yes. So don't use NSA implementations """STANDARDS""" if you are skeptical?
I for one completely trust my fellow United States citizens.

>>11770628
This is an outdated edition. It's been republished as a 1600-page 4-part brick.

https://www.springer.com/gp/book/9783319644332

It also looks like mostly academic fart-huffing nonsense. They spend entire chapters doing some kind of topology of hand gestures made by musicians.

>>11770638
What book do you recommend if I wanna learn advanced cryptography like you?

>>11770643
Gotcha. The good thing about maths is that many seemingly different things speak the same language, as we can see from the difference and similarity of music and internet.

>>11770646
There's a site where you can get a £159.99 discount on that. There is also a treatise by Euler, but it's in Latin. Are you up to the task?

>>11770655
Is this what you get when you go over board with category theory while wanting to be the next Deleuze and Guattari?

>>11770651
>we're both using wikipedia.
Where did I use Wikipedia?
My knowledge of cryptography comes from a Uni course I did in it where we derived things like RSA and especially ECC.

>I would agree that most people do it following the guidelines, but there are implementations that don't.
The guidelines still are not the implementation...

>>It asks whether every problem whose solution can be quickly verified can also be solved quickly
Please, for the love of god, if you have no clue about something do not take the extremely dumbed down Wikipedia literally.

>So factoring large primes is a "P != NP?????" problem.
P being not equal to NP says LITERALLY NOTHING about how hard it is to factor primes.

>No it doesn't.
You are trolling. How can you ignore the "Nothing" in front of my sentence?

>So don't use NSA implementations """STANDARDS""" if you are skeptical?
That is not how this works.

>>11770651
>Seeing as we're both using wikipedia.
I am unhappy that I share this thread with people who assume it's impossible to know any mathematics that you didn't just google and copy/paste

>>11770656
The book my professor recommended was "Elliptic Curves - Number Theory and Cryptography" by Washington.
I used it as preparation for my exam and I have to say that I found it to be pretty nice, although it obviously focuses on ECC.

I can't remember every getting anything out reading any online-discussion on P vs NP.

>>11770671
The only thing you need to know that P and NP have absolutely nothing to do with actually being hard/easy to solve.

>>11770655
what in the fuck

>>11770665
>There is also a treatise by Euler, but it's in Latin.
What's it called? I plan on learning latin by my 30s.

Why do Math Stack Exchange users seethe so much in general? There is no online community that seethes as much as they do. I thought mathematicians were rational thinkers.

>>11770686
Give examples. I dont hang out there much.

>>11770686
What do you mean? I honestly think there are better users and answers there than here. The rank of math forums is like:
1 - Math stack exchange
2 - Physics Forum
3 - maybe /mg/

>>11770666
>P != NP
your statement about this problem and how it doesn't apply to RSA, is wrong. I probably shouldn't have stated Asymmetric cryptography generally for primes, but that's either because you're unable to read between the lines or you're nitpicking.
I will admit I have no idea what the NSA provides in terms of numbers or libraries, I only know that a lot of libraries do use their standards.
>The guidelines still are not the implementation
Fine. There was literally no reason to loose your cool over distinctions like this. Guidelines are still away of determining an implementation.
>P being not equal to NP says LITERALLY NOTHING about how hard it is to factor primes.
The concept behind using large primes is based on the principle that some problems cannot be be calculated in polynomial time.
So using P != NP is descriptive of what a certain type of cryptography is trying to achieve.
>You are trolling
Yes. But you're also looking for an argument.

>That is not how this works.
Why?

>>11770668
There's literally nothing wrong with using wikis or websites as primers or confidence boosters to reaffirm what you already know.

>>11770679
Latin is a nice language. I was thinking about relearning it at some point. Here you go:
https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1032&context=euler-works
>in my 30s
So many years over 24... You are fighting against the very biology!

>>11770686
There are the actually good people like the pros and Tyrone, but then you have some over enthusiastic undergrad notation/terminology autism keyboard warriors who will start bombarding you with personal insults if you don't use the same words or notation as they would. I got so disappointed at those people that I actually deleted my account. So much cringe.

>>11770699
>confidence boosters to reaffirm what you already know.
sure
>as primers
there is something very wrong with pretending to be discussing something when you're just skimming the wikipedia article for something to say

>>11770687
https://math.stackexchange.com/a/563063/135291

first example that comes to mind - some comments have been deleted unfortunately. All of cleo's answers and comments from other people are like that.

But if you go there enough, you'll see people constantly complaining about downvotes, many meta questions related to them and overall extreme seethe

>>11770699
Yeah, even wikipedia has several sources for all their articles these days

>>11769296
Anyone else reviewing for the mGRE? The first exam is in about 100 days.

>>11770686
>>11770701
The question police are the ones that get me more than those. You absolutely cannot post an open-ended question on stackexchange, even if it's well thought-out and extremely interesting, without getting bombarded with seethe and downvotes about your question is "too broad" and "should be moved". This happens to everybody, even world-famous professors get shit on when they try to ask non-objective questions.

>>11770702
Oh so you agree.
There's nothing more to be said. Great.

What's a good introductory book for number theory? Can assume any algebra, topo, analysis background.

>>11770687
>>11770704
kek this is just too good

>>11770716
Serre, A Course in Arithmetic

>>11770701
>relearning
Kek, you either know or you don't know, you can't "relearn" a language

>>11770721
This is certainly a nice book, but, like, what would be a good book beyond that one?

>>11770686
Active posters in math.SE are rarely working mathematicians, they're mostly people with way too much free time. MathOverflow on the other hand has very knowledgeable people but they're also anal about open-endedness; it's something to do with SE's format rather than the communities in particular — it's a Q&A site and not a forum first and foremost.

>>11770710
100 days of practicing calculus is major overkill. You really don't need more than a month of dedicated practice (meaning a couple hours a day) on the mGRE to get above 80th, and realistically as far as admissions are concerned, everything above 80 is basically the same score. There's no point in going full chinkmode and wasting your entire summer for a 970.

>>11770719
Based Cleo

>>11770725
Weil, Basic Number Theory
Davenport, Multiplicative number theory

>>11770692
>your statement about this problem and how it doesn't apply to RSA, is wrong.
No it fucking isn't. Explain to why, if N /= NP that would mean anything.

