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12:06 AM
it is algebra
2 hours later…
1:44 AM
@0celóñe7 $a^2 + b^2$ is indeed $c^2$ for a right angled triangle
1:58 AM
@Phase uh, ok?
2 hours later…
3:47 AM
@JohnRennie greetings
Amazon is sending me emails with SSDs now
I can get an 850 500GB for like 120 bucks
Morning :-)
@0celóñe7 I bet that's not an M.2 disk like your Samsung though.
no, but probably faster
That's be a standard SATA SSD.
Still fast but not as fast as the M.2 disks.
my M.2 is slow
I don't think you realise how fast that disk of yours is.
3:51 AM
when I open my download folder it takes a good 5 seconds to load
it's slow
I've never seen a standard SATA disk anywhere near as fast, even with the lower benchmarks speeds you get.
@0celóñe7 that's not the disk.
It can't be.
There must be something else causing that delay.
I get the same speed when I run it as secondary after booting on my HDD
I wonder if something is slowing Explorer down ...
@JohnRennie I have monitored disk activity and it's low
near zero
for some reason the explorer has a hard time grabbing disk usage from the ether
maybe I have a virus hogging my disk and masking it
4:00 AM
wtf instadownload
That's the old File Manager from Windows NT4.
you trying to give me a virus?
It's a self extracting executable. Run it and it will unzip itself.
Windows defender has blocked it
what is this
It doesn't tamper with your system at all, it is just a self extracting zip.
Windows Defender can be paranoid at times. I use that executable on all my PCs and servers. It isn't malware!
4:02 AM
ok it unzipped
what now
Run C:\win32app\winfile\winfile.exe
lmao what is this
You should see something like:
On the left pane double click Users, then your username then Downloads and see how fast it displays the files.
4:05 AM
instaintly, but so does inwdows right now
first one in a week I swear
random read is still terrible, sequential read is back up
Ah, OK. So opening Downloads is slow only when you've recently restarted Windows?
you think it's defender doing a scan or something?
That will be because Explorer is looking in each file to see if it has an icon and/or other metadata.
@JohnRennie but the disk usage is like 1%
it's clearly not trying very hard
Next time you restart try using File Manager to look at the Downloads folder before you open it in Explorer. I bet it will still be instant.
4:07 AM
Hmm, ok
@JohnRennie In other news, my professor scheduled a geometric analysis seminar without consulting me
I'd been asking for a year, he sent me an email with the time, and it conflicts
Scheduling is a damn mess
So is your team the skins or the titans?
Come to the dark empire
4:22 AM
@JohnRennie I'm taking a fluid mechanics class. Is that something you like/know about?
If I'm honest I never learned enough about fluid mechanics to really understand it.
From where I'm standing it looks like an ad hoc collection of random attempts to solve basically intractable equations.
Maybe there is an underlying elegance to the subject, but I can't see it from here.
there are 8 books for the course
oh god, Landau
Hopefully they're not 8 intractable books.
some of these books are quite advanced
I wonder if Balarka like this style of music:
4:32 AM
Do you? @Secret
It's weird, but its a whole new dimension I don't know about
my power went out
just in my room
4:51 AM
check the fuse box
@skullpatrol what kind of advice is that
Of course I turned the power back on
Advice to prevent it from happening again.
Check for what?
How did you turn the power back on?
Flipped the switch
5:24 AM
Sounds like you're overloading the circuit
5:40 AM
After a long negotiation, I have concluded that it is better to start the MSE University project. Currently a bachelor program in pure mathematics is offered (theoretical physics and phd programs would be offered in future). To people who are interested in participating this project as students or tutors, please go to Registration for the MSE University and inform their enrollments.
7:48 AM
@MathematicsAminPhysics I guess this is an informal program (?) and not a officially recognized one
8:42 AM
@Blue Yes
9:24 AM
@DonThomasReyes It seems to me that the force you want is going to be $\frac{20 \text{kg}\cdot g \cdot \sin 30^\circ - 80\text{N}}{ \cos 30^\circ}$. Think about breaking both gravity and your horizontal force into a component that's parallel to the slope and a component that's perpendicular to the slope, and ignore the component that's perpendicular.
