/sci/ - Science & Math

This /mg/ is reserved for HIGH IQ POSTS ONLY. If you have no degrees or certifications in math, don't bother replying.

Please someone start a better one.
Thanks anons

>>12377756
Well, clearly not you then.

>>12377756
My brein Don werk so we'll what u say?

[math]2^3+43-1[/math] is prime.
[math]2^{3^{5}}+435-1[/math] is prime.
Is [math]2^{3^{5^{7}}}+4357-1[/math] prime?

>>12377922
Sorry, I fucked up. Get rid of the bit that says -1 on each equation.
[math]2^3+43[/math] is prime.
[math]2^{3^{5}}+435[/math] is prime.
Is [math]2^{3^{5^{7}}}+4357[/math] prime?

Lads why is the quadratic formula so important and why is it taught literally everywhere? What makes it so special? It's just a freaking parabola, big whoop.

>>12377922
>23+43
51, that's divisible by 3.

>>12377990
lern 2 aim missile

>>12377941
No, 4357 is congruent to 1 mod 6
Odd powers of 2 are congruent to 2 mod 6
Every prime past 3 is congruent to 1 or 5 mod 6

>>12377990
lots of practical use cases accessible to a wide audience.
i wouldn't complain too much. only up to degree 4 polynomials can you even have such an equation. beyond that there are none. so don't hate on quadratic-chan so much.

11th for determinantal point processes

>>12378001
.... An easier way would be to just say that it's divisible by 3....

>>12377707
These generals are being made by undergrads now right?
>>12377756
The generals are being made by undergrads you fucking retard.

>he thinks it's possible to throw a 100 heads in a row.
Then why don't you do it and upload a video so we can see? Oh, you won't because you can't, because it's literally impossible. Stop pretending it's possible. You have no proof that it is.

>>12378030

>>12378047
epic

khanacademy singlehandedly made me understand math and got me through algebra 1,2 and precalc when i was in high school 10 years ago. Thank you, Sal, you're based.

>>12378030
What made you think that, was it the unnecessary edit to the text or the soulless image?

>>12377990
Literally just for engineers too dumb to derive it themselves.

>>12378090
Both of those things are indicators of reddit. It was just intuition.

If I started to flip a coin, then continued forever, is it possible for me to only get "heads" for an infinite span of time? Or only any finitely large amount of time?

>>12378289
No. You will not get a streak of heads longer than 100 no matter how many coins you flip.

I'm fairly sure I said at some point that you need to tell undergrads to go fuck themselves and fuck off if you want to keep high schoolers off these threads, but this seems like an opportune moment to say it again.

>>12378303
>>12378008
>False assumption. 1 head in a row is possible but not 100 heads in a row.
where is the threshold?

>>12378410
Somewhere between 1 and 100.

my coin has heads on both sides
only losers use fair trials

>>12378444
based lang

>>12378346
cool picture
>>12378444
so based

>>12377777

Are statistics math?
Why not?

>>12379315
Statistics is about interpretation, organization and visualization of data at it's core and so the use of mathematics will always come as something not actually fundamental to the field. As the field has developed, it has become obvious why is that probability is such a good language to formulate statistical models and to understand data in general, but that's a separate issue. It's not math in the same way physics isn't math even if math has become an essential part of it. Now that statisticians are sometimes informal and some refuse to learn probability properly is another issue. The mathematical theory behind the techniques is perfectly rigorous, and it is hard to find a proper statistician who because of their informality they say some incorrect shit, but there are plenty of researchers who's whole area depends on statistical models who cannot define what a p-value is lmao.

I really hate Galois Theory.

>>12379605
Why? Did you get filtered?

>>12379641
>dude what if we used the same notation as in groups for different things

>>12379605
Maybe you would like it more if you started calling it cool Galois theory. That seems to have worked with real numbers.

the further I get in math, the more I realize my professors aren't actually geniuses, they just know more about math and practice problems more... I feel like anyone could get a PhD in math just as long as they have average baseline intelligence and high work ethic. Am I wrong?

>>12379807
>I feel like anyone could get a PhD in math just as long as they have average baseline intelligence and high work ethic. Am I wrong?
You're going a bit too far with your conclusion. You've realized your professors aren't all that much smarter than the students, which is true. But once you get to a higher-level math class you're in an environment where everybody is at least reasonably intelligent. You really do need well above average intelligence to succeed in grad school, you're just overestimating what "well above average" looks like.

>>12377707
Infinitely many coin tosses would have [math]2^\aleph_0[/math] possibilities I'm pretty sure. Actually I'm not too sure.

Can someone explain the mandelbrot set to me? What is the function iterating towards? Is it an approximation of some sort?

>>12379951
I am not good at latex let me try again. [math]2^{aleph_0}

>>12379807
why did you ever think your profs are geniuses in the first place

>>12379957
here you go anon
[math]2^{\aleph_0}[/math]

Apart from that: Anyone familiar with the combinatorics of fans? I'm studying Ewald's "Combinatorial Convexity and Algebraic Geometry" and it's really nice. Stuff like that should be taught in a first AG course, not overly abstract shit like Hartshorne.

>>12379982
Add surreal numbers after.

Should I embrace the arnoldpill and study Physics with Maths? My main area of focus is in CS, and if possible I want a field or application that intersects Maths, CS, and Physics.

>>12379862
Makes me a bit sad. I mean, since I don't have the aptitude I won't ever fall in love with math enough to try to get a PhD. But it does make me think of how cool would it be to have all that math inside your head

I want to learn everything about the formal system of math.

Would figuring out if an arbitrary Pokémon battle be an example of the halting problem? Obviously you would need to understand the random number generator. I’m fairly certain it’s undecidable

I made an offhand comment about this to a friend but on further thought I’m not so sure.

>>12380424
You’d also need to know each player’s choices ahead of time, I forgot to mention that

Is there any [math]x\in\mathbb{R}\setminus\mathbb{Z}[/math] such that [math]n^x[/math] is an integer for every positive integer [math]n[/math]?

>>12380442
No

>>12380424
>Would figuring out if an arbitrary Pokémon battle be an example of the halting problem
what did he mean by this?

>>12380442
brainlet here, what does the backward slash between R and Z mean?

>>12380500
Take a guess as to what would make the question mathematically meaningful.
Hint: [math]\mathbb{Z}[/math] and [math]\mathbb{R}[/math] are sets with [math]\mathbb{Z} \subseteq \mathbb{R}[/math].

>>12380482
But why?

>>12380500
Difference of sets.

>>12380510
try using logs

>Didn't get an A on my topology midterm.
Hold me bros..

Someone recommend me books about Distribution Theory. I'm struggling with them too much and the textbook I have isn't sufficient for me.

>>12380518
Already did. To be honest I'm not convinced there're none yet, but it would be weird.

>>12380539
find dirt on your professor, use it against him/her/them

>>12377756
I will bomb your department faggot.

>>12380266
>I want a field or application that intersects Maths, CS, and Physics.

Quantum Computing/Information Theory maybe, I know nothing about it tho.

What is the longest proof in mathematics?

>>12380753
fermat's last theorem

>>12380682
based unibomber

exactly how "lateral" is the complex axis to the x,y axies? is it just the third dimention?

Why the fuck isn't there a widely agreed upon notation for vector derivatives ffs
When f: R^n --> R, it's derivative is not a fucking column vector. It's a linear map represented by a row vector.

We should just use [math] D_{x_0} f [/math] and [math] \frac{df}{dx} [/math] for DERIVATIVE.
And then use the name GRADIENT for the transpose of the derivative and notate it by [math] \nabla_{x_0} f := (D_{x_0} f)^\top[/math].

>>12380799
I don't get your point, didn't you just describe the widely agreed on notation?