>The concept behind using large primes is based on the principle that some problems cannot be be calculated in polynomial time.
No. The idea is to have a "one way function" a function which is computationally very expensive to reverse.
The actual growth rates might (and do) come into play, but they are no the general principle.
If you could do integer factorization for an n bit number in O(n^100000) that would mean nothing.

>So using P != NP is descriptive of what a certain type of cryptography is trying to achieve.
No.

This literally just isn't true. You can repeat it 10 times more, but it is false. FALSE.

>>11770735
Thanks!

>>11770725
Number Theory - Henri Cohen is one of the TSSB (Three Sacred Springer Books) together with Gorodentsev's Algebra and Fuchs-Fomenko's Homotopical Topology.

>>11770722
I haven't used it since 2013, so I have it in me, yet buried under a lot of other stuff. Something to do after I get out of school.

>>11770719
Must you hurt my brain with this?

>>11770742
>Cohen
Yet another new Cohen. Just how many are there?

>>11770728
I dunno, my application seems pretty shit otherwise.

>>11770736
>No it fucking isn't. Explain to why, if N /= NP that would mean anything.
>computationally very expensive to reverse
Exactly, N != NP, ?????
It's computationally expensive to reverse, but if you where to apply a prime to the cryptographic algorithm of increasing size, the harder and harder it is to reverse. N != NP (?)
And why the NSA is using an implementation of an algorithm by choosing specific numbers to input into it. Don't use those numbers is all I'm saying?
Maybe there's a reason you have to use those numbers other than their guidelines, but then that would be a flaw in the standard itself, if it were reversible by the NSA or anyone.

>No.
Yes?
https://projecteuler.net/problem=3

>but it is false. FALSE.
so you say.

>>11770742
Have you read it? Is it good? I looked it up and it's very long.

>>11770755
Then I'm sorry to inform you that your application is still going to be shit. The GRE is a pretty minor factor. It's mostly just to make sure that if they assign you to TA calculus you can actually do it without fucking up.
People don't get into schools based on "hey that guy scored 93rd on the GRE, get him"

>>11770752
>Yet another new Cohen. Just how many are there?
Yikes! Yet another neonazi implying with jewish authors, we don't need your kind here

>>11770773
I didn't, but it was recommended to me here long ago by someone who said it was a god tier book.

>>11770784
Why are you recommending books you haven't read?

>>11770765
>It's computationally expensive to reverse, but if you where to apply a prime to the cryptographic algorithm of increasing size, the harder and harder it is to reverse. N != NP (?)
This is incoherent.

>And why the NSA is using
Any evidence that the NSA is?

>Don't use those numbers is all I'm saying?
I am not implementing cryptographic software.

>Maybe there's a reason you have to use those numbers other than their guidelines
There isn't. Certainly not a reason anybody knows besides the NSA.

>https://projecteuler.net/problem=3
And?

>>11770778
>implying with jewish authors
Implying what?

>>11770785
I wasn't recommending shit, I was memeing.

>>11770778
>>11770789
this is /pol/ bait.
Freaking out when someone merely points out a name that is jewish is a common tactic.

>>11770773
The ratio of "posts about books" to "actually read the book" here is probably <5%, because you can give book recommendations without having a single scrap of a clue what you're talking about.
Namedropping big fat yellow books that take a year to read is a great way to try to appear knowledge without posting any mathematics.

>>11770792
Fuck off Nazi.

>>11770790
>act like a retard
>get called out for being a retard
>"j-just meming you bro"
kill yourself animenigger

>>11770778
>another neonazi
Please don't expose me to my supervisor! I need him to pay my rent.
>we don't need your kind here
Well this true, but for some reason I'm addicted to these threads.

>>11770773
It is good. Covers a lot of stuff and has good exercises & beautiful pictures.

>>11770789
Probably that finding a paper written by a Cohen is somewhat difficult when there are 5 or more wrong Cohens popping up. Unless you know a co-author or the name of the paper.

>>11770802
>Probably that finding a paper written by a Cohen is somewhat difficult when there are 5 or more wrong Cohens popping up. Unless you know a co-author or the name of the paper.
When would this ever happen?
People don't write their bibliographies as
>[4] Cohen (no first name) some paper somewhere idk

>>11770801
Kek, go back to mathstack exchange, retard. If I wanted to trick anyone I could have just said 'Yeah, I did read it, it's pretty good', not my fault you're too dumb to think a post calling Springer books sacred with an anime girl image is a serious post.

>>11770716
You've gotten quite a few meme answers. None of those are really introductory, except maybe to some degree, Serre.

A classical introduction to modern number theory - Ireland/Rosen is very nice
Algebraic number theory and Fermat's last theorem - Stewart/Tall is also good, but assumes at least some knowledge in elementary number theory (can be picked up relatively fast though)

As for analytical number theory, I don't know much, but heard Apostol is very good but very hard in the exercises.

>>11770809
>embarrassed tranny starts furiously seething

>>11770809
>>11770814
Stop fighting.

>>11770808
When my supervisor tells me that there is a paper by X, he doesn't specify too much. Any guesses how hard it was to find "a paper by James"? That made finding "this one paper by Cohen" look easy. At some point I decided that it is easier to just ask if he could send the papers or at least their titles to me.

>>11770786
>a prime to the cryptographic algorithm of increasing size
Me use big big prime as input for crypto thingy that does number thing. It get hard for big big prime.
>Any evidence that the NSA is?
I assume NSA use RSA. But they may not because it could be insecure as you say.
Who really knows.
>I am not implementing cryptographic software.
Your claim is that the NSA do and they have the "backdoor" to all RSA because of this.
The only way they would know your AES plaintext is if they know your key, or maybe have the computation power to brute force to get to your key.
>There isn't.
Then don't use those numbers lmao. That's my whole point.
Great, the NSA provide numbers. That doesn't mean RSA or any other crypto algorithm is insecure. Which is another point I made, which I did make vague to wind you up a bit.
>And?
It explains the point about using large primes.
If you have solved this problem you know that it's quite expensive, and that increasing large primes get hard to calculate through a brute force method (the only known way currently). So calculating infinitely large prime and knowing it's factors is a P != NP (?) problem.

Let me take you back: I said >Using specific numbers is implementation based.
You said
>That is false. The NSA has standards for Cryptography
But you don't agree that the standards (using specific numbers) that have been used to make libraries for crypto, are optional.
Which you further indicated that they are.

