/mg/

[Anonymous No.11913994]

Cauchy Schwarz edition

No aggressive retards edition

[Anonymous No.11914001]

>>11913994
>no agressive retards
EXCUSE ME FORGOT-TO-LINK-THE-LAST-THREAD-KUN?

[Anonymous No.11914003]

>>11914001
the last thread turned out to be very unaesthetic so it shan't be counted

[Anonymous No.11914017]

>>11909838
Old aesthetic thread

[Anonymous No.11914023]

Why are sine and cosine only each other's derivatives when you use radians? What makes degrees and other measures useless?

[Anonymous No.11914029]

Who here likes Group cohomology?
Wanna chat a bit a bit about it. I have to speak of it a tiny bit this week.

[Anonymous No.11914037]

>>11914023
>why is [math]a \exp (bx)[/math] its own derivative if and only if [math]b=1[/math]? What makes other real numbers useless?

[Anonymous No.11914040]

>>11914029
Pedo faggot

[Anonymous No.11914041]

In the Cauchy–Schwarz (CS) inequality |u⋅v|≤∥u∥∥v∥, let's assume v is a normalised vector, i.e., ∥v∥=1. Then the CS inequality becomes |u⋅v|≤∥u∥. Now, it's a trivial matter to show that these two forms of the CS inequality are in fact equivalent, in the sense that if |u⋅v|≤∥u∥ for all normalised vectors v, then the usual CS inequality holds for all vectors. So, let us restate the CS inequality as stating that |u⋅v|≤∥u∥ for all normalised vectors v. Now, the physical/geometric interpretation of u⋅v in this case is that it is the component of the vector u in the direction v (since v is assumed normalised, that's all it is, a direction), while ∥u∥ is the magnitude of u. So the CS inequality is merely stating the intuitively obvious fact that the component of a vector u in a single direction is bounded by the magnitude of u.

[Anonymous No.11914049]

>>11914037
Connecting the two is a good insight that is probably part of intro analysis but I am just on Calc ii. Can you post a link or book that explains why

[Anonymous No.11914058]

>>11914003
>unaesthetic
peak autism. don't bake the next one, dipshit

[Anonymous No.11914062]

>>11914058
this is the no-aggressive-retards edition, sorry sir, please step outside

[Anonymous No.11914068]

>>11914049
No, anon, the point was actually "the chain rule is a thing which exists."

[Anonymous No.11914072]

>>11914040
Reese Witherspoon was at least 20 in that movie, calm down

[Anonymous No.11914079]

>>11914062
Sure thing, boss. I'm going down with OP, though.

[Anonymous No.11914111]

>>11914040
Pretty sure Reese Witherspoon was 18 in that movie, but I'll not get mods spooked for /tv/ posting

[Anonymous No.11914114]

>>11914111
I was just being antagonistic, I don’t care what you want to stick your cock inside of. Just don’t CS post or /pol/ bait and I have no quarrel with you.

[Anonymous No.11914118]

>Elliptic curves

[Anonymous No.11914156]

>>11914118
based elliptic chad

[Anonymous No.11914158]

>Zariski topology

[Anonymous No.11914171]

>symplectic capacities

[Anonymous No.11914273]

>numerical analysis

[Anonymous No.11914296]

>d*screte """math"""

[Anonymous No.11914299]

any good lecture series for PDEs lads? my prof is a dork

[Anonymous No.11914302]

[Anonymous No.11914317]

>>11914296
>people who hate combinatorics are so dumb they can't even get the meme

[Anonymous No.11914318]

The last thread piqued my interest in Ramsey theory. Is a basic knowledge of combinatorics and number theory enough to get into it?

[Anonymous No.11914319]

[Anonymous No.11914321]

>>11914317
>He thinks I'm trying to follow /mg/ memes instead of insulting his pathetic "field"

[Anonymous No.11914324]

>>11914319
thanks for the stroke
ill be billing you for the damages

[Anonymous No.11914325]

>>11914319
Correct me if I'm wrong, isn't [math]0^0[/math] left undefined in most contexts?

[Anonymous No.11914335]

>>11914325
No, in most contexts it is defined as 1.

In >>11914319 though, 0 appears to be used as a variable, rather than a number. Which is pretty cursed, but it means the above doesn't necessarily apply.

[Anonymous No.11914338]

>>11914321
ok copemaster

[Anonymous No.11914342]

Is [math]57[/math] the building block for UFDs?

[Anonymous No.11914344]

>>11914338
you would think that people who are widely hated and criticized would avoid a place where they will be met with hatred and criticism, no? what do you get out of being a faggot and standing strong in the face of universal opposition from people that want nothing to do with you? just go to >>>/g/ or >>>/r/eddit, they will coddle and nurture your inferiority complex there without question.

[Anonymous No.11914348]

>>11914344
show me on the doll where the combinatorialist touched you

[Anonymous No.11914350]

Post kino Euclidean domains.
[math]E=\{\alpha+ \beta \omega \lvert \alpha, \beta \in \mathbb{Z}; \omega=e^{\frac{2 \pi i}{3}}\}[/math]

[Anonymous No.11914353]

>>11914344
Bold of you to assume I don't already have tabs open with both /g/ and reddit.

[Anonymous No.11914376]

If a chain complex has a chain homotopy to itself making the identity chain homotopy equivalent to 0, does that make the complex exact?

[Anonymous No.11914388]

>>11914353
>>11914348
I automatically assume all discrete math and CS posting is perpetrated by redditors. You don't have to announce to me or the thread where you came from.

[Anonymous No.11914400]

>>11914376
Yeah. A chain complex is exact if and only if all the homology groups vanish. If the identity and the zero map are chain homotopic, they induce the same map on the homology, which is simultaneously the identity and the zero map, so the homology is necessarily trivial.

[Anonymous No.11914430]

In linear algebra, and classes thereafter, professors always touted about the dual vector space being very important, however, they never actually showed its importance, I at most saw perhaps some forgettable exercises about it at most.

So what exactly is important about the dual space? What is its relevance? I'm particularly interested in the finite dimensional case

[Anonymous No.11914477]

>>11914430
It's just the space of functions to the ground field, which is isomorphic to your original space anyway so like 99.9% of the time it doesn't fucking matter. Geometers like to jerk off about contravariant vs covariant vectors, but in reality they are just weak at composing tensors and need their hands held to multiply n-dimensional matrices.