1 hour later…
10:52 AM
Q: Moving a question to a different forum or asking for more attention

ubuntu_noobI'm posting in meta for the first time so please don't mind some culturally outrageous thing that I would end up doing here and inform me about my peculiarities. So I have this question on stack exchange physics, about the fermi level of materials and I have not received any assistance on it as o...

11:06 AM
@MathematicsAminPhysics So how many have registered? Who are going to be teach? What will be taught ?
11:22 AM
@Blue You can see them in Registration for the MSE University. As I said, details would be determined by the committee.
I don't understand. If I take this course, I can say afterwards "I got my bachelor's degree from a chat room", right?
With a good approximation, yes.
And in the future, you'll be offering PhD's?
So I'd be able to call myself Dr Dawood?
11:33 AM
After defending your dissertation.
I see. May I comment that I find this whole concept incredibly insulting, to everyone who has a genuine degree from a genuine bricks-and-mortar university?
It is really important that you have a genuine phd degree?
If you like, I can call you Dr Dawood.
I have an actual degree, not a PhD, from an actual university. I worked very hard for it, for a number of years. It's really important that this is not devalued by a proliferation of people claiming they have degrees, when they really don't.
This project is not suitable for those people whose main concerns are earning degree at all.
11:39 AM
question: when referring to this chat room from the outside, do we prefer to say 1) "the h Bar", 2) "The H Bar", 3) "the hbar", or 4) "the $\hbar$"?
@Jim Do you mean "say" or "write"?
@DawoodibnKareem yes
Actually, Rob, I find those insulting too.
@0celóñe7 what's up?
11:40 AM
everybody says 1)?
I don't talk about H bar.
@Jim 1), since that's what it is actually named. 4) is also fine, I guess, but you have to expect confusion :P
First rule of H bar.
11:42 AM
@djsmiley2k That is why you failed the first rule
Wait, first rule is 'Gravity is not a force'
2nd rule then :D
4) requires rendering :P
I don't have whatever it is, that makes stuff appear properly, here :<
^Cheap hack
ℏ for the same price.
11:47 AM
@ACuriousMind Too late now
I ditched measure theoretic probability for Elven lord's modern algebra topics course
@skullpatrol not really, there's also something special about seeing $\$\backslash hbar\$$
@djsmiley2k here
$\bar h$ is just wrong for so many reasons
11:51 AM
@Jim I have used that symbol a few times, but not to denote Planck's constant :P
@ACuriousMind I've used $\bar H$, but never $\bar h$. It's like using goto in programming; a lot of languages support it, but any time you use it, you become immediately wrong
@Jim In CFT, the conformal weight is occasionally written $(h,\bar{h})$.
And it's lowercase because these two things are just numbers
Uppercase for a number would be just wrong
@ACuriousMind Uh, energy eigenvalue?
Why is it so "wrong" to use goto?
11:57 AM
@ACuriousMind That's too bad. I'm sad to find out that all this time, physicists doing CFT have been wrong
@skullpatrol taboo, look into the history of goto. Essentially it's more trouble than it's worth
@ACuriousMind here is the book www-users.math.umn.edu/~webb/RepBook
@0celóñe7 Yeah, that's wrong.
BASIC started it.
@ACuriousMind what? Everyone writes $E\psi=H\psi$
what should it be?
@0celóñe7 Still wrong.
11:59 AM
@skullpatrol I seriously doubt that.
@0celóñe7 nice
It's clearly named after Kelly.
@ACuriousMind I figured I could take probability next year but he won't be teaching this course again
R. Kelly?
Scott Kelly
@ACuriousMind so is this something you know about?
@ACuriousMind and you didn't say what the energy eigenvalue should be
12:02 PM
@ACuriousMind avogadro's number
@0celóñe7 that book includes a bit more than I know about finite group representations, but yes, it is something I know about
@Jim also wrong
atomic number, Z
@ACuriousMind I don't think there's any way we get through the book
it looks pretty dense
12:04 PM
@0celóñe7 My own group theory course covered roughly the first four chapters before we moved on to Lie groups.
@ACuriousMind wtf, in one semester?
Thnx @DawoodibnKareem :P
@0celóñe7 It was pretty intense (also, a lot of the time-intensive proofs/computations were outsourced to the exercises)
@ACuriousMind Elven lord loves taking one class for some ridiculous proof
CS ed.SE would be interested in the taboos of using goto @Jim
12:08 PM
hypothes.is a great web annotation service. works on arXiv papers too.