National banks for example use their own cryptographic standards and implementations that differ from the NSA.
So you can avoid using the NSA systems by adopting a different cryptographic standard resulting in a different implementation.
(decent) Banks implement their own crypto.
notice how I said implement? Well that's because algorithms are general and not specific. Meaning how I adopt RSA to my bank might use some variations in the components in RSA.

>>11770822
>At some point I decided that it is easier to just ask if he could send the papers or at least their titles to me.
Why was this not your immediate reaction? At least ask for the first name or something, fucking hell.
Even if he doesn't remember the precise title of the paper offhand he can at least tell you it was written by Gordon James sometime back in the 80s or something similar.

>>11770830
>So calculating infinitely large prime and knowing it's factors
I mean the other way around soz

>>11770830
>It get hard for big big prime.
And?
That is true for polynomial algorithms as well.

>Your claim is that the NSA do
No, they create standards.

>and they have the "backdoor" to all RSA because of this.
I didn't mention RSA before you brought it up.

>The only way they would know your AES plaintext is if they know your key
What has AES to do with this now?

>or maybe have the computation power to brute force to get to your key.
Or maybe they know a method to do better than brute force...

>Then don't use those numbers lmao
I DO NOT IMPLEMENT THE SOFTWARE.

>That doesn't mean RSA or any other crypto algorithm is insecure.
Yes, so what?

>If you have solved this problem you know that it's quite expensive
Under half a second on my phone.
It's quite trivial compared to many polynomial problems.

>So calculating infinitely large prime and knowing it's factors is a P != NP (?) problem.
Incoherent.

>But you don't agree that the standards (using specific numbers) that have been used to make libraries for crypto, are optional.
I said it about three times already but you weren't able to read it.

>National banks for example use their own cryptographic standards and implementations that differ from the NSA.
So you can avoid using the NSA systems by adopting a different cryptographic standard resulting in a different implementation.
(decent) Banks implement their own crypto.
notice how I said implement? Well that's because algorithms are general and not specific. Meaning how I adopt RSA to my bank might use some variations in the components in RSA.
And?

>>11770839
It's incoherent either way.

>>11770833
Asking for help in anything is extremely hard, even in small things like that. I just needed to do it myself for myself. Too much pride.

I just want to say that NSA has a proven track record of manipulating the cryptography community into using flawed systems.
https://en.wikipedia.org/wiki/Dual_EC_DRBG

>>11770866
>I'm so proud I can't ask my professor anything ever
This is not pride, this is nonexistent self-esteem. Please seek therapy.

>>11770869
You mean to tell me that an enormously large organization with the clear goal of global surveillance might be interested in having Backdoors to supposedly secure communication?
Shocking.

>>11770855
>polynomial algorithms
Yes, but not to the extent that the computation of primes is factorial and then checking if those numbers are prime.
https://en.wikipedia.org/wiki/AKS_primality_test
I assumed you understood computation, so my bad. If I generate a suspect prime and then check if it is a prime or not is definitely a P != NP problem, which should be assumed if I am trying to find large prime numbers or brute force it.
I'm drunk and should have explained that you have to check the prime after you generate it by factorizing it.
>No, they create standards.
I'm glad we agree on that.
>I didn't mention RSA before you brought it up.
RSA is the one the NSA would want to find mathematical flaws as it is the hardest to brute force and the public and private key is different, meaning less chances of it being exposed. Which is why most attacks against RSA are looking for implementation flaws, something the NSA might do.
AES Is easier as it's easier to brute force, it's conceivable that massive computer farms would be able to crack it over the years, and the same key to encrypted is used for decryption, so there's more opportunities to expose the key.

>I DO NOT IMPLEMENT THE SOFTWARE.
Neither should most people. Banks do, governments do, the illuminati probably do.

>Under half a second on my phone.
You might understand mathematics, but you do not understand computation.
If you do that, as I said, with larger and larger values, the time increases n!.

>I said it about three times already but you weren't able to read it.
No you deviated from your initial argument once you forgot what it is we were arguing over.

>>11770876
I doubt a foreigner would get free therapy in the UK, so I think I'll just try to keep myself together with willpower. Also, drinking helps a bit.

>>11770869
One reason for the Tor network to exist is for the US army to avoid the all seeing eye of the NSA, if I remember correctly.

>>11770869
Why didn't Obama end that shit? Especially after Snowden. I don't exect Trump to care about privacy, but shouldn't democrats stand for human rights and all? Snowden said it got worse under Obama, I can't even imagine how bad it is under Trump. Let's hope Biden puts an end to all that shit the NSA and CIA try to get away with.

The US is such a fucked up country, I live in a shithole, but I'm glad I don't live there.

>>11770896
>Yes, but not to the extent that the computation of primes is factorial and then checking if those numbers are prime.
>https://en.wikipedia.org/wiki/AKS_primality_test
>I assumed you understood computation, so my bad. If I generate a suspect prime and then check if it is a prime or not is definitely a P != NP problem, which should be assumed if I am trying to find large prime numbers or brute force it.
Totally incoherent.

>RSA is the one the NSA would want to find mathematical flaws as it is the hardest to brute force
For a given key size ECC is much more secure.

>and the public and private key is different
How many vodka bottles do you need to write stuff like this?

>You might understand mathematics, but you do not understand computation.
Because my phone can solve this problem in 0.5s?

>If you do that, as I said, with larger and larger values, the time increases n!.
No. GNFS is better than that.

>No you deviated from your initial argument once you forgot what it is we were arguing over.
My claim has always been that the NSA is aware of secret mathematics it uses to be able to attack cryptographic systems.

>>11770869
I agree.
But conceptually, there is nothing proven to be wrong with RSA, other then how it is implemented, or the amount of bits it can use for the number. Which is the whole argument I'm having.

>>11770912
>Why didn't Obama end that shit?
LOL
Why the fuck would he want to?
Is there literally any reason for it?

>democrats stand for human rights and all
First and foremost they stand for their own power and the power of the US government. Like LITERALLY any political party would.

>I can't even imagine how bad it is under Trump
Exactly how it has always been? Just because some person who says he is for the red party sits in the white house doesn't replace the entire US agencies...

>Let's hope Biden puts an end to all that shit the NSA and CIA try to get away with.
Are you insane?