[Anonymous No.11914480]

>>11914430
>I'm particularly interested in the finite dimensional case
You use the dual space to define differential forms on manifolds and thus integration, but I don't remember any other examples in the finite dimensional case. Its real importance becomes apparent in the infinite dimensional case where the identification between a space and its dual doesn't always hold.

[Anonymous No.11914483]

>>11913994
I see a infinite side voronoi like structure now

[Anonymous No.11914492]

>>11914477
>Geometers like to jerk off about contravariant vs covariant vectors
I don't think I've ever seen a geometer say covariant or contravariant vector.
I have seen stuff like m-covariant n-contravariant tensor, tho.

[Anonymous No.11914496]

>>11914492
Whatever dude, a tensor is just a vector in some tensor product.

[Anonymous No.11914498]

Am I dumb or is this just not a graph?
It feels like every math book I read has some dubious definition

[Anonymous No.11914523]

>>11914498
That kind of graph is directed simple with loops allowed. You may be more used to undirected simple with no loops. They are all called graphs.

[Anonymous No.11914541]

>>11914430
>I'm particularly interested in the finite dimensional case
This is the wrong case to be interested in. I guess either you're more uncomfortable with infinite dimensional vector spaces or you want instant gratification for your class, but any finite-dimensional example will necessarily be weak since you cannot _need_ the dual space to do anything; if you don't like it, bludgeon an isomorphic copy of V in there instead.
However one simple example that pops into my head immediately is that duality is the natural way of explaining what transpose maps are actually doing beyond "lol matrix entries go flippity floppity"

[Anonymous No.11914613]

Redpill me on boolean algebra

[Anonymous No.11914694]

>>11914613
Lattices and Boolean algebras are both finitely based.

[Anonymous No.11914708]

>>11914613
[math]R=(\mathcal{P}(x),\Delta,\cap)[/math] is a Boolean ring.

[Anonymous No.11914716]

Why are retarded and uncreative people so upset by infinite mathematical objects, do they understand nothing in mathematics is physically real and that the greatest physicists understand this and gladly make use or whatever tools are at their disposal to describe nature? Is it some kind of pathology, a kind of primordial fear of the unfathomable? Its Lovecraftian almost to them, but none of these things are substantial, they can no more harm you than you can reach out and touch the 2D pedo bait you jerk off to every night.

[Anonymous No.11914720]

>>11914694
hmmm..... yes... i see...

[Anonymous No.11914731]

>>11914716
its not pedo bait, its called loli you mong

[Anonymous No.11914772]

>>11914731
I call it like I see it.

[Anonymous No.11915154]

>pass all math courses with relative ease
>still cannot grasph basic physics
fuck the programme coordinator who though this was a useful subject for a math major

[Anonymous No.11915190]

>>11914273
I hate numerical dickheads so fucking much

[El Arcón No.11915194]

Yeah, on that other one I did not immediately notice the relevance of the multiple paths from A to B. Great problem.

[Anonymous No.11915344]

>>11914318
Yeah, at least at a basic level.
In fact here's a set of absolutely kino lecture notes that only require elementary topology knowledge and nothing else: https://tartarus.org/gareth/maths/notes/iii/Ramsey_Theory.pdf
I'm going through these myself right now and have just finished the first chapter. It's awesome but the problems are really hard ;_;

[Anonymous No.11915404]

>>11914477
>It's just the space of functions to the ground field, which is isomorphic to your original space anyway so like 99.9% of the time it doesn't fucking matter.
it doesn't fucking matter only when you actually have the isomorphism

[Anonymous No.11915430]

>>11915404
They asked about the finite-dimensional case.

[Anonymous No.11915441]

>>11915430
I'm talking about the finite-dimensional case. A vector space and its dual are isomorphic obviously, but they absolutely cannot be considered the same unless a specific isomorphism is given.

[Anonymous No.11915450]

>>11914716
>none of these things are substantial, they can no more harm you

Son, there are ineffable concepts that can hurt you way more than physical objects. Ph'nglui mglw'nafh Aleph K'hard'hinal wgah'nagl fhtagn.

[Anonymous No.11915468]

>>11914430
Basically linear algebra concepts should come in pairs: a computational/matrix form, and a coordinate free/vector space interpretation.
Duality gives you an abstract interpretation for the operation of transposition and for row vectors that is compatible with the usual interpretation of matrices and column vectors.
Another point is that it lets you look at vector spaces «from the outside». Typically, you look at vector spaces as spanned by bases. Dually, you can look at them as being cut out by hyperplans. Duality for finite-dimensional spaces tells you that all of this is nice and well-behaved: p linearly independent equations define a codimension p
subspace.
This result is pretty much all there is to finite-dimensional duality, but duality can be useful conceptually nonetheless.
Basically it is the appropriate language when you think of linear algebra in terms of hyperplanes, which can be useful eg. when studying convex geometry.
It is also very useful to have in mind when studying bilinear algebra (a non-degenerate bilinear form being nothing else than an isomorphism between a space and its dual).
Finally, it also prepares you for the study of duality in infinite-dimensional spaces, which is no trivial matter

[Anonymous No.11915514]

>>11915344
Thanks.

[Anonymous No.11915582]

[math]x^2+k=y^3[/math]

[Anonymous No.11915585]

>>11915582
Easy. [math]x = 0, k = 1, y = 1[/math].

[Anonymous No.11915589]

>>11915582
>>11915585
Dumb frog poster.

[Anonymous No.11915593]

>>11914376
iirc, the converse is also true

[Anonymous No.11915594]

>>11915582
Easy if [math]\sqrt {-k}[/math] is in a UFD

[Anonymous No.11915600]

Tell me about [math]\mathbb{Q}[/math]-vector spaces.
Why does nobody talks about them? In linear algebra courses all you care about are [math]\mathbb{R}[/math] and [math]\mathbb{C}[/math]-vector spaces, sometimes you talk about finite vector spaces for stuff like applied math and other things, and then in commutative algebra you start talking about modules.
So where is the theory of [math]\mathbb{Q}[/math]-vector spaces?

[Anonymous No.11915604]

>>11915582
That's an elliptic curve if [math]k[/math] is non-zero, yes, is that what you wanted to know?

[Anonymous No.11915605]

>>11915600
galois theory

[Anonymous No.11915613]

>>11915605
This.