>>12380489
I meant to say will end, sorry for strokeposting

>>12380424
>>12380427
Hello susu

>>12380914
Literally who?

The J language was designed for mathematicians. Give it a try! It transcends all modern languages by decades, while removing the need for understanding complex subjects such as OOP, iteration, types, generics, resource management, etc.

It's free and open source.

Rather than those antiquated ideas, J has even more complex replacements! Apparently they are easier if you don't already know the shit ones. All functions are monadic and optionally dyadic as well. All operators are rank polymorphic, some even for data structured by infinite rank. J functions can be algebraically equated, and composed arbitrarily. There are no references to function parameters. Why would you need to? They're rank polymorphic. This also takes care of the loops. Iteration can be further controlled with the fold operator.

These properties allow J's compiler to produce more efficient code than is possible with other languages. The compiler can do function checking and make better decisions. Too bad it's single threaded and interpreted. It's for convenience!

If you look into J, you will notice it's ugly as sin, and looks a bit like regex or some joke language. I can assure you, it is no joke. If you master J, which can take years unlike OOP iteration #38484. Despite this, after just a few days of learning J, you will be able to do more than you could with Java after 10 years of learning.

And lastly, all J operators support complex numbers. They're a main feature of the language by design. You can define them like "2j5", which would be 2 + 5i. Yes it's a j not an i. The language is called J what do you expect.

>>12380928
I was talking irl about generating a pokemon battle program that could determine optimal play

I thought you were one of those people

Also the game can go on indefinitely if theres no time limit, and the players just switch pokemon in each turn instead of using moves

>>12380931
Is this a meme language? I kind of feel like implementing something in a meme language as a learning experience.

>>12377756
logic and science are just a narrative like any other and it is not proven at all it investigates anything

This is because formalized science is based on logicized maths, hyped as ''the language of the universe'' by posci addicts lol

and logicized maths is based on logic

all mathematicians are logic babies addicted to ZFC and they are all platonist, ie ''numbers are real bro not social construct''.

By the way truth is not found in logic. Logic is just a field by autistic pedants about well formed formulas and valuations, ie sending a formula to 1 or 0 and asking what are those valuations which are stable under inference rules. Zero truth in this, especially truth in the casual sense. Tarski truth is moronic, meaningless. Peak atheist.
Just like there is no truth in science, just some stats and a stat convention for saying ''if p value is XXX then the result is''true''''

This is why science is shit for politics and even for daily life.
At best scientists can come up about some stats about some formal system. Like ''this material has such and wear and tear, therefore our backlog of such conditons lead to 60% of breaking in the next year''
That's the pinnacle of the scientific claim and all their claims remain phrased as uncertainty.

all of this because the brain got bigger while the cranium stayed the same.

>>12377990
Most recently for me, finding roots of the characteristic equation for second degree differntial equations

>>12377990
>>12380975
And it's easy as hell, just learn the song

>>12380931
All ascii languages are garbage. When will someone release an interpreter for legit mathematical language? Math is 2.~D by the way, so you can't just give a shitty answer like
>muh utf8
>in 2020 it's now actually possible to program with emojis!
>my emoji ridden code is totally me!

>>12380500
>backward slash
It's read "minus".

>>12380442
[math]\frac{2}{1} \in \mathbb{Q}[/math]

>>12380853
Nope
Everyone uses whatever he wants. It's a mess.
Vector to vector everyone agrees (numerator layout, the usual Jacobian)
But Vector to scalar, and Matrix to Scalar everyone uses whatever he feels like.
The things is though, not being consistent completely fucks up the chain rule and yiu constantly have to thing wether you have to transpose and reverse the order.

https://en.wikipedia.org/wiki/Matrix_calculus#Layout_conventions

>>12380753
Classification theorem of simple finite groups. Tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
(On side note, how much paper would it be necessary to print a multiplication table of Monster group?)

I'm trying to study numerical linear algebra but I keep getting distracted by concepts in advanced linear algebra and then abstract algebra...
Any tips for how to manage this ADD-behaviour?

>>12381155
i've learned to embrace it by simply following the path that it shows. For example you're getting distracted by advanced LA and Abstract Algebra - then go learn THEM instead. That's what I do, I follow where it leads me and it's actually working pretty well. I'm learning a lot. it's basically your brain telling you: NO, i don't care about this shit, go learn this instead, it's better for you.

I dumped my thought about Benford's law, >>>/qa/3644690
The thread contains script and data in OP.
Discussions from math or programming view are welcome.

>>12381287
there are many people debunking the benford law election fraud stuff. So, from my perspective you:
- know more about benfords law than professional mathematicians
- you didnt know professional mathematicians have debunked the benford law - election fraud bullshit
- you have some weird conspiracy theory about professional mathematicians being in on the fraud
- you are schizophrenic
please tell me which one you are so I can put you in a mental category

>>12381311
Sorry sweatie that has been rebunked.

>>12381349
ah so the schizo or conspiracy category
thanks for playing, kiddo, but you are the weakest link

>>12381311
> know more about benfords law than professional mathematicians
No. I knew the law at the first time in this election.
> you didnt know professional mathematicians have debunked the benford law - election fraud bullshit
Yeah, I don't know all of them, but I've read an article as I've stated in <a href="//boards.4channel.org/qa/thread/3641850#p3641908" class="quotelink">>>>/qa/3641908,</a> and I don't understand why they don't try to remove errors.
> you have some weird conspiracy theory about professional mathematicians being in on the fraud
No, but I just understand why they wouldn't struggle to remove errors.
> you are schizophrenic
I don't know.

>>12377707
Can I get a degree or something in math after I graduate? How do I display competency in the subject? I'm a senior CS student and regret not learning it.

>>12381364
Sorry, typo.
>No, but I just DON'T understand

>>12381372
It's always possible to go back to school if you have the money for it. You can also just take some occasional math classes and get good grades if you're still a student. If you haven't taken an analysis class you should try that.

>>12381364
Your post in /qa/ is terribly written. Is english your second language?
Also, you seem to THINK you have accounted for the "orders of magnitude" problem, but i dont think you have. But you're probably not mentally capable of understanding that

>>12381378
Yeah good point, I'll give it a shot. Idk if I'll have time or money to go back, maybe I'll just take classes at a community college or something.

>>12381381
> Is english your second language?
Yes.
> "orders of magnitude" problem
What is that?
> But you're probably not mentally capable of understanding that
Why?

>>12381395
>>12381395
>Why?
because if you had even the slightest intelligence you would already know what the "orders of magnitude" problem is.

If i take 100 people and ask them to vote on two choices. If the two choices are approximately equal. Then I'll get approximately 50:50.

If I take 10,000 people, and separate them into 100 groups, and ask them to vote, I'll get mostly numbers in the 40s and 50s.
This is because the order of magnitude is 2. From 1 to 100. or 10 * 10. or 10^2.

This is an order of magnitude.
If you can't see why this would be a problem from an election in places where the average number of votes is between 570 and 1200, then you're beyond hope

>>12380780
>unibomber
kek

>>12380943
It is not a meme language. In fact, there are very few people who know this language, or anything in its family if languages, exists. The ones who do are likely autistic geniuses. I'm not totally sure what you mean by meme language, but maybe Shakespeare or Brainfuck is what you're looking for?

>>12380993
You might like J's predecessor, APL (A Programming Language). I'm not sure if it's "2.~D", but it uses non ascii characters and requires a special keyboard.

>>12381436
Actually that's better. I'm going to learn some J, thanks anon

>>12381436
>In fact, there are very few people who know this language, or anything in its family if languages, exists. The ones who do are likely autistic geniuses.
I now know it exists. So I'm an autistic genius then?