>>11770918
>But conceptually, there is nothing proven to be wrong with RSA, other then how it is implemented, or the amount of bits it can use for the number.
Literally nobody disagrees with that.
It's trivially true.

>>11770927
>LOL
>Why the fuck would he want to?
>Is there literally any reason for it?
Because it's a violation of one's individual rights? Dude, there's literally no defending that shit, even if you're american, they're spying constantly on you, on everything you do, no matter who you are. On the Snowden movie, they even wanted to get Japan on board of that shit and the japanese refused because spying on their own citizens is disgusting, so don't try to normalize this shit.
>First and foremost they stand for their own power and the power of the US government. Like LITERALLY any political party would.
Explain to me why Japan didn't want to spy on their own citizens then. Believe it or not, but not everyone is as dirty as the US government
>Exactly how it has always been? Just because some person who says he is for the red party sits in the white house doesn't replace the entire US agencies...
We don't know about that, all we know is that Trump doesn't give a shit about anyone but himself and maybe his family and he'll do anything to maintain his power.
>Are you insane?
I'm not expecting it, since he doesn't strike me as a really good candidate, but I hope he puts some limits on the NSA at least

>>11770913
>Totally incoherent.
I choose a possible prime. by using i*6, +1 or -1.
>either side of a multiple of six is one or two primes
This is one way of doing it.
I must then check if that number is a prime or not. How do I do that?
Well, I divide by increasingly large numbers until I reach more than 1/2 of the suspected prime. Basically. As you showed, there are better ways of doing this, but the concept is similar enough for our purposes.
This is a N != NP because of the decision between prime or not prime.
>many vodka bottles
Not enough. Plus RSA using a public key system, there's nothing wrong about what I said.
>Because my phone can solve this problem in 0.5s?
What is 0.5!
What is (0.5!)!...
>GNFS is better than that.
Not likely that much better that it makes a difference in RSA. Otherwise RSA would be flawed if that were the case.

>The NSA has probably backdoors to some cryptographic systems
is what you initially stated.

RSA I don't think so, definitely those backdoors are implementation based and all CVE's to do with RSA are related to being implemented in a bank or government institution, or the private key being exposed. AES maybe, but again, most of the CVE's are implementation based or exposure of the key.

The other weakness off these systems is their key size, but so long as the implementation is fine, using a larger key size will fix that issue.

I'm pretty certain that the NSA hasn't cracked RSA if you do RSA right.

>>11770949
> reach more than 1/2 of the suspected prime
less than.

>>11770946
>Because it's a violation of one's individual rights? Dude, there's literally no defending that shit, even if you're american, they're spying constantly on you, on everything you do, no matter who you are. On the Snowden movie, they even wanted to get Japan on board of that shit and the japanese refused because spying on their own citizens is disgusting, so don't try to normalize this shit.
I am not asking why it is bad. I am asking why any head of state would want to get rid of it?

>Explain to me why Japan didn't want to spy on their own citizens then.
Because that is tremendously unpopular. The US got away with it during the aftermath of 9.11., but since it is already here no politician has any good incentive to get rid of it.

>We don't know about that, all
We kinda do.

>he'll do anything to maintain his power.
Certainly not. He seems to care much more about his self image then power.

>I'm not expecting it, since he doesn't strike me as a really good candidate, but I hope he puts some limits on the NSA at least
Biden will not be president. And this isn't just a matter of the election. This guy clearly is totally unfit to be a president, if he didn't die during his term certainly he would become otherwise unfit.
He also has literally zero incentive to change anything.

Again. The Democrats were in power for quite some time, yet the Patriot act is still there. If you believe that either party wants to get rid of it you are delusional.

>>11770949
You are totally incoherent.

You also aren't arguing, you are trying to explain to me first semester landau symbols while either totally clueless or severely drunk.

>>11770776

>>11770970
lmao.
you salty boy, you.
You're right. I wasn't arguing with you initially, but you wanted to start an argument.
RSA is fine,
AES is probably fine.
The NSA might have secret mathematics, but it has nothing to do with magically reversing your public key to your private key. Unless you can reason as to how they would do this.
I can safely say that secret RSA crackers is classified so non-existent and requires a schizo security clearance to access this information.

>>11770982
You brought up RSA and AES. I never mentioned a specific scheme.

>The NSA might have secret mathematics
So you literally agree with me?

>>11770965
>I am not asking why it is bad. I am asking why any head of state would want to get rid of it?
Maybe you're misinterpreting me, I'm not advocating for the end of the NSA or the CIA, I'm advocating for ending their spying on normal people.
>This guy clearly is totally unfit to be a president
Same as Trump then. I don't care who wins, but from the way I see it, it's almost impossible for Trump to win right now, especially after the Coronavirus and George Floyd's death, his popularity is extremely low and people call him Bunker Boy these days on social media.

>>11770987
>So you literally agree with me
Yes. But I don't agree it has anything to do with mathematical flaws in Cryptography as that leads people to false conclusions.

when does it make sense to take the digital root of a number and when doesnt it?

>>11770989
>I'm advocating for ending their spying on normal people.
For the third time. Which politician would have any interest in that?

>Same as Trump then.
Trump is physically healthy and doesn't seem to be suffering from age related decline. He also isn't sniffing young girls.

>his popularity is extremely low and people call him Bunker Boy these days on social media.
Nobody gives a shit what people call him on social media. What kind of argument is this?
Honestly, I really do not give a shit about the US election. It seems pretty clear that both the red part and blue party will do the exact same things.

>>11771005
>But I don't agree it has anything to do with mathematical flaws in Cryptography
Why not?

>>11770812
Thanks anon!

>>11771009
*sigh*, sure, continue to be an unreasonable jerk, I won't post about it anymore since it's not maths

>>11770989
Are you seeing people coming out of the woodwork to criticize Trump though? Pretty uch all of my social media friends who do so have been bitching about Trump since the beginning. And on the other hand, I'm hearing a lot more anti-protest anti-lockdown sentiment from people in real life than on social media

>>11771011
That's what I've been arguing about.
How do you have a public key and reverse it to a private key assuming the asymmetric system is using a 2048-bit key?
Even if you look at the research the top universities are doing on calculating primes, or utilizing quantum shit, they are no where near reversing that.
You don't need to do any of that anyway, people are very clumsy with their keys. The focus would be, if you wanted to decipher a message encrypted using something like RSA, would be to find the key through other methods. In the cases this doesn't work, there are much better avenues for finding out their desired information, other than finding a mathematical flaw in RSA.
I'm sure agencies like the RSA do try to do this, but it's a poor allocation of budget as there are far better methods for gathering intelligence.