[Anonymous No.11915648]

>>11915600
While I personally think this would be a cool and good idea, the trigonometric functions that form the matrix elements of orthogonal transformations have transcendence properties that quickly fuck you up. This probably goes for all compact smooth transformation groups.

[Anonymous No.11915651]

>>11915600
algebraic topology with rational cohomology

[Anonymous No.11915669]

>>11914716
The issue isn't an unreasonable ontological fear, the issue is too strong axioms lead to the theories making claims that aren't "realizable", where the definition of realizable may come in various more or less strict forms.

>>11914273
Numerical analysis isn't gay, it's actually important and based.

[Anonymous No.11915697]

>>11915651
This. Rational (co)homology and homotopy lets one consider the torsion-free parts.

[Anonymous No.11915753]

I read once a list of mathematical ways of catching a lion (Bolzano-Weierstrass method, Inversion method etc.) Any link?

[Anonymous No.11915757]

>>11915753
http://www.gksoft.com/a/fun/catch-lion.html
Literally just googled "mathematical ways of catching a lion".

[Anonymous No.11915798]

>>11915757
But which method is the best? That’s the real question here.

[Anonymous No.11915810]

[Anonymous No.11915962]

>>11915810
>Symplectic geometry
>Hamiltonian dynamics
BIG FUCKING YIKES.

[Anonymous No.11916027]

>>11915962
>year of our Lord two thousand and score
>disliking symplectic geometry and Hamiltonian dynamics
Bruh moment.

[Anonymous No.11916029]

>>11915810

[Anonymous No.11916031]

>>11916027
This is /mg/ - Mathematics general
fuck off somewhere else like to the physics general or something
>>11916029
now this is based

[Anonymous No.11916034]

>>11915450
It's believed that Cantor suffered from being bipolar.

[Anonymous No.11916038]

>>11916034
>bipolarity bad
>mental illness bad

[Anonymous No.11916081]

>>11916029
Doing that is comfy.

[Anonymous No.11916098]

>>11916029
なぜ背骨がこんなに広いのですか?

[Anonymous No.11916104]

>>11916098
It's a big book.

[Anonymous No.11916156]

>>11916098
Because it's a magazine/leaflet in the original drawing.

[Anonymous No.11916157]

Hello /mg/ did I miss anything?

[Anonymous No.11916164]

>>11916157
yes, some anon solved the langlands program

[Anonymous No.11916182]

How do I remember things?

[Anonymous No.11916192]

>uni is top 100 in math in shangai ranking
can I make it nonetheless?

[Anonymous No.11916199]

>>11916192
Make what? Hey I hit my toe on my desk, can I make it?

[Anonymous No.11916211]

>>11916199
make it into academia

[Anonymous No.11916216]

>>11916199
Based.

[Anonymous No.11916219]

>>11916211
Choose your supervisor(s) correctly and you will.

>>11916216
Based on what?

[Anonymous No.11916220]

>>11914023
the other guy gave you a sort of brainlet circular non answer. the reason is because radians correspond directly to distance along the circle circumference times circle radius. hence the name. this fact plays out when you consider differential angles, the differential arc length is directly related to the angle. draw it out for yourself, sin(x+dx) on a unit circle.

[Anonymous No.11916264]

>>11914023
assume that the curve [math]c(t) = (\cos(t),\sin(t))[/math] is a counter-clockwise parametrization of the unit circle by constant speed (this is basically the high-school definition of sine and cosine). draw a picture, convince yourself that the tangent vector [math]\dot{c}(t)[/math] must be [math]c(t)[/math] rotated by 90 degrees counter-clockwise, but perhaps scaled by some constant: [math]\dot{c}(t) = k(-\sin(t),\cos(t))[/math]. since [math]k = |\dot{c}(t)| [/math], this constant is 1 if and only [math]c(t)[/math] is parametrized by arc-length. this means exactly that the parameter (i.e. the "angle") is the displacement on the circle.

[Anonymous No.11916278]

>>11913994
>proof question on a test
>student's proof is just a reassertion of the claim
I just taught my first proof-based class and was amazed at how common this was. Anyone else taught a proof-heavy course before? How did it go?

[Anonymous No.11916283]

>>11916278
>>student's proof is just a reassertion of the claim
seems like you did a great job, your students are already adopting manner of accomplished mathematicians

[Anonymous No.11916307]

>>11916278
>student's proof is just a reassertion of the claim
Name three (3) times this happened.

[Anonymous No.11916354]

>>11916278
Wait until you have to try and teach them induction

[Anonymous No.11916384]

>>11916278
have you tried not being an amerinigger? eurochads learn proofs in school so you never run into this problem

[Anonymous No.11916387]

>>11916354
amerimutts don't do induction in highschool?

[Anonymous No.11916391]

>>11916384
>guy posts anything at all about education
>I'M EUROPEAN I LEARNED THIS IN PRESCHOOL FUCK AMERISHARTS
errytiem

[Anonymous No.11916406]

>they don't teach algebraic K-theory in american highschools

[Anonymous No.11916420]

>point out the obvious fact that american hs is worse than european
>amerimutts seethe and get triggered
>amerimutts throw the we have the best unis card
>europeans get triggered and start arguing about how the ranking favours american unis
etc. Now, let's make bets how long this show will last this time.

[Anonymous No.11916426]

>>11916420
ahem.

[Anonymous No.11916429]

>>11916354
https://en.wikipedia.org/wiki/Problem_of_induction

[Anonymous No.11916431]

Does replacing all instances of divisors b in [math]\frac{a}{b}[/math] with multiplication by [math](b)^{-1}[/math] make everything commutative?

[Anonymous No.11916437]

>>11916431
what everything? divison? no.

[Anonymous No.11916439]

>>11916426
Ah, yes. I forgot prizes altogether. Please proceed.

[Anonymous No.11916455]

>>11916278
>student uses the word 'clearly'

[Anonymous No.11916461]

>>11916426
>Frenchies win normal proportion of of Abel prizes compared to other countries
>win normal amount of Wolf prizes
>3x as many Fields as you'd expect
is the Fields, dare I say, rigged?

[Anonymous No.11916492]

>>11916461
more like wolf and abel prices are memes

[Anonymous No.11916503]

>>11916492
>everything that's not set up by our circlejerk to make ourselves look good is a meme
cope

[Anonymous No.11916507]

>>11916503
IMU is based in germany
seethe harder medalet.