>>12381424
Oh, it's the same matter in the article of >>>/qa/3641875 and indeed, I THINK I took it into account. If you require mathematical correctness, it's a very huge problem, but it's not for statistical approach. And I tried to check the difference between Biden and Trump, not Biden and the theory as others did.

>>12381424
And the matter can be investigated using files in <a href="//boards.4channel.org/qa/thread/3641850#p3644703" class="quotelink">>>>/qa/3644703,</a> >>>/qa/3644704. The right graph shows that, though the data is all states.

>>12381436
I know about APL, and it's still a "1D" language. Math is 2D because of things like fractions, slightly 3D if you say that small operators like indices have a z-index of 1 and big operators like summation symbols have a z-index of -1.

>>12381378
>If you haven't taken an analysis class
If that frog is from Europe, there should be no fucking way to get a CS degree without having passed an exam on Analysis.
If he is from North America, I have no idea. How the fuck do you even want to do CS without basic knowledge in Analysis?

>>12378037
the thing here is to specify the time period. if you keep tossing the coin for an infinite time it will eventually land 100 heads in a row and every number thereafter.

>>12381643
>. if you keep tossing the coin for an infinite time
lol
> it will eventually land 100 heads in a row and every number thereafter
No it won't. You will never get more than 100 heads in a row. That's physically impossible.

>>12381657
it's a thought question you dimwit
>it's physically impossible
it's possible even without infinite tosses just very less probable.

>>12381539
Many CS programs in North America have an “algorithms” course that covers most of the necessary mathematical concepts for CS. If you do a B.A. you don’t need to take any supplemental math courses.

It’s completely possible to avoid most higher level math and graduate with a CS degree.

>>12381668
"Algorithms" usually isn't even considered "Theoretical CS" in Europe. That's "Applied CS".
Are US colleges really only meant to produce code monkeys?
The more I hear, the more it looks like it.

>>12377707
>triple integrals in cylendrical coordinate?
>no problem!
>chapter 1 page 1 of Rudin analysis.
>I understand nothing.


Why is it like this? Calculus is so fucking easy.

>>12381788
can I interest you in some categories my friend?

>>12381790
no thanks, I maybe autistic but even I am not that autistic.

>>12381665
>it's a thought question you dimwit
I also have a thought experiment for you. Imagine that in my room there is an infinitely large monkey whose body is made of radioactive cheese and when it claps her hands her hairs turn red. Question: if I shrink her with a shrinking laser beam, will she still be able to walk on her hands?
>it's possible even without infinite tosses just very less probable.
No it's not lol. Nobody has ever gotten a streak of 100 heads in a row when flipping a coin, nor they ever will. It's as impossible as jumping 100 meters high. Sure, each time you jump the height of the jump is a little bit different, in this sense it's random, and nobody is able to give a precise least upper bound to how high one can jump, and still everyone knows that jumping 100 meters is impossible. It's the same with flipping coins. You have to be delusional to think that it's possible to flip 100 heads in a row.

>>12381818
infinite monkey with infinite typewriters will eventually write all the works of Shakespeare

>>12381818
>nor they ever will
prove it

>>12381826
Yes and also if it chews infinite gum it will make an infinite bubble that will subsume the infinite universe and we will all fly infinitely far away to the land of infinitely small gnomes who will greet us with cakes and songs.

>>12381829
The proof is by observation. Try flipping coins on your own, see what's the longest streak you get. You will soon realize that it's impossible to get a 100 long streak of heads.
If that doesn't satisfy you, then do you believe it's possible to jump 100 meters high? If so, you're retarded. If no, can you prove it?

>>12381830
No because infinite bubble is physically impossible.

>>12381839
Why is infinite monkey with infinite typewriters possible but not infinite bubble?

>>12381679
>Are US colleges really only meant to produce code monkeys?
Ok so I did the B.A. track for this reason.

The B.A. and B.S. are essentially the same degree, just that the B.S. may or may not require more math courses. The B.A. is allowing you to take humanities/art courses and count them towards the major. You can front load a shit ton of those courses bc a lot don't have pre-reqs and aren't sequential. So I'm able to graduate full year earlier by doing this.

The state of U.S. education is very bad, I'm doing this to save money. Trust me, I know how shitty this is, but my family isn't rich and I need to get a job. Even if I passed all the math courses on my first try, it would still keep me there a year longer just because of the sequential nature of it.

Most jobs also don't care, they just want to see internships, projects, and GPA in that order. Financially, the B.A. just makes more sense. At least for me.

Worst comes to worst I can just take some summer math courses at a local university and I will still be saving thousands.

>>12381835
proof by observation that RH is false: try proving RH for yourself. you'll soon realize that it's impossible.
proof by observation that RS is true: try disproving RH for yourself. you'll soon realize that it's impossible.

>>12381835
retards like you are the reason why /sci/ is dead. your whole argument is "you cant do it yourself that is why its not possible." but if you had passed high school maths you would know that there is 1 in 2^100 chance of 100 consecutive heads occuring in a row meaning that if you were to toss the coin 2^100 times it will occur. It could even occur well before that. And i said in the start that there is very low probability of it occuring but it still IS possible.

>>12381875
>meaning that if you were to toss the coin 2^100 times it will occur.

>>12381899
You are female? Why anime? Are you gay? You are either female or gay? Maybe both? Wanna kiss? Maybe cry? Shit up

>>12381858
The two are completely different. One is done by a well-defined algorithm and the other is a problem of pure creativity.

>>12381919
>You are female?
>Wanna kiss?
yes to those questions

>>12381956
>female
guess that explains it
the whole debate was whether its possible or not, not how long would it take. fucking retard

People haven't heard of parallel cointossing. It can take microseconds.

On PRNG it won't ever happen to have cointoss that lands on same side, even with infinitely many tries you don't get trully what is sequence of same coin tosses due to nature of machine.

I've tried, don't bother to.

Also on real lottery numbers, I proved that Gambler's fallacy is not fallacy but correct.

Platonic idealistic random number generators unfortunately doesn't exist.

After 300 coin tosses you can choose which side you land, if you toss a coin.

Or you can predict, if computer tosses coin for you, with great accuracy.

If you don't trust me get us some discord, and I'll write you software for that again.
(It's been deleted in accident)

All right, finally gonna start my Analysis and Abstract Algebra studies, do you guys have any tips? I'll use Tao's book for Analysis and Dummit&Foote for Algebra.

>>12381962
>the whole debate was whether its possible or not
It's not.
>not how long would it take
It's impossible so this is the wrong question. How long would it take to jump 100 meters in the air? Stupid question, it's impossible.
>>12381978
How does parallel cointossing work?

>>12381987
this has to be bait
fuckwit i just told you that 1 in 2^100 times it will occur. That means it IS possible. if you cant understand that i've got no hope for you, just tell me this why are you in /mg/ if you lack even the basic capacity to do math

For those who've tried Amann & Escher analysis textbooks (I guess mostly Germans), how do you like them?

>>12382007
I heard some german anon saying here once they're better than any english books on the subject, but then again you can't blindly trust people on this site. Here's a crazy idea though, why do you try it for yourself and compare with other ones? Genius idea, right?

>>12381995
>fuckwit i just told you that 1 in 2^100 times it will occur
Wrong.
>That means it IS possible
Wrong. It's impossible.
Just because the probability is nonzero doesn't mean it's actually possible. Probability is an idealization that does not always translate to the real world, especially when the numbers you're dealing with are very small or very large.
Analogously, in St. Petersburg paradox just because the expectation of the game is infinity doesn't mean that you should expect to win an infinite amount of money, that's stupid, and betting anything more than $100 on the game is stupid.

>Here's a crazy idea though, why do you try it for yourself and compare with other ones? Genius idea, right?

>>12381987
You have multiple participants and they toss coin at once.