@skullpatrol betcha they know all about it
Yeah, that's a good bet.
Thanks for sharing @heather
You should clarify whether the "divergence" of the stress-energy tensor is the ordinary or the covariant divergence. — ACuriousMind ♦ 12 mins ago
@ACuriousMind What?
@0celóñe7 $\partial_\mu T^{\mu\nu}$ vs. $\nabla_\mu T^{\mu\nu}$.
The former is not even defined.
It's defined in the minds of physicists.
12:22 PM
@0celóñe7 Exactly, and this is a physics site, which is why we got questions like physics.stackexchange.com/q/346793/50583 and its followup physics.stackexchange.com/q/348085/50583, where one point of confusion was precisely this.
The second seems to have nothing to do with this
And not sure about the first one either
Partial derivatives on tensors are not necessary tensors themselves, but can we really do contractions like that?
goto is still a keyword in Java, but will throw a compile-time exception if you use it
@0celóñe7 is it not defined on the jet bundle or some shit
12:32 PM
that's some serious hate if you make the program crash for even using it once in your code
@Jim A compile error is not the same thing as an exception.
@DawoodibnKareem yeah, my brain thought error but I wrote exception because I'm used to complaining about the latter
@Slereah I think it is - it's not a proper tensor field, but it does live somewhere
The jet bundle is basically Detroit
Where all the worst things live
$\partial_\mu T^{\mu\nu}$ this is not mathematically defined?
12:38 PM
@DawoodibnKareem Excuse me if I have disturbed you.
@MathematicsAminPhysics Well, if it's not a bachelor's degree, you shouldn't call it a bachelor's degree. Similarly, the so-called PhD.
@ACuriousMind How exactly? Partial derivatives depend on coordinates.
Thanks @Jim I also found this :-)
It lives in some function space over an open set defined by the chart
Pretty meaningless
We can have derivative of matrix functions, so what prevents having a derivative of a tensor function which are multilinear maps?
12:42 PM
@0celóñe7 You claim that e.g. the LL pseudotensor is meaningless?
If so, then you have finally abandoned physics for pure math entirely :P
Are partial derivatives of tensors pseudotensor, I do recall back in the undergrad GR course whenever we tried to transform it into another coordinate, it does not just having a change of sign plus the jacobian matrices that showed it indeed transforms like a tensor?
@ACuriousMind that thing is defined wrt an explicit asymptotic coordinate system
I'm saying that just saying "ordinary divergence equals zero" is completely meaningless
@Secret what
What's what a covariant derivative does
@0celóñe7 So LL's third requirement is meaningless?
I think all physicists should take a good hard look at the axiomatic definition of connections
@ACuriousMind I just said there's an explicit coordinate system in play.
@0celóñe7 No, I am not talking about the covariant derivative, "$\partial_\mu T^{\mu\nu}$" seemed to be the ordinary partial derivative of a tensor that is formally resembling a divergence, That cannot even be a pseudotensor?
12:49 PM
@ACuriousMind But you should know that there are sufficiently bad coordinate systems, like ones where the metric decays as $1/r+1/r^\alpha$, where "invariant" quantities defined by such pseudotensors are in fact not invariant
there are some physicist shenanigans with integration by parts that fail
@0celóñe7 Well, no one is claiming the pseudo-tensor is invariant. What they are claiming is that its 4-divergence is invariant (I think)
And I am willing to believe you that there are some mathematical issues with actually defining these objects to a mathematical standard of rigor
@ACuriousMind Yes, ordinary divergence in a chart, which is specified
I'm saying that writing down $\partial_\mu T^{\mu\nu}=0$ without further commentary is meaningless
right ok
Because you can just go to polar coordinates or something and that's no longer true
You see the same issue in $\Bbb R^n$, really. It's a danger of the compact notation we use for GR
@0celóñe7 True. Still, I think it would be good to explicitly state that the divergence that's meant in Slereah's answer is a covariant divergence. You know how bad physicists are at this, why make it even harder for them? :P
12:53 PM
nvm, I misread, it's not the covariant derivative, all is good
@Secret Actually no, you're right. $\partial_\mu T^{\mu\nu}=0$ has no meaning in special relativity either
You have to specify a Cartesian coordinate system
@Secret y'know, since you deleted the message that refers to, someone reading this after the fact is not going to know what you're taking about at all :P
People like to say that $\partial=\nabla$ when the metric is flat, but that's only true in a flat coordinate system
@ACuriousMind Opps sorry, that's how it is commonly done in the maths chat, looks like I inherited one of their cultural habits...