[Anonymous No.11916513]

someone familiar with the serre spectral sequence here? I have a question about the theorem about it.
Recall that it goes like this:
>Let (F, X, B) be a fibration with some extra condition
>then there is a spectral sequence that converges to the homology of X
>and the second page is blablabla
My question is now: Is there some implicit canonical filtration for H(X) or does this theorem mean "for every filtration of H(X), there is a spectral sequence with the above properties"?

[Anonymous No.11916527]

Is lattice theory /mg/ approved?

[Anonymous No.11916530]

>>11916513
For a graded module, there is the filtration where you filter it as if you were dealing with the skeleta of a CW complex. More precisely, your filtration degree (can't think of a better word at the moment) n tells you you take the "n-skeleton" of your module - the degrees up to n. Now, notice that you have the graded module [math]H_*(X; R) = \sum_{n\in\mathbb{N}} H_n(X; R)[/math], and apply the filtration above to this.

[Anonymous No.11916534]

I don't want to pollute this thread with offtopic discussion, but can anyone give me some input?:

>>11916456

[Anonymous No.11916538]

>>11916530
>the filtration degree n
what about the second index?

[Anonymous No.11916539]

>>11916538
The second index comes from the Hodge structure.

[Anonymous No.11916545]

>>11916538
Serre is [math]E^2_{p, q} = H_p(B; H_q(F; R))[/math] or maybe the the super and subscripts in E reversed I never remember how they go, but yeah, the second index is the degree of the homology module of F as your coefficients. Have you seen how SSS is derived for CW complexes? You start with a CW complex and filter it using the skeleta. This will then filter the total space of your fibration, and hence the fibre also. I don't remember the exact details, but the filtration on B corresponds precisely to [math]F_nH_*(B; R) = \sum\limits_{i=0}^nH_i (B; R)[/math].

[Anonymous No.11916546]

>>11916038
>bipolar
Woman
>mental illness
Low iq

[Anonymous No.11916551]

>>11915669
That doesn’t explain why people who are shit at math are also the people most afraid of infinity.

[Anonymous No.11916554]

>>11916545
yea, but you also have the SSS if you don't have a CW complex. how do you do it then?

[Anonymous No.11916556]

>>11916554
I don't remember. I only use it on CW complexes. That's the filtration for your homology module, though, in both cases.

[Anonymous No.11916560]

[Anonymous No.11916572]

>>11916560
1.

[Anonymous No.11916575]

>>11916572

[Anonymous No.11916579]

>>11913994
do you guys use sypy?

[Anonymous No.11916580]

>>11916575
Oh, it was starting from 2.
1000, then.

[Anonymous No.11916581]

>>11916560
depends on the order.

[Anonymous No.11916586]

>>11916581
>>11916580
Wrong

[Anonymous No.11916591]

>>11916586
it literally is you absolute buffoon

[Anonymous No.11916594]

>>11915585
this is so sad. Anime is a asian subversion propaganda to make men girly. seeing anime image in lit, pol, heck even g is enough;but, finding an anime poster in /sci, that too in /sci/mg/ is beyond humanity. if you keep watching that propagand one day mathematicians will be researching in drawing best looking anime waifu than actually doing something useful for humanity. Not just saying, meaing it. The whole maths community condemns such behavior. Pls refrain or kys.

[Anonymous No.11916595]

>>11916591
Prove it.

[Anonymous No.11916601]

>>11916595
Proof:
int main()
{
int a = 2;
for (int i = 3; i <= 1000; i ++)
{
a = (a+i)/(1 + a*i);
}
cout<<a;
}
You can now compile this and run it to see it print out 1000, q.e.d.
You can also do some algebra or something I don't really care.

[Anonymous No.11916603]

>>11916594
Anime signifies the perfect ideal that everyone should chase. Just like an anime drawing represents the platonic ideal of a girl, I struggle to become the world's strongest mathematician.

[Anonymous No.11916604]

>>11916601
>int
idiot

[Anonymous No.11916610]

>>11916604
Good point.
I've swapped to double and it returns 1 now.

[Anonymous No.11916616]

[Anonymous No.11916618]

>>11916610
That's not correct.

[Anonymous No.11916622]

>>11916618
Then at least tell me where the program is wrong, because the problem statement is near ilegible.

[Anonymous No.11916626]

>>11916622
The answer is close to 1 but not 1. Your computer program rounds it to 1, which is not the answer.

[Anonymous No.11916627]

>CStard is retarded and rude when called out for being retarded

[Anonymous No.11916629]

What does the ideal undergraduate curriculum consist of / what was your undergrad like /mg/?

[Anonymous No.11916631]

>>11916627
What if I told you that CS is math?

[Anonymous No.11916636]

>>11916631
Then you would be mistaken.

[Anonymous No.11916637]

>>11916595
gladly. Apply the exact same procedure but to the set [[5,1000]] instead of [[2,1000]]. Call the resulting number [math]x \in \mathbb{Q}, x>0[/math].
[math]\frac{2+3}{1+2\times3} = \frac{5}{7}[/math] and [math]\frac{4+\frac{5}{7}}{1+4\times \frac{5}{7}} = \frac{33}{27}[/math].
Conversly, [math]\frac{4+3}{1+4\times3} = \frac{7}{13}[/math] and [math]\frac{2+\frac{7}{13}}{1+2\times \frac{7}{13}} = \frac{13}{17}[/math] .
Therefore, the final result is either [math]\frac{x+\frac{33}{27}}{1+x\times \frac{33}{27}}[/math] or [math]\frac{x+\frac{13}{17}}{1+x\times \frac{13}{17}}[/math] which are different for any [math]x \neq 1[/math].

[Anonymous No.11916639]

>>11916631
It would be extremely copeful

[Anonymous No.11916640]

>>11916637
lmao

[Anonymous No.11916647]

>>11916455
>student uses the word 'trivial'
I literally never saw a student use that word (when used as a substitute for the proof of a claim) and get full marks for the exercise. What on earth were they thinking? (I guess they probably weren't)

For another proof-by-intimidation statement, try "by the fundamental laws of mathematics"

[Anonymous No.11916649]

>>11916640
>[Deleted]
uh oh, coping much?

[Anonymous No.11916650]

>>11916637
> [math]\frac{2+\frac{7}{13}}{1+2\times \frac{7}{13}} = \frac{13}{17}[/math]
Sure thing buddy

[Anonymous No.11916659]

>>11916639
[math]4 \mu[/math]

[Anonymous No.11916669]

>>11916560
500501/500499

[Anonymous No.11916673]

>>11916669
Bingo!