Or multiple running PRNG on multiple processors.

The things happen when it's out of a row. Even for PRNG, somehow world is related to those computer generated series of numbers.

It's almost like god.

Assuming normal distribution among realities or this reality, happening that you've got 100 head series, means, there's somewhere 100 non-head series.

>>12382040
It's still impossible even if you have all the people in the whole earth throwing coins at parallel. Nobody will ever get 100 heads in a row.

>>12382040
yeah and if i get a trampoline i can jump higher
it's still never gonna be 100 meters

cannonball/projectile motion
any other good motivations for students to be interested in parabolas

>>12382047
based

>>12382044
any ordered element of {Head, Tails}^100 is impossible to attain
all coins are designed to explode before they are tossed 100 times in a row

>>12382061
>any ordered element of {Head, Tails}^100 is impossible to attain
Wrong. A lot of sequences in {Heads, Tails}^100 are very possible to attain, you will inevitably attain one such sequence if you flip a coin 100 times. That said if you pick any particular sequence randomly, then it's very likely that it's impossible to attain, just like flipping heads 100 times in a row.

>>12381986
Read chapters through first while trying to capture general ideas and link together different concepts. Then you can go back through with a fine-toothed comb and be really autistic about every little definition. If you try to perfectly understand everything the first time you come across it you will go through material much too slowly.

>>12380931
oh shit based. always good to see a fellow Jchad. Just made me realize I lost all my Project Euler solutions in J when my old phone bricked. Now I am sad.

>>12381024
this is in R tho

>>12381472
so fish lol. (or ><>) not sure you'd want to do anything to serious in that language tho.

>>12381956
wtf you just kiss anyone who asks? What a slut.

>>12381986
My tip is don’t have an IQ below 130.

i don't feel so good mg

Guess I should make coins with two heads.

>>12381875
I'm on your side of the argument, but the way you argue is weak.

>meaning that if you were to toss the coin 2^100 times it will occur
There is no finite number of throws that guarantees you 100 consecutive heads.
You can't even guarantee a single heads. There is a probability of 0.5^(2^100) (non-zero!) that you won't get even one heads.
If you keep throwing, it will eventually happen, but only, because in concept you keep throwing an infinite number of times.

>>12377707
>On linalg class for math majors
>Were using Axler
>Ask professor about tensors
>"Well anon for tensors you'd have to master pretty much everything in the book were using before starting"
>"It really isn't simple at all, in fact that study is hard as nails"
>"Why do you want to know about tensors btw? That's usually something only physicists care about?"

I don't doubt my prof as he knows his shit but is he exaggerating a bit? I keep seeing tensors being used everywhere so I wanna know the gist of them.

>>12382403
>You can't even guarantee a single heads
Yes you can. If you flip a coin 100 times you're guaranteed to get a single heads.

>>12382405
>I don't doubt my prof as he knows his shit
Actually, he doesn't.

>>12382413
>[math]\frac{1}{2^{100}}=0[/math]
You read it here first!

>>12382439
>Probability nonzero therefore it's possible

If you toss coin 100 times, you'll be determining what side it lands. I people learn.

>>12382448
The probability doesn't even need to be non-zero for an event to be possible, it just needs to be on the sample space.

How many times do I have to flip an array of numbers to be guaranteed to win the lottery, become a category theorist, and have a more stable career than 50% of mathematicians despite being only 13% of their IQ

>>12382436
Why do you say so?

>>12382448
>>12382471
Let's say you have a train that arrives at a station anywhere between 5pm and 6pm with even probability distribution.
The train arrived at 5:51pm, what was the probability of this occurring?

>>12382513
>The train arrived at 5:51pm, what was the probability of this occurring?
Exactly at 5:51:00.0000...pm? Exactly zero. It could still happen, though.
Between 5:51 and 5:52? p=0.01666...

>>12382405
Fuck it, I'm bored and sick of grading right now. What do you want to know about tensors?

>>12382547
Not him, but could you go over the basic definition and why they are so useful in physics? Better than Google

>>12382547
What they are basically.
What is a multilinear map and so on?
Can you go over the basic ideas, preferably from a mathematically rigorous perspective?

>>12382546
Yes exactly. An event of probability 0 does not mean the event is impossible.
It just needs to be an element of the sample space for it to be possible.

>>12382574
Basic definition: If V,W are vector spaces over a fixed field, [math]V \otimes W[/math] is a new vector space that is sort of like the cartesian product. It contains ordered pairs [math]v \otimes w[/math] of a vector in V and a vector in W (these are called pure tensors), but it also contains all finite linear combinations of these things (i.e. not every tensor is pure). You're also allowed to freely move scalars around; i.e. [math](cv) \otimes w = c(v \otimes w) = v \otimes (cw)[/math]. So anything tensor a zero vector is the zero vector of [math]V \otimes W[/math]. It's also not hard to see that if V,W are finite dimensional, then [math]\dim (V \otimes W) = (\dim V)(\dim W)[/math].

Physicists care about tensors for the same reason differential geometers care about them. They will usually be working with the tangent/cotangent space of some manifold, so the tensors they consider will be elements of some (n+m)-fold tensor product [math](TM)^{\otimes n} \otimes (T^*M)^{\otimes m}[/math]. You can describe vector fields and many related constructions in this way. If you fix a basis for these spaces, you can write these as multidimensional arrays which is how tensors are usually described to laymen. Indices in this array corresponding to copies of [math]TM[/math] would be called covariant and indices corresponding to copies of [math]T^*M[/math] would be called contravariant. I do neither diff geo nor physics, so I can't say much more beyond that.

>>12382546
>Exactly at 5:51:00.0000...pm
This is nonsense, it has no actual meaning in reality. All of our time measurement instruments are finitistic, so it's impossible to determine what real number a given time moment corresponds to.
>>12382513
You have to understand that probabilities are merely a mathematical tool and not an actual thing that exists in a real world. One person could select a probability distribution that gives the probability of the train arriving at 5:51pm a nonzero probability and another could select a different distribution where the probability is zero. There is no way to determine empirically whose distribution is correct.
Probability theory is a tool that sometimes helps us understand some particular events when the numbers in play are small and uncomplicated. To extend it beyond them and give some metaphysical primacy to probability is foolishness.
I never said that probability 0 implies an event is impossible, that would be a foolish thing to say. What I am saying is that just because probability of some event occurring is nonzero is not enough to infer that the event is possible. One understands from everyday experience that it's actually impossible.
You can take any discipline in physics or otherwise that uses probability theory. There will very often be events that have nonzero probability yet understood that they are impossible and they view the probabilities as negligible. According to statistical physics, the probability of all the air molecules in a room arranging themselves into one small corner of the room is nonzero but everyone understands that this is actually impossible. This doesnt break the theory because the probability is small enough to be negligible and at reasonable sizes the probabilities assigned are helpful to understand what's going on. But as I said, to assign ontological primacy to the mathematical approximation that you use to help you understand what's actually going on is complete foolishness.

>>12382607
More rigorous description: [math]\otimes[/math] is a bifunctor on [math]\textbf{Vec}_k[/math]. It lets me build "lists" internally in [math]\textbf{Vec}_k[/math] and do computations in parallel. A multilinear map is precisely that; i.e. any morphism in the image of [math]\otimes[/math]. As a concrete example, if I have three linear maps [math]f:V_1 \rightarrow W_1[/math], [math]g:V_2 \rightarrow W_2[/math], [math]h:V_3 \rightarrow W_3[/math], then I get a new map

[eqn]f \otimes g \otimes h: V_1 \otimes V_2 \otimes V_3 \rightarrow W_1 \otimes W_2 \otimes W_3[/eqn]

defined on pure tensors by [math](f \otimes g \otimes h)(x \otimes y \otimes z) = f(x) \otimes g(y) \otimes h(z)[/math].