The principle RSA is based on works provided someone doesn't find a really dope prime pattern that makes calculating primes easy.
But I doubt it.

mfw dumb /pol/ niggers derailing /mg/ yet again

>>11771027
>*sigh*
Lol

>unreasonable jerk
How am I a jerk or unreasonable?
Literally everything I posted seems to be factually correct.
Or do you believe that Obama wanted to repeal the Patriot act for 8 years, but just forget?

>>11771037
Which dumb /pol/ nigger?
The one who brought up surveillance seems to seriously believe that US Democrats have an interest in stopping spying.
That certainly is a bigger conspiracy theory then believing that the Nazis are still roasting Jews on the (flat) moon.

>>11771037
>mfw dumb /pol/ niggers derailing
And CS niggers
No one cares about computation.

>>11771034
>I'm sure agencies like the RSA
Please go to sleep, you clearly are drunk and incoherent.

>>11771045
It's the same guy being an unreasonable jerk about politics and cryptography, it seems he finds pleasure by offending people who don't agree with his views.

Also, don't use the word 'nigger' here, man, with all due respect.

>>11771048
>incoherent
no you

>>11771054
>unreasonable jerk about politics
I don't think I am. I pretty much only state facts.
Or do you seriously believe that Obama wanted to get rid of the Patriot act and just forgot.

>it seems he finds pleasure by offending people who don't agree with his views.
If stating facts offends you, then maybe the problem is you.

>>11771054
>Also, don't use the word 'nigger' here
Apologies.
I have also been arguing about cryptography. But I am starting to doubt that he actually knows anything or if I'm being baited.

>>11771060
I don't call the NSA the RSA.

>>11771068
OH SO IT IS THE SAME GUY.
Lmao.
Okay, now I know you don't know anything about cryptography.
Go back to your schizo cave.

>Also, don't use the word 'nigger' here, man, with all due respect.

>>11771075
>Go back to your schizo cave.
That is very unkind of you.
Please do not use rude language.
I feel offended.

>now I know you don't know anything about cryptography.
You are confusing me, I am not the drunk one.

>>11771074
>clutching at straws
heh heh.
I maybe drunk, but you are fucking retarded.

>>11771076
Shut the fuck up Nazi pig. Or get you fucking head bashed in #antifa.

>>11771081
The argument you made was what I suggested to you roughly 10 posts ago.
Do you want an actual response?

bros,
[math] |a - b|^k /leq a^p - b^p [/math] holds for nonnegatice a,b and p >1, r...right?

>>11771089
Your only argument is that I am incoherent.
The entire time you were only looking to detail this thread because you don't like people to have nice things.
I'm sorry that I assisted in the derailment.

>>11771103
>Your only argument is that I am incoherent.
That is not actually an argument. Many of your sentences have been malformed beyond being comprehensible.
That makes it basically impossible to argue anything with you as you are currently not able to express your thoughts.

>you were only looking to detail this thread
No. I try to detail threads usually only when I am drunk posting.

>I'm sorry that I assisted in the derailment.
That's how drunk posting works.

>>11771102
Suppose b > a.

>>11771102
a=0, b=1, p=1?

I am gonna say the n-word.


NON-EUCLIDEAN GEOMETRY

>>11771111
I'm going to ask the NQ. How did you get such nice quads?

>>11771111
I'm going to say the n-word too.
NUMBER THEORY

>>11771106
I'm not a good writer at the moment but my sentences are comprehensible.
Give me an example of where you had trouble.

>>11771109
>>11771110
shit.
[math] |a - b|^p \leq |a^p - b^p| [/math] for a,b nonnegative and p > 1.
I think this is wrong but I can't find a counterexample!

>>11771120
See >>11770765

>>11771126
Maybe quote a specific sentence, as there is nothing incomprehensible, or wrong, about what I said.

>>11771116
No clue. And frankly I am proud about this random number.
>>11771119
YOU CAN'T SAY THAT, IT HAS NO APPLICATIONS!

>>11771129
And why the NSA is using an implementation of an algorithm by choosing specific numbers to input into it.

Yes, but not to the extent that the computation of primes is factorial and then checking if those numbers are prime.

If I generate a suspect prime and then check if it is a prime or not is definitely a P != NP problem, which should be assumed if I am trying to find large prime numbers or brute force it.

This is a N != NP because of the decision between prime or not prime.

So calculating infinitely large prime and knowing it's factors is a P != NP (?) problem.

>>11771124
it's true bro...

>>11771142
hallelujah, my thesis is saved.

>>11771139
this is gibberish

>>11771146
Oh really?
I haven't noticed that!

It's just some posts by the drunk anon in this thread who still claims to write comprehensibly.

>>11771139
It seems pretty comprehensive to me.
Generate a possible prime that might not be a prime. Check if it is by factorizing.
For any given potential prime, I have to work out if it has any factors until I reach less than 1/2 the number, to determine if it is prime.
>P != NP (?)
I meant N != NP. But it makes no difference as you know what I meant.

>>11771154
>It seems pretty comprehensive to me.
Anon...

>I meant N != NP.
And what would that mean?

>>11770198
Do the exercises the videos give you.

>>11771156
>N != NP
Well it's either prime or not prime isn't it, by factorizing I can determine that, but for increasing large prime values, it becomes improbable to calculate.
>Anon...
You seem to not be able to provide reasoning or a break down of my sentences, to show why they are incoherent.

>>11771165
>Well it's either prime or not prime isn't it, by factorizing I can determine that, but for increasing large prime values, it becomes improbable to calculate.
How does that relate to the famous N=NP conjecture?

>>11771154
What is this N, supposedly not equal to NP?

>>11771173
N=/=0, therefore N=/=NP.

>>11771173
It's obviously a part of some elaborate Nakayama argument.

>>11769296
I would like to study mathematics, because I have a genuine interest in it (not because /sci/ bullied me into studiying it). However, I am put off by the reputation of the people that study math. Do I have to be a genius? I have never done a mathematics competition. I found out that I really like math after starting to read "How to Prove It" by Velleman. I also taught myself C++ and I can use FLTK (a graphics library). I started to read "Structure and Interpretation of Computer Programs." I know I would definitely like to pursue a minor in computer science potentially a double major if I have good grades. This leads me to my question: If I treat mathematics as if it is a job, by allocating 40 hours per week to school, do I need to be a genius to get a good grade? Getting a good grade is kinda important becuase I would like to do a double major in math/CS or go to gradute school.