In some fields (good fields like analysis) it is always assumed to be a Cartesian coordinate system, so we can get away with it
But in GR where people like to have coordinate freedom, they also need to shoulder some responsibility and be more precise
@ACuriousMind trying to come up with some flippant Marie-Antoinette response to that
12:57 PM
I don't think I've ever seen anyone but Straumann fully commit to the coordinate-free notation.
@ACuriousMind Wald
Abstract index notation is a beautiful thing
I think I recall in my undergrad GR course, we wrote something like this for the covariant divergence: $\nabla_{\mu}T^{\mu\nu}$
He does it incorrectly, but still :P
@Secret Yes, and that has meaning
@0celóñe7 Ohhhh, I think many physicsts will defend their notations by claiming it was abstract index notation all along!
That's the "same" object regardless of coordinate system, so you can be forgiven for not explicitly stating that you're using one.
@ACuriousMind But the problem is that $\partial_a$ has no meaning in the abstract index notation.
You need coordinates to define it, and it's a patently different object than another derivative in a different coordinate system.
12:59 PM
I wasn't saying they would be right ;) (I don't really know about abstract index notation :P)
Wald does write things like $\nabla_a V^b=\partial_a V^b+\Gamma^b{}_{ac}V^c$, which is horrendous
@ACuriousMind In abstract indices you denote the argument of a tensor (seen as a multilinear map) by an index. So if $\omega$ is a 1-form and $v$ a vector, then $\omega(v)=\omega_av^a$, etc.
You can use the flexibility of indices without having to define coordinate systems
That being said, often you want to work with Christoffel symbols, etc., and those are not tensors
And this being a notation, you can't really "prove" anything about it, but it's certainly against the spirit to write something like $\Gamma^a{}_{bc}$.
@ACuriousMind Now Straumann is also dishonest. He doesn't use special symbols for abstract indices. He does everything with indices in one style, then says "interpret the indices as abstract when applicable"
Christoffel symbols is like one of the most horrible looking thing under a change of coordinates
$\partial_{\mu}\partial_{\nu}-\partial_{\nu}\partial_{\mu}$ is one of the terms you can get if I recall
1:16 PM
@Secret That's...not a term that appears in the Christoffels at all.
@MathematicsAminPhysics It'll be better if you don't call those "degrees". You might just say that the MSE University you are designing will offer "courses" (like MOOCs do) by voluntary teachers in chat rooms. You can set course timing as events. Anyone who is interested can register themselves there. However, since it is a virtual chat room we can't do anything if the teachers fail to turn up at the right time.
Q: Christoffel symbol transformation law

ArthurIt is known that the transformation rule when you change coordinate frames of the Christoffel symbol is: $$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^\nu}{\partial x^\gamma \over...

Sorry I misremembered, it appears I only remember there are 2 terms and one of them is a product
It seems ridiculous when you say the MSE university will be offering "degrees" like "Bachelor/Masters/PhD" :P
@Blue There is a dedicated room for feedback where most of what you want to say has probably already been said. Maybe have a look there before you start a new discussion
1:19 PM
@ACuriousMind Oh, thanks. I'll see. I don't think I have much more to say about this though.
The whole MSE university idea is a big utopian nonsense. It has already been settled in various lengthy conversations in chatrooms and piles of downvotes on the MSE meta
I don't see why the OP is pursuing this far fetched trash anymore, but if he so wishes.
@BalarkaSen I'm taking a course on finite group rep theory
what's a group
Not to mention, I have not even have the chance to launch the ICBM yet! (metaphor)
1:36 PM
Please confine discussion of the "MSE university" to its dedicated chatroom. Thanks everyone.
8 messages moved to Trash
@ACuriousMind What the hell?
I think it is a dirty work that one wants to criticize an idea in this way.
Yeah, I am surpsied @ACuriousMind
Has someone told you to censor this?
@0celóñe7 What?