[Anonymous No.11916679]

>>11916669
Is it the sum of all multiplications with an odd number of terms divided by the sum of multiplications with an even number?

[Anonymous No.11916683]

>>11916679
what

[Anonymous No.11916686]

How does one cope with being a good student but lacking creative thinking?

[Anonymous No.11916687]

>>11916686
vodka

[Anonymous No.11916689]

>>11916687
I thought alcohol makes you dumber?

[Anonymous No.11916691]

>>11916683
The formula is that for two terms (a and b are one-term multiplications, 1 is zero-term, ab is two-term), and it's also that for three terms, so I'm extrapolating.

[Anonymous No.11916707]

>>11916689
Exactly.

[Anonymous No.11916715]

>>11916616
How do you solve this?

[Anonymous No.11916722]

>>11916715
Please don't spoil it until at least after 2 days.
Let people figure it out on their own.
>>11916669
Looks like that was a bit too easy for you. Here, try pic related instead.

[Anonymous No.11916728]

>>11916722
Fuck, left some typos. Just take x=a and y=b.

[Anonymous No.11916739]

>>11916722
>Looks like that was a bit too easy for you.

[Anonymous No.11916744]

>>11916679
I've confirmed that this works by induction.
Doesn't exactly work for computation tho.

[Anonymous No.11916855]

>>11916722
this one isn't even well-defined dipshit

[Anonymous No.11916858]

>>11916855
Yes it is.

[Anonymous No.11916873]

>>11916722
That would be -1, sir.

[Anonymous No.11916878]

>>11916669
you just used a computer didnt you

[Anonymous No.11916898]

>>11916858
No it's not
If you evaluate (2 (3 4)) you get -13/7, ( (2 3) 4) gives you -7/2

[Anonymous No.11916910]

>>11916878
>It is beneath the dignity of excellent men to waste their time in calculation when any peasant could do the work just as accurately with the aid of a machine.
>Gottfried Leibniz

[Anonymous No.11916912]

>>11916898
Shit you're right. I made a typo. Here's the actual question.

[Anonymous No.11916920]

>>11916912
>forgets again about the a and b
Bro...

[Anonymous No.11916930]

>>11916873
>solve anon's problem with computer
>gives 0.6 something
>hmmmmmm, why did he think it equals -1, is the floating point arithmetic completely fucked?
>anon posts the correct version
>rerun the algorithm
>-1.02
Very cheeky of you.

[Anonymous No.11916937]

The gauge of a convex set [math]C[/math] is defined as [math]\gamma(x, C) = \inf \{ \mu > 0 : x \in \mu C \}[/math]. Apparently, when the set is not closed, the gauges [math]\gamma(x, C)[/math] and [math]\gamma(x, \overline{C})[/math] (where [math]\overline{C}[/math] is the closure) are in general different.

But I don't understand how this is possible. Say that you have some convex, bounded set [math]C[/math] and a point [math]x \in 2 \overline{C}[/math] but [math]x \notin 2 C[/math]. Then clearly [math]\gamma(x, \overline{C}) = 2[/math]. But then we have a sequence of points in [math]2 C[/math] converging to [math]x[/math], so wouldn't it hold that [math]x \in (2+\epsilon) C[/math] for any [math]\epsilon[/math], and thus also [math]\gamma(x, C) = 2[/math]? Am I missing something here?

[Anonymous No.11916969]

>>11916937
Are you assuming that [math]0 \in C[/math]? Because I can definitely picture why they could be different otherwise.

[Anonymous No.11916985]

>this faggot has a 145+ IQ
I wanna kms

[Anonymous No.11916995]

>>11916985
stop advertising yourself on /sci/ faggot
nobody likes your cringe channel

[Anonymous No.11917003]

>>11916969
Ah yes I guess this needs to hold. But is that sufficient to make [math]\gamma(x,C) = \gamma(x,\overline{C})[/math] for all x?

[Anonymous No.11917006]

>>11916930
You're giving me too much credit, i fucked up in a very dumb way and got exactly -1.
On a second try, I got a rational number which is roughly -1.00202, so I guess our computations match at least.
Also, one could argue that the problem is still a bit fucked since f(2, 5) gets you zero in the denominator.

[Anonymous No.11917019]

>>11916995
Flammy is /ourguy/, newfriend.

[Anonymous No.11917030]

>>11916995
Not him, I'm just a salty hatewatcher
>>11917019
He's a /b/tard at most

[Anonymous No.11917034]

>>11917003
Probably not.
I thought about this a bit further, and the issue probably happens at infinite dimensions, which is why my geometric intuition is screaming at me that those functions are equal.
To give a shitty example, the set of all series in [math]l^2[/math] with finitely many non-zero entries forms a convex set, and its closure is [math]l^2[/math], but the function probably wonks because for series with infinitely many non-zero entries there is no fucking mu.

[Anonymous No.11917057]

>>11917034
Ah, you're right that it gets fucky when there is no [math]\mu[/math] whatsoever such that [math]x \in mu C[/math].

I was looking for an example where the two functions are both well-defined and finite but give a different value. Wouldn't the argument in >>11916937 show that such a case is not possible?

[Anonymous No.11917062]

>>11917006
Ok FUCK here's the TRULY FINAL version. Now it's completely well-defined and easily solvable (if you're smarter than a 12th grade Zimbabwean).

[Anonymous No.11917068]

>>11916930
Retarded code monkey uses floating point numbers when all this problem can be stated and solved in fractional integer arithmetic. Good job, Anon-kun.

[Anonymous No.11917072]

>>11917034
Alright, I think I figured out a proper example.
We consider the set of all sequences [math]a_n \in l^2[/math] where [math]|a_n| < 2/n^2[/math] and, for at most finitely many entries, [math]|a_n| \geq 1/n^2[/math].
Then, for the sequence [math]b_n = 2/n^2[/math] we should have that [math]\gamma (b_n, \overline{C} ) = 1[/math] but [math]\gamma (b_n, C) = 2[/math].
>>11917057
Don't think so.
Could be wrong tho.

[Anonymous No.11917074]

>>11913994
Which book should I spend my dollars on in order to self study real analysis?

[Anonymous No.11917079]

>>11917074
Lang

[Anonymous No.11917087]

>>11917006
>I got a rational number which is roughly -1
What is it exactly?