>>12382612
>Yes exactly. An event of probability 0 does not mean the event is impossible.
That 0 probability requires that time is ultimately not quantized, though.

>>12382547
Go back to grading faggot. My graders in two of my classes are two months behind on homeworks.

>>12382704
>>12382574
Tensors are what makes the study of multilinear maps into linear algebra.
Take some multilinear map
[math]f: V \times V \to W[/math] where [math]V,W[/math] are vector spaces. Being multilinear means it's linear in each argument. Now note that [math]V \times V[/math] is not actually a vector space, so [math] f[/math] is not actually a linear map. You want it to be linear because you know a lot about linear maps and how they behave, so the natural question is to ask whether it's possible to view somehow [math]f[/math] as a linear map?
[math]V\times V[/math] can actually be viewed as a vector space where the addition and scalar multiplication is componentwise: [math](v_1, v_2)+(v'_1, v'_2)= (v_1 + v_1', v_2 + v_2')[/math]
and
[math]\lambda (v_1, v_2) = (\lambda v_1, \lambda v_2)[/math].
This is a linear structure on [math]V\times V[/math] but it's not what we're looking for because with this structure you can see that [math]f[/math] is not linear.
It seems like there is no other option but to resign to a new study of multilinear maps. Not so! It's actually possible to view [math]f[/math] as a linear map and that's precisely what tensor spaces are for.
Let [math]S[/math] be the free vector space on [math]V\times V[/math], i.e. the vector space whose elements are formal linear combinations of [math]V \times V[/math].
This is a good vector space and you can easily realize that [math]f[/math] is actually linear when viewed as a map from [math]S[/math]. However, there is a lot of redundancy. The redundancy is not particular to [math]f[/math] but all linear maps.
For example [math](\lambda v_1, v_2) \neq \lambda (v_1, v_2)[/math] in [math]S[/math]
however [math]f((\lambda v_1, v_2))= f(\lambda (v_1, v_2))[/math] for any given linear map.

>>12382769
So we want to form a quotient space where these two vectors are identified.
Similarly we want to identify
[math] (v_1 + v_2, v') [/math] with [math] (v_1, v')+ (v_2, v')[/math]
[math](v, v_1 + v_2) [/math] with [math](v, v_1)+ (v,v_2)[/math] and so on.
When you quotient out [math]S[/math] by the subspace generated by these relations you get a new vector space that is called the tensor product
[math] V \otimes V[/math] and a natural map (this time LINEAR)
[math] f: V \otimes V \to W[/math]. We also have the canonical multilinear map [math]\phi: V \times V \to V \otimes V[/math] that sends [math](v_1, v_2)[math] to its coset in [math]S[/math].
To see that we haven't missed anything and this is the right construction, we have the universal property that for ANY bilinear map
[math]h: V \times V \to W[/math]
there is in fact a UNIQUE linear map
[math]\bar{h} : V \otimes V \to W[/math] such that [math]\bar{h} \circ \phi = h[/math].
This is the universal property. In fact, just using this universal property you can prove many basic theorems about the tensor products, for example that
[math](V \otimes W) \otimes Z = V \otimes (W \otimes Z)[/math], the associativity.
This is the basic gist of the tensor products.

>>12382547
>sick of grading
do your fucking job you little pussy

Y O O L E R
O Y O O L E
O O O L
L
E L
R

>>12382704
>>12382733
>>12382769
>>12382774
Thank you very much. this is very interesting.
We were just learning about duals, dual spaces, quotient spaces, product spaces, etc, in this linalg class I was talking about. This tensor stuff seems to me like just going deeper into these topics the class has already covered, so why was my professor saying tensors are much more advanced than what we've seen?

>>12382405
>>"Why do you want to know about tensors btw? That's usually something only physicists care about?"

Holy fuck I hope this guy doesn't have a math degree.

>>12382808
The standard applications are more involved maybe, but it's pretty much basic linear algebra to start playing with them.

Imagine a Timer is set to count down by one
every second from a random natural number
Does the Timer ever hit 0?

One way to solve this is to input the random number in backwards, from right to left
then you can run an algorithm that finishes in
finite time, to see if the remaining unimputted
numbers are all 0's so you can stop inputting
the number into the Timer.

>>12382823
>random natural number
This is meaningless expression.

>>12381024
2/1 is an integer, so not an answer.

>>12382831
View a random number as a measurable function on the reals which takes a constant value at some natural number almost everywhere.

>>12382856
What measure are you using on the reals? The lebesgue measure is not finite.

>>12382811
He has a phd lmao, from a good uni too. Very involved in research also.

>>12382819
Oh I see, thanks.

>>12381818
Your retardation is painful to see.
If every human keeps tossing a coin 100 times once a day, in 100 years the chances that 100 heads in a row has been one of the daily results for someone is actually very high.
Keep in mind that the population can keep on growing too.
kys retard

>>12381986
if you're the kind to value rigor way more than "intuitiveness" (if that's a word), then there's a good chance you'll hate Tao's book.
He got some nice problems, though.

>>12382938
>If every human keeps tossing a coin 100 times once a day, in 100 years the chances that 100 heads in a row has been one of the daily results for someone is actually very high
No it's actually very low. It's so low as to be negligible.

>>12382978
Tao's book doesn't grant you intuition it's pure uninspired, and unremarkable autism. Dreary and boring, without anything to distinguish it from dozens fo other textbooks. You'd be better off learning most of it's contents from an Advanced Calc text.

>>12382823
if you start at any number and start counting down, you end up hitting zero
so yes, the timer hits zero, since it doesn't matter what you mean by "random"

>>12382808
by the way, they didn't tell you, but you can learn tensors in greater generality (and yet still quite grounded and familiar) by learning tensor products of modules. Modules generalize vector spaces. Instead of a field of scalars, you have a ring of 'scalars'.

In this context, tensor products are quite useful and used often by algebraic geometers and algebraists in general.
Tensor products fly above geometry or physics.

Why the HECK are you not allowed to say the n word on /mg/!?

>>12383068
Even non-left 4chan is banning words now?
jesus christ

>>12382093
>pick any particular sequence
thats what i did
this is the case for any chosen element
i.e. all outcomes are zero
so exploding coins

>>12383068
which one?
number, numerical, nowledge, neildegrasstyson, noetherian, nilpotent

>>12383068
[math]\mathbb{N}[/math]?

>>12382448
You're wrong because you can literally obtain any fixed sequence of head/tails in 100 tosses in the first try. Doesn't matter how unprobable it is, it's possible.
After all, your argument is literally the same for any fixed sequence, and yet you'll obtain one if you try it.

What you're actually arguing is that it's impossible to, humans, even if with robots, toss coins enough times to 'essentially guarantee' 100 heads in a row. Meaning, more than 75%, or 90%, or 99% chance of achieving it.

We could try building a trillion robots such that each one tosses a coin every 3 seconds nonstop. We'd need 100 trillion years to have tossed 100 coins in sequence more than 2^100 times.

It's not the same claim, you can't say it's literally impossible.

>>12383068
numerics?

>>12383160
>>12383177
>>12383182
Nigger

It’s redditors and tranny janny. You can say nigger as much as you like, don’t let tourists stop you, nigger.
>>12383358
No, he means nigger. As in a stupid smelly greedy violent black african nigger ape.