You said yourself that some of what I just moved was not "Nice".
And since there is already a room full of discussion about this topic, I see no purpose in having it also spill into here.
Case closed.
1:44 PM
@ACuriousMind Really? What tool do you use to do simple quick stuff with pictures (like cropping) then?
@Mostafaııııııııııııııııııııııı GIMP
Does it have a GUI?
or scripting only?
ahhh I missed the AMA with @Mithrandir24601 :(
@Mostafaııııııııııııııııııııııı Me too. I was sleeping then. :P
It was around 2am-3am over here
Seems you guys have forgotten that bookmarking a conversation is feature that exists @ACuriousMind
Going to bookmark it now
1:50 PM
@Mostafaııııııııııııııııııııııı It has a GUI
@Mostafaııııııııııııııııııııııı Gimp's got a gui
You still there? @0celóñe7
going to bookmark from the beginning of the AMA to this:
yesterday, by AccidentalFourierTransform
later alligator
@skullpatrol sure
1:57 PM
@Mostafaııııııııııııııııııııııı Heh, tradition must be upheld :P
OK, for the sake of tradition:
I have bookmarked the AMA with Mithrandir
Thanks @Mostafaııııııııııııııııııııııı
@skullpatrol no problem ;)
consider a free particle in 1D. Is its Hilbert space separable?
@Mostafaııııııııııııııııııııııı Yes
What do all those lines on your username mean? @Mostafaııııııııııııııııııııııı
2:12 PM
@ACuriousMind but separable means it admits a countable orthonormal basis. considering that to each and every real number is assigned an eigenvector of this Hilbert space, how is this countable then?
@Mostafaııııııııııııııııııııııı Ah, that's because the $\lvert x\rangle$ are not actually vectors in the Hilbert space!
Other than being unique :-)
@skullpatrol nothing. Just a totally random thing
@ACuriousMind What??
Can you explain more?
@ACuriousMind What is the Hilbert space of that system then?
@Mostafaııııııııııııııııııııııı See e.g. physics.stackexchange.com/q/249851/50583, physics.stackexchange.com/a/90261/50583. The Hilbert space is $L^2$, but the $\lvert x_0\rangle$ would correspond to $\delta(x-x_0)$, which isn't even a function, let alone square-integrable. A countable Hilbert basis of $L^2$ is given e.g. by the stationary states of the harmonic oscillator.
@ACuriousMind Seems pretty square integrable to me. Clearly $\delta^2=\delta$ by evaluating at points, so $\int\delta^2=\int\delta=1$.
@ACuriousMind Hmm, are the dirac deltas a basis of the space of tempered distributions in some sense?
2:25 PM
From that answer: a countable set can span a vector space of uncountable dimension. wow!
what are some of the holes in the standard model?
@ACuriousMind but Shankar never talked about these ::feeling betrayed::
@Mostafaııııııııııııııııııııııı That's wrong.
@0celóñe7 that quote? (from ACM's answer)
@Mostafaııııııııııııııııııııııı I guess it should be a "countably infinite set" rather than any "countable set" ? (Pardon me if I'm saying something silly)
2:28 PM
@ACuriousMind Uh, did you really say that $L^2(\Bbb R)$ has uncountable dimension?
@Blue yes he means a countably infinite set
@Mostafaııııııııııııııııııııııı Yeah. Thanks
It's just that span means something different in infinite dimensional vector spaces.
Yeah, ACM is mixing terminologies.
He is using dimension in the Hamel sense but span in the Hilbert sense.
2:30 PM
I think ACM is aware of these. He's just saying it's uncountable dimensional if we take span in the Hamel sense, but countable dimensional if we take span in the Hilbert sense.
I have a possibly silly question about Feynman diagrams if anyone can help ...
"a countable set can span a vector space of uncountable dimension" is just a slogan for that. He's writing an answer for physics.SE, not math.SE, after all.
Wow. This is the first time I'm seeing JR asking a doubt. :P :)
Yes, ACM is aware of these, but I think that most doubts in physics are because of poorly explained math
If you take a diagram representing Compton scattering then you have a photon and electron coming in and a photon and an electron going out. If you rotate the diagram 90º, i.e. swap the time and spaces axes, then you have another diagram with an electron and poitron coming in and two photons going out.