[Anonymous No.11917089]

>>11917087
it's like 900 digits long
OP is being a fag per usual

[Anonymous No.11917095]

>>11916686
Going into industry like everyone else who doesn't belong in academic research institutions (should do)

[Anonymous No.11917099]

>>11917089
How did you get -1.002002? (this is roughly correct BTW)? Did you use a computer?

[Anonymous No.11917102]

>>11914302
Extremely based

[Anonymous No.11917105]

>>11917074
Tao

[Anonymous No.11917107]

>>11917074
Abbott or Pugh if you're feeling adventurous. Cummings is also pretty excellent and cheap, which makes it a good supplement.

>>11917105 is also good

[Anonymous No.11917110]

>>11917099
yeah just 15 lines in python

[Anonymous No.11917113]

>>11917072
ok that's a cool example, thanks.

I'm still not quite sure what exactly makes it break down. for example I don't think there is anything about this definition which suggests a different behavior in infinite dimensions, so perhaps there is also a finite-dimensional counterexample

[Anonymous No.11917115]

>>11917110
You're not supposed to use a computer. You're supposed to work it out by hand (there is a closed formula)

[Anonymous No.11917129]

>You're not supposed to use a computer. You're supposed to work it out by hand

[Anonymous No.11917131]

What is even going on in this thread?

[Anonymous No.11917133]

>>11917129
Zimbabwean highschoolers don't have computers. You're not smarter if you need a computer to solve such an elementary problem.

[Anonymous No.11917135]

>theres a closed formula
yeah sure there is mr chaos

[Anonymous No.11917144]

>>11917099
Yes, I doubt this one is reasonable without a computer. It does have a closed-form, and I could conceive that somebody with way too much time to waste could maybe get it by hand from torturing the recurrence relation you get by always leaving the largest digit until last (assuming it's a well-defined process, which I'm not entirely convinced of given how much trouble he had even posting it), but I'm skeptical this one has a fast "trick" way of doing it like the first one. Problems that collapse with a trick generally don't have gross messy answers.

[Anonymous No.11917156]

>>11917144
There is a simple formula in terms of well-known functions.
>"trick" way of doing it like the first one
Can you tell me what the trick is? Perhaps a similar trick can be applied to this one.

[Anonymous No.11917163]

>>11917144
>I can't conceive of how someone would do this using creativity so it must be alright to brute force the problem with a computer gaining no insight into the problem or any other related problem
t. eternal CStard

[Anonymous No.11917169]

>>11917163
>i cant conceive of how anyone could find problems that i like to be total faggot shit thats just a waste of time

[Anonymous No.11917173]

>>11917163
What if I told you that CS is math?

[Anonymous No.11917174]

>>11917156
>There is a simple formula in terms of well-known functions.
I don't consider some gay fraction with harmonic numbers that spits out fractions hundreds of digits-long "simple". In fact I don't consider anything that returns a 900-digit fraction when fed an integer "simple"

[Anonymous No.11917183]

>>11917174
>I don't consider some gay fraction with harmonic numbers
Have you found such a fraction? Can you share it?

[Anonymous No.11917185]

>>11917183
>Have you found such a fraction?
Yes
>Can you share it?
No

[Anonymous No.11917188]

>>11917169
>total faggot shit that's just a waste of time
That's not the point of the problem, retard. If this is your approach to anything that looks alien to you imagine how you solve actually compelling problems with no precedent for handling them. Actions become habits, habits become character and so on.

[Anonymous No.11917191]

>>11917185
What was the trick you used for the first one? I want to compare solutions.

[Anonymous No.11917193]

>>11917188
ok cool, but i just found a rational function that works for all the terms in desmos
you have fun unraveling the recursive trash or whatever

[Anonymous No.11917199]

Reminder.

[Anonymous No.11917201]

>>11916912
Answer: The man starts with 2 and 5, and the division isn't defined so the world explodes. □

[Anonymous No.11917202]

>>11917113
>so perhaps there is also a finite-dimensional counterexample
I really don't there is, because in that case gamma should be basically a norm (coincidentally equivalent to the euclidean one) and then we get the continuity hypothesis that you used in your proof.

[Anonymous No.11917208]

>>11917199
This isn't CS vs math.png it's CS vs stats.png
Name your files correctly

[Anonymous No.11917223]

>>11917133
Elementary =\= worthwhile

There are absolutely rote computations that no one without a lot of time to waste wants to do by hand.

[Anonymous No.11917224]

>>11917208
Stats is math.

[Anonymous No.11917225]

>is that a...a finite sum?
>oh godDD I'M COOMPOOOOOOOOOOOOOTIIIIIIIINGGG AAAAARRRRGGGHHH

[Anonymous No.11917229]

>>11917224
Stats isn't math

[Anonymous No.11917234]

>>11917225
lel, too supid

[Anonymous No.11917253]

>>11917225
No bully, I need to practice programming to get a job code monkeying.

[Anonymous No.11917254]

>>11917144
to check that the problem is well-defined, you just need to check that [math]f(x,y) = f(y,x)[/math] and [math]f(f(x,y),z) = f(x, f(y,z))[/math]
>>11917115
>>11917156
Indeed, there is a way to solve this without a computer.
Let [math]F(x_1, \dots x_n)[/math] be the result of the operation described in the problem, on numbers [math]x_1, \dots x_n[/math]. Let's also denote by [math]\sigma_n, \sigma_{n-1}[/math] the nth and (n-1)th symmetric polynomials of those variables; i.e. [math]\sigma_n[/math] is the product of all x_n's, [math]\sigma_{n-1}[/math] is the symmetric polynomial of degree n-1.
Then, it is easily verified that for any [math]n \geq 2[/math],
[eqn]F(x_1 - 1, \dots x_{n} - 1) = - \frac{(n+1)\sigma_n - 2\sigma_{n-1}}{(n-1)\sigma_n - 2\sigma_{n-1}} [/eqn]
The problem asks to evaluate [math]F(2, 3, \dots 1000)[/math], which the interested reader can easily do using the formula above.

[Anonymous No.11917263]

>>11917254
Can you share how you found that formula?

[Anonymous No.11917275]

>>11917229
>measure theory isn't math

[Anonymous No.11917283]

>>11917275
>measure theory
You mean probability theory which is built from measure theory and real analysis. That is math, but statisticians are not probability theoriests or analysts, they don't do pure mathematical work.