>>12383326
>You're wrong because you can literally obtain any fixed sequence of head/tails in 100 tosses in the first try.
Wrong. You can't obtain the sequence of all heads.
>Doesn't matter how unprobable it is, it's possible.
It's really not lol. If you think it's possible go do it and show me the video.
>After all, your argument is literally the same for any fixed sequence
Not really. HHH...H is a special sequence because it's all heads.
>What you're actually arguing is that it's impossible to, humans, even if with robots, toss coins enough times to 'essentially guarantee' 100 heads in a row
Yes. No matter how many coins you flip, you will never get 100 heads in a row. This is a simple empirical fact.
>Meaning, more than 75%, or 90%, or 99% chance of achieving it.
I'm not talking about the probability or chance here. I'm talking about actually doing it. i.e. actually getting 100 heads in a row. It's impossible.
>We could try building a trillion robots such that each one tosses a coin every 3 seconds nonstop. We'd need 100 trillion years
You can try. Good luck. I'll be waiting. But you won't succeed since it's impossible.
>you can't say it's literally impossible.
I just did, because it is. It's really not hard to understand this, I don't see why it triggers you people so much.

>>12383439
>Not really. HHH...H is a special sequence because it's all heads.
Ok I won't waste my time with you

>>12377756
>tfw has master's in Math
>tfw still dumber then a brick
I'm going over my old Calculus textbook because we moved so fast through the material that I never got a real chance to get a solid understanding

>>12383371
>Nigger
Why the racism?

>>12383487
>masters
>revisiting calculus
what the fuck nigga
what did you learn in your masters? calculus XVI?

>>12381668
Where I go (US) CS students don't need to take many math classes. They have to take like two freshman level calc courses and then a couple math electives, but nothing past sophomore year.

>>12383478
You think getting all heads is the same as getting any other sequence so why don't you go on tour surprising people when throwing coins? After all, getting the sequences that you get when flipping a coin should, according to you, be just as miraculous as getting a sequence of 100 heads in a row.

>>12383513
Why would you assume he's racist? That's very prejudicial of you.

>>12383527
if you tell monty hall problem to thousands of people who never heard of it, the vast vast majority will get it wrong.
Public perception is meaningless.

Btw, why are you so bored today?

>>12383487
Did they not require analysis and algebra?

>>12383672
>Btw, why are you so bored today?
Probably the same reason as yours.

>>12383517
>>12383695
It was a State Uni, so the Cal 1~3 line was geared towards Engineers and Scientists, which means they just had us do the entire course mechanically, 'Here's power rule, chain rule, integration by parts, don't ask how it works, just use these tools to solve this problem set and fuck off'

They never asked for a proof, never asked for understanding

>>12383744
I refuse to believe a regionally accredited university gave a math master's student a calculation based education in mathematics. That is fucking outrageous. My university isn't even ranked in USnews or with the AMS and I still got to see enough analysis and algebra to be able to understand what's happening in lang and rudin.

>>12383744
"Mathematics is just a tool."

>>12380507
>>12380500
finishing up my first course in abstract algebra and I definitely was reading it as [math]\mathbb{R}[/math] mod [math]\mathbb{Z}[/math]. Don't think that would even qualify as a group (no identity)

>>12383695
Yeah they did require us to do Analysis and Abstract Algebra, but Analysis was effectively just taking the absolute most important bits from the cal 1~2 line and demanding proofs. We never really went over say, why implicit differentiation works, or how trig substitution works for integration, or why Green's theorem works. For algebra you don't really need a solid understanding of calculus to do proofs on abelian groups and so forth

>>12383785
Ok then that is potentially not as bad as you think. Building the math you see early on (e.g. the fundamental theorem of calculus) with analysis takes time. There are even deeper abstractions to go from there. Just take a deep breath and make sure you cross reference.

>>12383764
Believe it, what this guy said right here >>12383767
The school I went to is 'engineering cucked.' Even the computer scientists there couldn't do fundamental proofs by induction because the Major was more 'Software Engineer' then computer scientist. Imagine your school's entire cash cow being held up by Engineering Tentpoles, but you still force nearly every Engineering major to go through the initial Calculus line from functions to 2nd Order Diff Eqs. The tent steaks math courses keeping the tent cloth rooted to the ground are not important in the slightest, as the moment the students leave the math department, they're going to just have programs do most of the heavy calculations anyways. What does that mean? Cramming one of the most important undergrad disciplines into as tiny of a time as possible. If they asked any engineer to say, 'prove The Divergence Theorem', over 4/5ths the class will shit themselves and complain to the Engineering departments, who have the power to dumb down the math curriculum, so not only do you get students who think they passed cal 3, but just passed a 'memorize and apply' course, but you get exceptionally poor math majors (such as myself) who think that this is what its like for the rest of the path from the Bachelors and Masters to the PHD, when it absolutely isn't

>>12383487
>>12383744
>>12383785
After reviewing your posts, the committee has decided to invalidate your master's degree. We hope you the best acquiring a real degree from an acceptable institution.
Please inform the institution that accredited you that their permission to hand out masters degrees in maths has been revoked.

https://pastebin.com/evhpYAjb

I've created game. I wonder how many % can /sci/ get.

>>12383890
Its really weird you know. During my first year of the master program at that same place, I met a transfer from a top 100 school. There were several complaints on him from the engineering students so my professor had me to sit in on one of his lectures to see what the problem was. He was able to make me understand arc length in possibly the most beautiful mathematics lecture I had ever been in. It turns out his skill was so far beyond the institution he was getting his PHD in that he had to be told explicitly to dumb it down for the engineers. It was at that moment I realized I had been tricked and deceived. I'd never be a real mathematician. The degree I had was little more then a participation trophy. 'Congrats, you paid real money for a piece of paper saying your good at math but the truth is all you're good at is getting suckered' and here I am. Unemployed hiding in hole away from the rest of the general population, knowing that I was lied to for 6 years by an accredited university. The truth is that the department only passed me and my peers because if they didn't, there would be no more math majors at that school. There are moments like today when I go and open my old Calculus textbook and tell myself 'maybe if I can just understand, really understand one more thing, then I might be a real mathematician', but even I know that's a false statement. I'll never be a real mathematician, I'll never contribute to the body of knowledge that is mathematics. The best I can hope for is to warn other potential REAL mathematicians not to follow the path I took

>>12383837
If you're going to review calculus instead of going right into analysis, you could take on Spivak's book and try to solve a bunch of its problems. His book is basically calulus done like analysis is done.

But then you should really consider going right into analysis already.

>>12383764
>>12383744
even my professors in community college tried to explain the proofs behind everything we did. My old calc 1 professor spent a lecture explaining Dedekind cuts (without proofs, just the intuition behind it) to a bunch of 19 year old engineering students.

>>12384062
Did he explain how they are fake and don't work? Most textbooks and lecturers tend to miss this part.

>>12384121
no. do explain what you mean by this.

>>12383972
I feel for you, anon. How many years you have?
Maybe you can do something yet, if you really, really wants mathematics.
If you're above 25, then you'd better give up and do something else, unless there's a sort of middle path for you instead of starting all over from 0. I don't know.

A friend of mine finished undergrad in computer engineering and then did a masters in mathematics. But I'm not from the US, things are different. He also didn't pay anything.

>>12384156
Teh mæmê.

>>12383764
I went to a uni in a shithole failed state third world "country" and I still learned formal analysis and algebra.

We took real analysis in one and several variables, complex analysis, measure theory, functional analysis and probability theory in the analysis courses and basically all of abstract algebra from group theory to galois theory + formal lingalg in the algebra courses.

We also had a bunch of formal diff eqs classes but sadly no topology classes as the degree was oriented towards a more applied side of things.

Can't imagine what sort of joke uni you would have to go to in order to supposedly be a math major and not even take real analysis

What applications do you use for graphing functions?

>>12383890
Some Mathematical body needs to start independently vetting math degrees, even if they come from an accredited institution. Its just way too easy for even overall good schools to have piss poor math departments in this era
>But who's going to be the arbiter of good enough?
Good point, something the body is going to have to vote on I'd assume, but the point is to protect people like retard anon from getting a subpar math skills from accredited institutions that have let their math programs to fall by the wayside. He would have been better served picking another degree then wasting his time

>>12384316
They should just be prohibited of giving any more degrees before reforming the whole course.