What is it with /mg/ and anime?
Why does anime and math go so well together?

>>11770314
They behave as a "product". If we restrict to maximal ideals for a moment then this is literally true. It can be a little more subtle. If you suppose that all these rings are over C, then spec of them gives you complex manifolds and the complex manifold associated to the tensor product is exactly the fiber product of manifolds. Note that since the zariski topology is so coarse once we get away from closed points there are way more prime ideals in the tensor product.

>>11771204
anime girls go well with literally everything

>>11771199
>I would like to study mathematics, because I have a genuine interest in it
Good start.
>(not because /sci/ bullied me into studiying it)
Don't lie.
>However, I am put off by the reputation of the people that study math. Do I have to be a genius?
You can get pretty far with hard work. Train your butt so that you can sit for hours.
>I found out that I really like math after starting to read "How to Prove It" by Velleman. I also taught myself C++ and I can use FLTK (a graphics library). I started to read "Structure and Interpretation of Computer Programs."
Can't comment on the books per se, but its good to be prepared.
>If I treat mathematics as if it is a job, by allocating 40 hours per week to school, do I need to be a genius to get a good grade?
No.

>>11771204
Autism and/or self insertion.

Is there some natural structure that can be built out of the Pythagorean triples? They have some nice properties, e.g. a scalar times a triple is itself a triple. There exists exactly n triples of the form (p^n, b, c) where p is prime, primitive triples remind me, but perhaps only from the name, of primes, etc.

I find it extremely disturbing that there is a gif of a woman waving at me every single time I need to check some quick thing from a paper using scihub, and so I am literally forced to download them all. Why am I being tormented like this?

>>11771345
Right triangles with fixed angles.

>>11770314
It corresponds to pairs (p,q) with p a prime ideal in B, q a prime ideal in C such that the inverse image of p and q in A coincide.

>>11771345
you should be able to do something based on the following formula:
(a^2 + b^2)(c^2 + d^2) = (ad + bc)^2 + (ac - bd)^2
or if you know a bit more, use the fact that pythagorean triple a^2 + b^2 = c^2 corresponds to a factorization c^2 = (a+bi)(a-bi) in gaussian integers

How difficult is functional analysis?

Does anyone have any good references for sheaf cohomology? I've been reading the appendix to Hubbard's Teichmuller book but it's not making sense to me and I feel like a fucking brainlet lmao
All good with ordinary cohomology, but I'm trying to understand Kodaira's vanishing and embedding theorems better
Already reading Wells' complex manifolds book for the proofs, but I feel like I'm stuck on sheaf and cech cohomology

>>11770245
Extremely based Alashposter

>>11770719
lmao Cleo got the fucking Ramanujan syndrome or what?

>>11771383
idk but the stein and shakarchi book is pretty swell

>>11771397
I can't guarantee this is good for sheaves, but the Alexander & Poincaré duality parts are good, so I will make the daring guess that it extends to the sheaf section: https://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf
>Extremely based Alashposter
Why thank you! https://www.youtube.com/watch?v=YNJ3xb8fphQ

>>11771173
fake and gay midwit autism, ignore it.

>>11771454
Thanks! I'll read through this, much appreciated.

>>11770689
physics forum is very low iq when it comes to math.

>>11771472
Enjoy!

>>11771482
What else is new

>>11770701
>https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1032&context=euler-works
ERROR: This is an invalid URL. Please reenter the URL, or if you clicked a link in an email message to get here, make sure the link was not split across two lines.

>>11771482
?
You know that there are fields medalists posting there, right?

>>11771616

we have superpermutation anon

>>11771641
that was a decade ago, besides, his 'proof' was incomplete.

>>11770809
You're autistic, stop trying to meme

Pointwise continuity is a pointless concept.
Compact patches must always be involved to speak of continuity.
Accept my authority.

>>11771204
It was a pieceful vibe to it.

That's my take anyway. I only like Lain.

>>11771790
I tried to watch Lain but the first episode was very off putting. Should I stick with it?

>>11771917
>very off putting. Should I stick with it?
It's clearly not meant for your double IQ brain

>>11771917
sure

what happens in the first episode again? someone an hero and Lain tries to talk with her dad?
Anyway, sure do

>>11771938
Yeah. It just seemed like they didn’t do much to convince you to continue watching. Besides I guess being curious about her allegedly dead classmate.

>>11771962
>Art needs to convince anyone to continue watching
Holy fucking shit, don't even bother, I already told you it's not for your double digit IQ, Lain is not your typical YA anime, it's a deep analysis of internet effects on youth and society, you understand nothing of art, go back to Naruto or whatever the hell you watch "for fun".

>>11771987
>it's a deep analysis of internet effects on youth and society
its about funny little anime girl do a dance :)

>>11771987
I really enjoyed Bakemonogatari and Katanagatari. Any recommendations of anime with based MCs?

>>11771987
kinda mean but yes

I found there's a lot of theories that don't have EP, even after droping LEM and Powerset.

Can't write down shit.

>>11772003
Shaft Animes, other works by NisiOisin. These days Kaguya-Sama pretty much follows Shaft philosophy, Bunny Girl is a Bakemonogatari clone with pseudo-science throw in.

>>11772025
Can you give an example (preferably double digit IQ friendly)?

>>11772003
City Hunter.

Any book recs for someone going into undergrad study?

>>11772106

>>11772123
Thank you, I'm not sure which I should read though as I'm not american

>>11772132
start from the top.

dont really feel like doing maths today...

>>11772106
Start with Apostol's Calculus I and II and Axler's Linear Algebra Done Right, those two books will give you the very foundation of all modern mathematics. Then, you should start Pugh's Real Mathematical Analysis and Aluffi's Algebra: Chapter 0, the first one will give you knowledge of proofs, while the second one will provide you with tools that appear in almost all fields of math. After that, Do Carmo's Differential Geometry of Curves and Surfaces and Tu's Introduction to Manifolds for a little geometry and then you can end with Conway's Functions of One Complex Variable and go into the advanced PhD topics,

>>11772171
>One Complex Variable and go into the advanced PhD topics
SEVERAL
COMPLEX
VARIABLES

>>11772089
It's about how strict you interpret "[math] \exists x [/math]".