2:33 PM
So the second diagram looks like pair annihilation to two photons.
Is it a general rule that if you rotate a Feynman diagram 90º then you'll get another diagram that represents something physical?
Presumably this applies to diagrams with any number of vertices since we're only considering what comes in from infinity and what goes out to infinity.
what the heck is a free module
"In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles."
@0celóñe7 But why doesn't Shankar mention this??
2:40 PM
"You can rotate an vertex into eight different orientations, each describing a completely different physical process. Included are electron-positron creations and annihilations, which contain the essence of Einstein's mass-energy relation ." Interesting. Got to read about this
@0celóñe7 I expressly said "as a vector space over $\mathbb{C}$", in which case the claim is true.
@Mostafaııııııııııııııııııııııı because the math is too hard
@Mostafaııııııııııııııııııııııı It's not Shankar's fault. Almost no physicist discusses these issue unless they come up - otherwise everyone just pretends one is doing linear algebra. Not saying that's the optimal state of things, but that's how it is
@Mostafa loool
2:41 PM
@BernardoMeurer look ^
@Blue hmm, thanks. That implies that rotating a Feynman diagram does always produce something sensible, though it doesn't actually say so explicitly.
@Mostafaııııııııııııııııııııııı there are much better uses for physics books
such as keeping fans away from carpeting
Though note that putting books on string theory near a computer will cause any SSD disks inside it to mysteriously run slow.
@JohnRennie Maybe some of the data is escaping into extra dimensions.
2:47 PM
@JohnRennie I haven't read this well, but it seems to discuss the rotation of Feynman diagrams.
You could check out the book. But well, I don't think I know enough to help you with this. My knowledge is limited to pop-sci Feynman diagrams. I hope to take up a serious particle physics course someday in the near future.
You're not really "rotating" the diagrams because a Feynman diagram doesn't really have an orientation in space - it's just a type of graph
@ACuriousMind pedant
OK so i want to learn about quantum mechanics and GR. I know single variable calculus and classical physics. I know newtonian mechanics and a modest amount of electromagnetism and optics. Can anyone recommend me path to achieve my goal stated above. thx
What this is likely referring to is the different "channels" one gets from diagrams that look as if they are rotated, but which should be thought of as different assignment of "in/out" to the external lines
2:49 PM
@Черенки I'm in the same situation as you. Start with the "for dummies" books and start learning Linear Algebra side-by-side (at least).
I'm doing the same
QM is more about math and less about physics
Same goes for GR
@Blue That's news to me :P
@Черенки read in this order: Abbott, Munkres, Jost (Postmodern), DiBenedetto, Conway (One variable, volume 1), Lee (smooth), do Carmo (Riemannian), Evans (PDE)
2:50 PM
shouldn't i have a solid background in classical physics
@ACuriousMind well, OK, you can take the rotation to mean a reassignment of the in and out channels. Anyhow the book @Blue linked says that yes rotating Feynman diagrams (in the sense I mean) doe always result in a diagram that makes physical sense.
@ACuriousMind Sorry. I shouldn't have said....more "about" math
I do not think QM is any more "about math" than classical physics is.
It includes a hell lot of math
At least not when you understand "classical physics" to consist of Lagrangian and Hamiltonian mechanics, as I do.
2:51 PM
That's what I should have said ^ :P
@0celóñe7 what is postmodern, smooth, riemannian and PDE
@ACuriousMind r/iamverysmart
@ACuriousMind the intended meme lord understood
@ACuriousMind do you know how to use google
You would get a kick out of it
2:54 PM
@0celóñe7 No.
You have infected me with the inability to use it :P
@0celóñe7 can you properly state your recommendations without abbreviating
i wonder what happens if i lmgtfy lmgtfy
shit is bound to get real
@ACuriousMind there's something wrong if you think Functional Analysis is the same level of math as analytical mechanics
2:57 PM
@0celóñe7 There is something wrong if you think QM as done by physicists actually requires functional analysis ;P
@Черенки 0celo is renowned for giving the worst book recommendation to beginners. Don't listen to him. :P
Actually those are the books i used as a beginner
@0celóñe7 so who is renowned for giving the best book reccomendations
@0celóñe7 so what are the names of the books that you used
I am
I'm taking a PDE course that has 6 books, and I own all of them already. Coincidence?

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