Fuck, what do you do when a task is going to take a lot longer than you estimated? I have some work I planned to do over this week. I planned to do 2 hours of work on it a day, but two days have gone by. And I did 6 hours of work on it yesterday, and 6 hours of work on it today and I've only just now begun to put a dent into it. It's taking 3 times longer than my estimation.

https://pastebin.com/KLwKtTkX

Fixed error, now it actually can know which is prime and which not.

https://pastebin.com/P9ZuL7XL
>> this is correct working script

>>12384590
Even the Eratosthenes sieve would have been less retarded.

>>12384608
No it wouldn't because you'd have to compute it every time you start a script.

Pre-computed values is way to go in data science.

>>12384608
literally you can rewirite first paragraph for
"let primes = require("prime-generator")(10000)"
Anytime you want to change anything, but now it's best solution available.

There's a function to get divisors, which is basically it.

But then program wouldn't take over 1000 lines.

>>12384608
Actually is memory game, not a math game, sorry.

[math]\bullet[/math]What textbooks, papers or text did you read today?
[math]\bullet[/math]What non-textbooks did you read today?
[math]\bullet[/math]Did you write something today?
[math]\bullet[/math]Did you do some programming today?
[math]\bullet[/math]Did you build something today?
[math]\bullet[/math]Did you clean up something today?
[math]\bullet[/math]Did you plan something today?
[math]\bullet[/math]Did you manage to work off some bureaucracy/paper work today?
[math]\bullet[/math]Did you practiced any skills today? If so, which?
[math]\bullet[/math]Did you do sports/cardio/weightlifting today?
[math]\bullet[/math]What were you eating today?
[math]\bullet[/math]How much sleep did you get yesterday?

>>12384608
You are right. But in fact it doesn't matter.

>>12384619
False. Compiled regex is pretty fast tbqh.

x.matches(/\d+/) ? that's a real number! : that's a natural number.;

>>12383920
1. Why the heck did you put the entire list of primes in the "game", just use rabin-miller twat
2. This is not a game

>>12384647
> What textbooks, papers or text did you read today?
Algebraic number theory by Milne.
> What non-textbooks did you read today?
Three body problem, I'm quite liking it.
> Did you write something today?
Only an alternative proof to a lemma and cleaned some notes.
> Did you do some programming today?
Ye did a stupid whatsapp spammer for a chat meme group
> Did you clean up something today?
Myself counts?
> Did you plan something today?
What cake to cook tonight.
> What were you eating today?
Doritos, cheese and some meat, not that healthy but still better than not eating.
> How much sleep did you get yesterday?
too much, about 8:30h.

>>12385009
Are you the anon who talked about going deeper on studying number theory on the last thread?
Are you an undergrad or graduate?
What do you think of Milne's book so far?

>>12383772
Are you saying R/Z isn’t a group? It certainly is and is isomorphic to R.

>>12384992
>the entire list of primes
Haven't clicked the link, but I highly doubt it's there.

>>12377756
Fear not, for I consume Rick and Morty.

>>12384316
they also should do this to the cunt professors that work at those universities. They are no better then their students.

t. lectured undergrad mathematics for 5 years at a state university in USA

>>12385009
8h30 isn't too much

>>12384647
huh, I was asleep when this was posted.
Thx for taking over a bit

>>12385025
((1/2)+Z)+((1/2)+Z)=1+Z=0+Z
Show me a real number of order 2.

>>12377756
If you are so smart, how many 10x10 MOLSes can be found?

>>12378026
But it isn't

>>12377990
Many calculus problems require you finding roots of functions. Many functions allow you to simplify to a quadratic one.

>>12377941
How in the hell is 50 a prime?

>>12385258
I concur
Why the fuck do we have Algebraists teaching teaching mutivariable calculus when they themselves only touched on the subject once in their academic career? I mean, even if they do great research, their ill understanding of specific undergraduate courses propagates through the system

>>12385496
Any half decent algebraist can teach calculus well. It takes no more than a couple hours max to review any aspect of basic calculus to a level higher than most undergrads will ever reach when you're an actual mathematician.

>>12385513
Students need to be taught calculus by geometers so they can properly grasp the geometry behind the subject instead of falling into plug and chug formulas and repetitive contradiction proofs.
They also need to be taught algebra by geometers so they correctly understand the geometry behind symmetries and groups of symmetries.
Additionally, it's also best if they're taught set theory by geometers, so they can get the geometry behind it.

Basically if at some point in time you were taught by someone who isn't a geometer your education is subpar.

>>12385535
Okay, but maybe we'll give some time to other areas of math too. Try to play nice with others.

>>12385548
NO
THE RUSSIANS LAUNCHED SPUNIK
SO NOW WE ONLY TEACH CALCULUS
ONLY CALCULUS
ONLY
CALCULUS

>>12385025
>It certainly is and is isomorphic to R.
No, it's isomorphic to U(1).

I need one ticket to new mexico

>>12382405
that depends a lot on what you mean by tensors
people care about tensors, but only within certain contexts, and there is absolutely no reason to learn about tensors until you're learning about those contexts (commutative algebra, differential topology).

>>12385022
im not that anon, but I'm an undergrad and Milne's book are the only one with full explaination that my prof at uni don't give (and also some paper from keith conrad are actually useful material)

>>12382808
again, because people only use tensors for much more advanced fields of math than linear algebra

>>12385265
Dunno, I have the best profit sanity/study when I sleep between 7:10h and 7:40h, if I sleep a little more than that I don't want to get off my bed.

>>12383371
you will never succeed as a mathematician, disgusting anti-intellectual moron

>>12384250
desmos and only desmos, if it can't be done in desmos then i need matlab or numpy/matplotlib for it

>>12384535
You hyperventilate to make your brain work faster to make up for the lost time.

>>12385025
>R/Z
>R
It's not well-defined!

>>12384422
I think a form of accreditation would be more efficient as long as the general public know. Regional accreditation does a solid job of punishing for profit intuitions by being an effective warning for not only employers, but students and other schools. It also warns others if a student was stupid enough to get a Math degree from a school without a vetted Math program. Many math departments will let incompetent students attain all the course work required for a degree primarily because either the Professors can't or won't fail over half the class, or there will be an unofficial mandate from the dean to only fail a specific % of students in order to either pass state guidelines or to warp the school's statistics.

>>12385535
>the geometry behind set theory
What geometry?

>>12382796
it's actually Oi-ler

>>12382796
the correct pronounciation is "yooler" it should rhyme with "ruler"

>>12386073
Geometry over Fun.

>>12385489
You mean 51? Still not a prime.
Is that Grothendieck's prime? No remembur now

>>12385535
>HURR GEOMETERS BECAUSE ONLY THEM CAN TRULLY GRASP AND TEACH INTUITION IN MATH DURR
kys, you lot are almost no better than mediocre physicists

>set theory by geometers
Dude. Just off yourself, really.
Most retard post in the last 5 threads, congratulations.

>>12386073
Uhhhhh lattices of subsets are geometric-like.

>>12381443
Good luck anon - you'll need it.

>>12381444
Not necessarily.

>>12381472
Fractions can be represented in 1d with parentheses and division. One language I know of with actual fractional representation is Clojure. It's a pretty good language too. Not sure what you're talking about with operator z-index. Do you mean operator precedence? That shit is not good.