[Anonymous No.11917284]

>>11917263
turn on your favorite computer algebra software
make it compute the function F(x_1...x_n) for you, for some few small n's
stare into the expressions until you notice a pattern

[Anonymous No.11917285]

>>11917254
>simple formula in terms of well-known functions
>symmetric poly shit
the evens are [math]\displaystyle\frac{x^2+x+2}{x^2+x-2}[/math]
and the odds are the reciprocal of that
proof by graph

[Anonymous No.11917288]

>>11917254
>>11917284
>there is a way to solve this without a computer
>how
>turn on your favorite computer algebra software

[Anonymous No.11917290]

>>11917285
that's wrong

[Anonymous No.11917302]

i'm going to sleep. /gnmg/

>>11917288
you got me...
if it's an excuse, I would do exactly the same if i didn't have a computer - just that it would take me 3 hours instead of 20 minutes

[Anonymous No.11917303]

>>11917302
Night, lad.

[Anonymous No.11917321]

its all downhill from here bois

[Anonymous No.11917333]

>>11917321
>already too spatially retarded to take a proper picture
Indeed it is over for you.

[Anonymous No.11917336]

absolute state of this thread. here's a question for Ramseychads:
given a coloring of the natural numbers with finitely many colors, must there always exist a set of arbitrarily large monochromatic arithmetic progressions which all have the same ratio (i.e. common difference)?

[Anonymous No.11917338]

[Anonymous No.11917413]

>Fucking hate the anime trannies
>Can't talk shit to them because they're smarter than me
Existence is suffering.

[Anonymous No.11917423]

>>11917413
>they're smarter than me
You could just break their necks with one hand and take comfort in the knowledge that they and most of the faggots in this thread are physically inferior.

[Anonymous No.11917429]

Threadly reminder to ask physicists about Seiberg-Witten theory

[Anonymous No.11917440]

>>11917429
I'm not much into physics, though.

[Anonymous No.11917445]

>>11917336
Isn't that just Van der Waerden's theorem?

[Anonymous No.11917449]

https://youtu.be/FRFfkbvKKdI

[Anonymous No.11917485]

>>11917321
Nice D&F nerd. Do you have a Foote fetish yet?

[Anonymous No.11917496]

>>11917445
Van der Waerden's theorem only guarantees that if you want a single monochromatic arithmetic progression of length n you can find it. But the difference may depend on how long you demand your progression to be. He's asking if there exists a single fixed difference that admits arbitrarily long monochromatic progressions.

[Anonymous No.11917527]

>>11917423
I am 6'1 but I don't lift so I'm not sure if that's true either.

[Anonymous No.11917539]

>>11917527
>Be me
>Anime tranny
>6'3
>Workout legs often
~ Hi ~

[Anonymous No.11917547]

>>11917539
>tranny at 6’3”
Just rope. You look like a freak guaranteed.

[Anonymous No.11917552]

>>11917547
Wow do you hate the entire wnba then?

[Anonymous No.11917557]

>>11917321
What's wrong with natural transformations?

[Anonymous No.11917585]

>>11917552
Yes. They should all be barefoot and pregnant.

[Anonymous No.11917589]

>>11917585
Hey, Peterson recovered

[Anonymous No.11917613]

>>11917413
There isn't anything wrong with insulting your betters.
But you still probably shouldn't insult them, since they get off to it.

[Anonymous No.11917615]

>>11917589
who?

[Anonymous No.11917616]

>>11917615
https://en.wikipedia.org/wiki/Camille_Jordan

[Anonymous No.11917617]

>>11917527
>>11917539
Sad!

[Anonymous No.11917734]

>trannies
>smarter than anyone
If they were so smart, why weren't they born women?

[Anonymous No.11917755]

what the fuck is a Coxeter–Dynkin diagram and why should i care?

[Anonymous No.11917758]

>>11917734
But maybe they were though

[Anonymous No.11917766]

>>11917734
We are better than women because we have more options in bed ;)

[Anonymous No.11917782]

I know I can multiply with i "above and below" to get the desired result, but what am I doing wrong (in I assume the marked step) here?
[eqn]
{1 \over i} = {1 \over \sqrt{-1}} = [\sqrt{-1}]^{-1} \overset{?}{=} \sqrt{(-1)^{-1}}
= \sqrt{1 \over -1} = \sqrt{-1 \over 1} = \sqrt{-1}
[/eqn]

[Anonymous No.11917800]

>>11917782
Right, you can't do the marked step, since that equality isn't necessarily true with complex numbers. A similar argument of this kind shows that [math]i^2=i\cdot i= \sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)\cdot (-1)}=\sqrt{1}=1.[/math]

[Anonymous No.11917827]

>>11917734
>Phoneposter
You have to go back.

[Anonymous No.11917832]

>>11916031
>fuck off somewhere else like to the physics general or something
Physics IS mathematics. Every physical phenomena will eventually be mathematically formulated.

[Anonymous No.11917841]

>>11917800
I see

[Anonymous No.11917853]

>https://en.wikipedia.org/wiki/Ethnomathematics
Would this be a good field of study? I'm brown, could I ride out tenure in a mathematics department on this?

[Anonymous No.11917864]

>>11917782
The base of your problem is that square roots come in pairs. [math]\sqrt{1} = \pm 1 and \sqrt{-1} = \pm i[/math] , but you've implicitly chosen to use 1 and i here, which there's no justification for doing, and because you forgot that the sign is ambiguous you get a sign error.
In the reals you can make square root expressions not ambiguous by saying "I always want the positive root, always throw away the negative one" but the complex numbers don't have an ordering so that rule doesn't work.

[Anonymous No.11917938]

>>11917832
Math is axiomatic; physics is not.

[Anonymous No.11917957]

>>11917938
i mean, physics is pretty axiomatic at this point

[Anonymous No.11917963]

>>11917758
They weren't
>>11917766
You will never be a woman

[Anonymous No.11917969]

>>11917963
But maybe they were tho

[Anonymous No.11917972]

[Anonymous No.11917986]

>>11917972
this colouring seems incredibly random

[Anonymous No.11917992]

>>11917969
Biologically impossible to transition between sexes. They weren’t, aren’t, and never will be women.

[Anonymous No.11917996]

>>11917972
>the gods understand English, but will answer all questions in their own language
why are these gods such dickheads

[Anonymous No.11917997]

>>11917992
Yeah whatever but what if they were

[Anonymous No.11917999]

>>11917997
Die faggot.