Example: Let [math] O(T, X) [/math] be the basic predicate that says that T is an total ordering relation on the set X.
The set theory ZFC proves (mostly via axiom of choice) that all sets can be ordered, and in particular it proves
[math] \exists t. \, O(t, {\mathbb R} ) [/math]
But that's icky, since it's also provable that we can't describe an ordering on [math] {\mathbb R} [/math].

That's an extremely non-constructive theory in that sense, the existence [math] \exists x [/math] in this theory is super weak.

There's those at least 3 schools of conservatism in mathematical logic for mathematics, distinguished by what they still allow to be valid principles of inference. (Most of their principle will be valid in classical math.)

E.g. the Russian school of Markov Jr. (the son), adopts the arguably non-constructive Markov's principle
https://en.wikipedia.org/wiki/Markov%27s_principle
for decidable predicates of numbers:
If
[math] \forall n. \ P(n)\lor \neg P(n) [/math],
then
[math] \neg \forall n\, \neg P(n) \implies \exists m\, P(m) [/math]

In words:
>For decidable P, if it's not the case that for all n in N, we have that P(n) is false, then there exists an m such that P(m) is true.

I.e. if you know P(n) can be evaluated (to true of false) for any n, and if you know - by whatever means - that there has to be an example m, then (even if you don't know it), the math and logic is such that you're allowed to conclude [math] \exists m\, P(m) [/math].

The idea behind allowing this, is that if you got a decidable (i.e. computable) predicate over the (inductive) set of natural numbers, then you know that in principle you can turn on a computer and run through all n till you find m. Markov's principle is a promise principle in that sense.

Pic related are properties that would be nice to have of a theory T (but something like ZF if far away from having those)

>>11772160
Why not.
Do 20 minutes.


>>11772089
Bishops theory makes use of some choice and Brouwers intuitionism doesn't even have a formal logic, pic related (but it a student of his said fuck this guy and formalized it)

The number 3 >>11772216 is the Existence Property, and a bit harder to prove metalogically for some theories.
If you read it it says:

A theory has this property, if when
[math] \exists x. P(x) [/math]
is provable for some predicate, then there's actually a describing predicate [math] X [/math] (a mathematical statement that expresses a property), such that the theory T also proves
[math] \exists !x. X(x)\land P(x) [/math]

I.e. for a nice theory that has the existence property, it can't happen that it's provable that [math] \exists x. P(x) [/math] but this [math] x [/math] that's provable to exist can't actually be described (such as some quirky orderings that "exist" by the axiom of choice, but can't really be talked about as such, here's a SE question relating to that example btw., but of course that's just one example - https://math.stackexchange.com/questions/6501/is-there-a-known-well-ordering-of-the-reals))

If you want to know a weak variant of ZF that has the existence property, here's the axioms (no LEM assumed):
- Extensionality (having equal members implies equality) - this is also part of ZF
- Pairing (the pair {x,y} exist for each set x and y) - this is also part of ZF
- Union (the union in a set exists) - this is also part of ZF
- Bounded Separation (cut out new sets from bigger ones, but don't use \forall ranging over the whole universe) - ZF has a strong version of this (not predicative)
- Strong Infinity (an infinite set exists) - a variant of the infinity axiom that ZF also has/proves
- Strong Collection (generate sets with relation predicates) - a strong variant of replacement, I think ZF also proves this
+ Exponentiation (function spaces exist) - a weak version of power set

>>11772256
any books for that?

>>11772256
Oh I forgot Set induction, which without LEM is a weak form of Regularity (otherwise it's equivalent). That's basically ensuring all sets have a bottom and stuff.
I'm not sure if that's needed to show the Existence Property.
Also I don't know why my last bullet point is a +, that has no secret meaning.

>>11771987
To be fair, you need to have a very high IQ to understand Lain. The message is very subtle, and without a solid understanding of art most of the deep analysis will go over the typical Naruto watcher's head.

>>11772275
Depends what you mean by "that"?

plato.stanford.edu is nice most of the time and should have references
https://plato.stanford.edu/entries/intuitionism/
https://plato.stanford.edu/entries/set-theory-constructive/

Reality is that most people interested in such questions use typed systems nowadays, because of dependent type theory introduced in the 70's.
https://en.wikipedia.org/wiki/Dependent_type

You can actually strengthen Exponentiation (the function space axiom) to what's called Subset Collection axiom and you still got a theory that has a model in type theory (which is arguably more "real" on a computer). Subset Collection is a weaker variant of power set, still.
But this theory already doesn't have the existence property anymore.

Anyone here doing a graduate degree/post-grad research in cryptology? How is it?

>>11772294
this thread is for the discussion of pure mathematics. if you would like to make a thread about computers you should go to /g/ which is a board expressly dedicated to this subject.

>>11772297
school bully material right there

>>11772216
>>11772256
Danke schön! I think I understood at least 95 %. Did you come up with this EP-ified ZF yourself? If not, then maybe you could try finding the redundancy in the list of axioms and try to find a version! Not my expertise, so I'll just pay homage to people tackling these things instead of saying anything dumb.

>>11772277
The cloud of negativity covering you moved and you found happiness again. And that is because you realised you are a good poster.

>>11772294
Where can people actually even do that? I used to study them on my own as a kid and wanted to do them properly when I grow up.

>>11772317
go fuck yourself tourist nigger

>>11772322
Them = ciphers using Wikipeda and its sources.

>>11772322
No, it was proven here

https://arxiv.org/abs/1303.3408

I can't bother too much with all the (ugly imho) model constructions that prove or disprove this or that. Currently I mostly want to see what's provable in 3D space geometry with guaranteed "implementability."
E.g. where does Bourbaki style Lie group theory jump this point. Probably linear algebra alla root systems is fine, but huge enveloping algebra constructions are probably out of the realm already.

>>11772293
>Depends what you mean by "that"?
I meant the things on your image.

>>11772336
>Bourbaki
Not a person.

>>11772336
Model theory is an acquired taste, or maybe it's a calling. It's good to have some people do it, so the rest of us can do things far less tedious.
>Probably linear algebra alla root systems is fine, but huge enveloping algebra constructions are probably out of the realm already
Sounds reasonable. I see no reason for root systems to be too much. Any nice results yet?