>>12382254
Wow you're the first person I've seen on /g/ or /sci/ who knows J. I wouldn't say I know it, I try to write some J code and pretty quickly get confused. The forks, rank polymorphism, fuckin dyads... Confusing to me. Plus every operator does like 5 different things depending on context. I feel like I'm learning japanese

>>12382254
Same anon again sorry for double post, but how did you hear about J and why did you learn it? Do you know every operator and all of their intracacies? Have you tried benchmarking against other languages? I'm curious about how it performs. And lastly, do you write J in point-free (tacit) style?

A bit surprised by this:
[math]A, B[/math] are commuting differential operators. Trying to solve the ODE (or PDE, it doesn't matter)
[eqn] A B \psi = B A \psi = 0 [/eqn]
If [math]B = A-C[/math] and [math]C[/math] is invertible, then (with abuse of notation)
[eqn] \psi = C^{-1}(B^{-1}-A^{-1}) 0 [/eqn]
indicating that a general solution to the initial ODE can be written as [math]\psi=\psi_1+\psi_2[/math] where [math]A\psi_1=0[/math] and [math]B\psi_2=0[/math]

Except of the part where I ignore [math]C^{-1}[/math], do you know a fault or if anybody knows where I can find more information about this.
I've never seen this discussed in books or forums.

>>12386347
>Fractions can be represented in 1d with parentheses and division
Our retinas are 2D, why are mathematicians allowed to take advantage of that fact while grunt coders have to put up with a literal mental taxation.
>operator z-index
I just mean that the size of operators has some meaning. Also, the emphasis can have meaning. Mathematics is much more readable than "code" as a consequence.

>>12386314
seething algebraist

>>12378289
For any specified finite subsequence of coin flips, there is a nonzero probability of its occurrence, which means that given enough time and flips it will eventually occur within the larger sequence.

However, if you specify an infinite sequence of flips, then the limit of the probabilities of heads or tails on each flip tends to zero as the number of flips grows to infinity, since this is an infinite product of (1/2)s. So the probability is zero in that case.

>>12386439
>mathematicians allowed to take advantage of that fact
Not really, we still write latex linearly. Software people have 2d diagrams and stuff, which is more analogous to handwritten math.

>>12381155
The other response is good if you have time to burn and are a math student. If you need to learn numerical LA because you're a scientist/ML person who needs the info as soon as possible, then the ONLY way I can get myself to learn really applied, kind of boring topics like this is by working on small projects applying the math to my field, or by making some nice Jupyter notebook explanations with demos – e.g. take an interesting dataset from physics or neuroscience and fuck around with interesting factorizations. This also has the added bonus that you're immediately tying theory to practice.

>>12381539
Analysis in US means something different in Europe. US splits up the foundations and theory from applications (calculus vs analysis). Where I'm from at least, this was all just bundled into "analysis"

>>12386642
The code to generate those diagrams is 1d too

Can you have a scenario where the iterated integrals do not exist but [math] \int f(x,y) dm^2 [/math] exists and is finite? Or is it necessarily true that if the iterated integrals are not equal, then [math]\int f(x,y)dm^2[/math] automatically does not exist?

>>12386684
fubini theorem says "if the 2d integral exists, then the iterated integrals exist and are equal"

>>12386671
Right, it's the same situation.

>>12386741
Perfect. Thank you! It wasn't particularly clear from the textbook or from lecture that this was the case but that's what I thought it was.

Well, /mg/?

>>12386782
other envelope has 50% chance of having $5 and 50% chance of having $20 => expected value of $12.5. So you would be expected to gain $2.5 by switching

>>12386792
Ok good now think about it a little bit harder.

>>12386314
seek help, moron

>>12386761
note that what fubini's theorem says is "if f is in L^1 of the product, then the iterated integrals make sense, i.e. for almost every fixed x, y -> f(x, y) is integrable, and these integrals are integrable in x. and the reverse. and all these things are equal."
then tonelli's theorem says the product integral and iterated integrals are equal when the function is positive (whether or not you know it's in L^1 of the product).
these theorems are extremely useful, but they're much more useful if you know when you can apply them (because it's very very very often) as long as you use the right one.

>>12386782
bro what the actual fuck is this cancer, I'm scared

>>12386800
>now think about it a little bit harder
Why? He is correct.
There is no paradox in that problem.
The image is grossly mistitled.

>>12386823
>>12386792
The actual answer is quite simple but most people, especially those mathematically trained have a very hard time finding it or accepting it as correct before it's properly explained.

>>12386838
Think about it a little bit more until you understand why it's actually a paradox.

>>12386846
>>The Two (2) Envelope Paradox
>>You have one (1) envelope, but you receive another one (1) in the mail. How many envelopes do you have now?

>You may immediately find an almost trivial answer, but think a little bit more and maybe you'll understand why it's actually a paradox.

>>12386846
suppose you have 2 envelopes and yours contains $1 and the other either contains 1 millionth of a penny or $100 billion. Do you switch?

You might think you're supposed to switch, but WRONG you should actually just take the $1 because it's a paradox.

if one envelope contains A and another envelope contains B, does this mean my expected value is C

>>12386907
Whatever envelope you pick and whatever the money in it, your answer will always be that it's advantageous to switch. That means your decision is not based on any information that's particular to what you got, which means something is wrong here. There should be no benefit to switching because of symmetry.

>>12386819
Thank you anon. From the homework I've had to do so far, it seems like one of the iterated integrals is almost always easier to compute and, as a result, means that you can get some neat identities. We justified the computation of the Dirichlet integral using it.

>>12386941
the money is in a superposition which collapses the second it's observed. As long as you don't know how much money is in either envelope you can switch as many times as you want and nothing will change.

>>12386955
But you know that $10 is in the envelope that you picked.

>>12386907
>but WRONG you should actually just take the $1 because it's a paradox
proof by 'this is a paradox so I don't have to produce a proof'?

>>12386959
exactly, you know the amount so the superposition collapses and switching becomes beneficial. If you don't know the amount it follows heisenberg's uncertainty principle so you switching does nothing as the money exists in a quantum superposition.

>>12386973
The problem is still here. Whatever envelope you pick and whatever the amount of money you find, by your logic it's always beneficial to switch. But that's clearly absurd. For it to be beneficial to switch in some situations, there need to be situations where it's not beneficial.

>>12386986
the situation where it would not be beneficial is if you knew how much was in the other envelope but not your own. Then you stand pat

>op tripfags
>we're 308 posts in and he still hasn't made another post
What the fuck did he mean by this.

>>12386782
one envelope contains x
other contains 2x
the expected value of a random envelope is thus 1.5x
the other random envelope also has expected value 1.5x
so it doesn't matter!

>>12387046
But you know that one envelope contains $10. So the other one contains $5 with probability 1/2 and $20 with prob 1/2, so the expected value is $12.5

>>12387087
oh in that case you should switch obviously

>>12387110
Why? Whichever envelope I pick and whatever I find in it I could perform the same calculation which shows that allegedly switching is better. But surely switching can't be better in every single situation regardless of what I pick. Something is wrong here.

>>12387118
dude those calculations check out so i think you should switch

>Bayes' rule
more like GAY'S rule lmao amirite

New thread
>>12387355

>>12380554
duistermaat and kolk, havermath, or treves

>>12385535
>MUH SHAPES LMAO
>MUH COOL SYMMETRIES OF LE SNOWFLAKE WOAH SO COOL RIGHT??

>>12377707
I just encountered Norman Wildberger's concept of rational trigonometry, but I find his chromogeometry even more interesting and have been reading about it nonstop as well as watching his videos. What's the verdict on this guy? Is he out of his mind or is his chromogemoetry legit

>>12388067
The rationals are the initial ordered field so there is a method to his madness, even if it's a bit unconventional to deny yourself even the theory of a real closed field.
What's his chromogeometry about?

>>12381826
>will
You meant to say might. They also might all type nothing.