[Anonymous No.11918009]

>>11917996
Most people prefer communicating in their native tongue.

[Anonymous No.11918010]

>>11918009
not with foreigners who don't understand a single word of it

[Anonymous No.11918021]

>>11917999
Nice trips, also rude
I'm just asking what if these transpeople were born women, why did you not think of that

[Anonymous No.11918034]

>>11918010
Someone's never been to Japan.

[Anonymous No.11918044]

>>11918021
he's microbrain, unfortunately

[Anonymous No.11918059]

Does this kind of curve has a name?
[math]
\sqrt{(x-x_1)^2 + (y-y_1)^2} + c = \frac{1}{a}\sqrt{(x-x_2)^2 + (y-y_2)^2}
[/math]
[math]c>0, a>0 [/math]
If c = 0, it's just a circle. Otherwise, it looks like pic related.
It came up in my work and I'm wondering if there is a name for it so that I can search for it.

[Anonymous No.11918068]

>>11916455
>>11916647
>NOOOOO YOU HAVE TO JUSTIFY EVERY TINY DETAIL
fuck you TA niggers, I hate you

[Anonymous No.11918070]

>>11918059
try x1=0
x2=4
y1=0
y2=0
a=0.8
c=5

[Anonymous No.11918072]

>>11918068
>NOOOOO YOU HAVE TO JUSTIFY EVERY TINY DETAIL

[Anonymous No.11918083]

>>11918068
Yes, you idiot. Leave it blank or incomplete instead of trying to oneup people

[Anonymous No.11918091]

>>11918070
I forgot to state that I only need a > 1 at the moment. Otherwise it's completely different.

[Anonymous No.11918093]

>>11918083
Is there a bigger cope than leaving a problem blank?

[Anonymous No.11918121]

>>11918093
embarrassing yourself by writing out some laughable nonsense in a desperate attempt to scrabble out an extra mark and a half you don't deserve

[Anonymous No.11918126]

>>11918121
My favorite is when they just restate the problem.

[Anonymous No.11918151]

>>11918126
My favorite is when the TA is a sniveling little insect faggot that sucks off corrupt academics all day long.

[Anonymous No.11918167]

>>11918151
Cope.

[Anonymous No.11918172]

>>11917938
We don’t know that

[Anonymous No.11918175]

>>11918167
Not even a mathfag and also a 4.0 student so suck my dick pussy.

[Anonymous No.11918197]

>>11914023
In radians ds = r*dtheta (ds being the arc length differential).

If you advance one interval of dtheta, how much ds have you advanced? In radians it is a direct 1:1 relationship.
Now to get the change in the height, sin(theta), you multiply that change in arc-length by it's projection along the vertical axis. Which should give you cos(theta). This gives you how much "extra" height that differential step added.

In degrees there is not a direct 1:1 relationship, so your arc-length will not correspond to the angle so easily. So there will be a normalization term out in front of your derivative.

[Anonymous No.11918208]

why did the csfag snap

[Anonymous No.11918215]

>>11918208
Low self-esteem.

[Anonymous No.11918219]

>>11918208
they hate themselves and know they aren’t doing anything with their tiny deformed brains worth remembering. when you remind them of this they lose their shit or dissimulate and project. Its very funny.

[Anonymous No.11918233]

>>11918151
Based. pretentious teacher pet TAs are cucks and you can see it from a mile away

[Anonymous No.11918295]

What are the prerequisites for algebraic geometry?

[Anonymous No.11918297]

>>11918295
commutative algebra, autism

[Anonymous No.11918302]

>>11918295
No idea, but I wanted to tell the story about the time I was looking for basic plane and space geometry stuff and ended up downloading Algebraic Geometry by Hartshorne. I should have known what was up but getting filtered at like page 3 was fun

[Anonymous No.11918317]

>>11918295
algebra
geometry

[Anonymous No.11918323]

>>11918317
>geometry
nope, that's the joke

[Anonymous No.11918331]

>>11918295
Why the fuck do you want to learn algebraic geometry?

[Anonymous No.11918395]

>>11917449
thx, this is good

[Anonymous No.11918493]

/gdmg/

If someone here likes manifolds:
>On Lusternik-Schnirelmann category and topological complexity of no k-equal manifolds
https://arxiv.org/pdf/2007.08704.pdf
>Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds
https://arxiv.org/pdf/2007.08899.pdf
Somebody also seemed interested in K-theory, but I can't remember which one. Here's to you:
>Computations in higher twisted K-theory
https://arxiv.org/pdf/2007.08964.pdf
>Semilocal Milnor K-theory
https://arxiv.org/pdf/2007.09044.pdf

>>11918331
What's wrong with wanting to learn something?

[Anonymous No.11918521]

>>11918493
Why would anyone like manifolds?

[Anonymous No.11918525]

>>11918521
Why would anyone not like manifolds?

[Anonymous No.11918526]

>>11918521
Some people do. Who are we to judge?

[Anonymous No.11918528]

>>11918525
manifolds are scary

[Anonymous No.11918548]

>>11918528
Eat
Sleep
Math
Repeat

That’s the path we’ve choosen. There’s nothing scary about mathematics.

[Anonymous No.11918549]

what's the best book about abelian varieties?

[Anonymous No.11918552]

>>11918525
Other than information geometry, manifold is useless in the field that I like, which is probability theory.

[Anonymous No.11918562]

will differential geometry help me with mad cnc skills

[Anonymous No.11918606]

das neuer: >>11918605

[Anonymous No.11918734]

>>11918295
Basic undergrad education, including abstract algebra, and real and complex analysis. Commutative algebra, at the level of Atiyah macdonald or Bosch. You will need more CA than that, but you can always reference later, don't spend too much time doing that. Field and Galois theory, at the level of Morandi. Homological algebra - hard to give a reference, and you shouldn't spend much time here until you need it, but to start, Bosch has a nice intro and Aluffi has an accessible course. I'd also like to mention that Aluffi has pretty much all of these algebra topics (perhaps not as in-depth, but a good starting point), either in the main body of text or in the exercises. It even has a brief intro to classical AG

[Anonymous No.11919404]

>>11918083
>he thinks avoiding a trivial proof for the sake of saving time and space is an attempt to one-up others
I can feel your inferiority complex through my screen