>>11772381
He wrote a lot of book, that's obviously prove he existed.
(Actually, I think I didn't even make the point he/she/xe was a person)

>>11772345
Mhm, yeah try
>https://plato.stanford.edu/entries/intuitionism/
and go from there.

What's fun to read is also pic related
https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy

Sort of the origin of why Russel made his original type theory (now pretty much forgotten, but strongly revived of course 50 years ago) is how the debate about the Russel paradox was had 120 years ago.
E.g. there's the impredicativity discussion, about which I however never found a good text myself.

Namely that people back then argued that "set of all sets" and notions like this have the underlying conceptual problem that they are not "predicative":
What would it mean to compute
[math] T := \{n \in {\mathbb N} \mid \forall(S \subset {\mathbb N}). \phi(n, S)\} [/math]
?
The issue is that to assemble all elements of T to a set, you'd have to check all subsets S of N, but T is one of those.
With unbounded comprehension, of course writing down this set is no issue in e.g. ZF, but logicans took (or take) issue with that.

The theory called IZF is just ZF in a context without LEM (and regularity replaced with set indcution, for the mentioned technical reasons), but the impredicativity issue is why people actually looked at CZF, which is roughly what I outline above.
It's IZF, but with power set replaced with a weaker axiom and separation make predicative (no forall over all sets.)

>>11772397
Not from me, but there's texts on topology without LEM and the prove are hard as far as I could tell. Things just get cumbersome there. So I think the Lie theory will look more like algebra and less like differential geometry.

whats the relation between the traditonal flairs of geometry (euclidiean, hyperbolic, ellipitic) and modern diff geo?

>>11772431
The former all have a very simple distance and curvature description.

Riemannian geometry in general has more volatile connections between the structures in their tangent spaces and "diff geo" doesn't even need to have a metric.

How many hours a day can one realistically practice doing mathematics? Does anyone here do 8+ hours a day? Also, if I am self studying should I be doing every exercise at the end or the chapter or just enough to feel like I have understanding?

>>11772442
4 to 6 hours is reasonable.
Anymore than that you can't really be efficient anymore.
>Also, if I am self studying should I be doing every exercise at the end or the chapter or just enough to feel like I have understanding?
Just pick about 30% of the exercise randomly.
If two problems have the same premise then don't do both and move on to the next one.

>>11772418
>(Actually, I think I didn't even make the point he/she/xe was a person)
You will probably be replied how the anon just helped with your ignorance or something. Happened to me when I mentioned Bourbaki in a non-personal way.
>there's texts on topology without LEM and the prove are hard as far as I could tell. Things just get cumbersome there
LEM is a tender lover when doing topology, and I can believe that one will miss his touch when trying to do anything without him. Okay this point is not in the complement so it must... oh wait no LEM,
>So I think the Lie theory will look more like algebra and less like differential geometry.
This seems like a natural consequence of dropping a lot of topological aspects by necessity. The fragile stuff most likely occurs when it is time to start embedding the structures into bigger ones. Is there even any reason to assume all the enveloping algebras would exist?

>>11772442
>Does anyone here do 8+ hours a day?
I'm a workaholic and can easily do 8-10 or more a day with a few breaks.
>if I am self studying should I be doing every exercise at the end or the chapter or just enough to feel like I have understanding?
Do most, proceed, return if necessary.

>>11772442
>Also, if I am self studying should I be doing every exercise at the end or the chapter
Maybe. Depends on the book. If you're doing a book like Stewart's calculus that has 3000 spammy exercises, you'll be there forever trying to do all of them. In higher level textbooks, most books only have a couple hundred exercises at most (some have fewer still) and you should really try to do almost all of them and not skip one without a good reason for it.
>or just enough to feel like I have understanding?
Trusting whether you _feel_ like you get it is a recipe for disaster. It's so incredibly easy to convince yourself you understand something way better than you really do that your real understanding is always substantially worse than it feels.

>>11772456
>topology
Not a person.

>>11771987
art still needs a hook if it wants to be appreciated

>>11772481
Thanks for correcting my ignorance.

/gnmg/

>>11772454
Okay thanks. I will keep that in mind.

>>11772477
I had a feeling in that regard. How do you really know you're understanding then?

>>11772498
>How do you really know you're understanding then?
You can do the exercises.

>>11772277
Holy shit checked!

>>11772454
>can't really be efficient anymore
kek, that's the biggest lie made up by geniuses who don't want to be surpassed. Sure, go on and study 6h a day, you'll surely be able to compete with a math55 student who's doing 12h+ a day of math without counting lecture hours.

>>11772569
I'd bet you a testicle you only know about math 55 through sensationalist articles and have never met a single sutdent

If I have one or two publications, can I expect to get into a PhD program?

Other than that, im probably about equivalent to most other math undergrads who apply to grad school

>>11772576
>Bourbaki in a non-personal way
from math55 no, but I've met people from honors classes in high level institutions and believe me, the guy was literally doing 12h a day of study, he's now doing research in one of the best universities in the country and guess where I am now? On 4chan...

hey look at this tookys back!
>>11769446

>>11769447
Fuck precalc. Spivak calculus.
I know it's a meme here, but you can learn all the necessary precalc from the first chapter by attempting to prove every theorem and by doing all the non-starred problems and watching lectures when you are stumped. Of course, you should read and attempt the starred problems, but don't feel pressured to solve them.
Doing all the (non-starred) problems may take more time in the start but it actually makes the later chapters ez.
For example I know a student who only did the assigned problems spent hours integrating sqrt (tan(x)) (starred problem) when if you wrote an attempted proof for every theorem and solved every normal problem you would have already put in the work to understand what you need to do (the proof) and you already applied it before (the normal problem).


I've applied this method (since our books arn't spivak level you can do every problem)in every undergrad math class and you really feel like you are learning which gives you the motivation to put in 6+ hours a day
cons: could hurt your hand, could end up in time crunch if you take too many rest days.
If that's the case you will have to bs it like you normally do.

>>11770175
>real and functional analysis before undergraduate algebra
But why

New Thread:
>>11772937

>>11772003
konosuba, bunny girl senpai, the rest of monogatari (nise is pretty based too, but it doesnt hit as hard if you didnt see kizu first), but imo it never gets as good as bake