/sci/ - Science & Math

>Newton's notation for partial differentiation is absolutely kino, but only really works for two variables

>>10376018
>not writing the ' with the variable being derived next to it in the upper part
dx notation is standard but for my notes i use a modified newton.

>>10376018
can I get more math related touhou?

>>10376052
ultra brainlet here. anyone help?

>>10376038
That just completely removes what makes Newton good.
>>10376049
Pic related is the only OC on the subject I've ever made.
It was largely inspired by faint memories of "I want to fuck Rimbaud in his boipussy" from /lit/.
>>10376057
Could you fix the formatting? Use the TeX button on the upper left corner of the post window to check if it's working.

>>10376069
Now that I properly think about it, the formatting is absolute trash.

Non-unital algebras: meme or not?

>>10376074
it all formats correctly if you visit the thread, does it not?

>>10376057
you can chose a x with |x|=1 wlog, so the norm is 2

>>10376138
nvm, i am not sure if 2 is correct

>>10376143
ok, i am sure now that 2 is correct. basically, wlog write (a,b)=(cos(x),sin(x)), then the norm is
sup(sqrt(4cos(x)^2+sin(x)^2))= sup(sqrt(3cos(x)^2+1))=2

>>10376115
Weird, it hadn't earlier.
Anyhow, another anon has already pointed out how you just apply the coordinate representation of calculus and then just optimization.
For general convenience, you can assume that the vector has norm 1.

This video is very interesting. The theorem is heavily combinatoric and gives good vibes. It reminds me a lot of the two proofs of the Hales-Jewett theorem I have studied.

https://www.youtube.com/watch?v=bjlK6HkOTNs

I have a question: what is [math]E_{x~y}[/math] here referring to? (This slide appears at 6:09)

>it's an anon spends his sunday night frantically doing all the work he was supposed to do during the week episode
on the bright side the pressure helps me think. and gives me terrible stress headaches

>>10375982
Quick, name 5 rings of order 4.

>>10376057
Analo-operator theory proof:
For a bounded Teoplitz operator [math]T(W)[/math] with "kernel" [math]W \in L^2(S^1,\mathbb{R})[/math] we have [math]|T| = \operatorname{sup}\operatorname{Spec}T[/math]. Since [math]W = \text{const.}\in L^2(S^1,\mathbb{R})[/math] trivially we see [math]|T| = 4[/math], the largest eigenvalue.

Geometric proof:
Notice that [math]\underset{{\bf x}\in\mathbb{R}^2}{\operatorname{sup}}\frac{|T{\bf x}|}{|{\bf x}|} = \operatorname{sup}\frac{R}{r}[/math] , where [math]R[/math] is the distance from the origin to the ellipse [math]E_R = \{{\bf x}\mid 4x^2+ y^2 = R^2\}[/math] and [math]r[/math] is that to the circle [math]C_r = \{{\bf x}\mid x^2 + y^2 = r^2\}[/math]. Now since we get [math]\frac{|T{\bf x}|}{|{\bf x}|} = 1[/math] for [math]{\bf x} = (0,y)[/math], we know that [math]E_R[/math] and [math]C_r[/math] intersect tangentially (at two points), and can hence be parameterized by the same variable [math]t[/math], via e.g. rational parameterization. Since [math]R(t)[/math] and [math]r(t)[/math] have the same quadratic scaling through this parameterization, for each [math]t[/math] we see that [math]\underset{{\bf x}\in\mathbb{R}^2}{\operatorname{sup}}\frac{|T{\bf x}|}{|{\bf x}|} = 4[/math], the ratio of the major axis [math]R_x[/math] of the ellipse [math]E_R[/math] to the radius [math]r[/math] of the circle [math]C_r[/math].

>>10376798
Fuck. Replace every instance of [math]|\cdot|[/math] with [math]|\cdot|^2[/math] in my post. [math]4[/math] is the square of the norm.

>>10376057
[math]||T|| \geq 2[/math] because you can look at vector [math][1, 0][/math]
[math]||T|| \leq 2[/math] because [math]\sqrt{4a^2 + b^2} \leq \sqrt{4a^2 + 4b^2} = 2 \sqrt{a^2 + b^2} [/math]

>>10376798
you need to sleep

Trying to teach myself differential equations - looking at these two books, anyone familiar with either, or can make a better recommendation?
1. Ordinary Differential Equations, Tenenbaum
2. Ordinary Differential Equations, Arnold
1 is half the price and longer, but I don’t know if that’s appropriate

>>10376798
way to use a sledgehammer to crack a nut, buddy
informative, tho

What is life like as a math postgrad? I just got a funded position.

>>10377561
uhh they're both the same price when i checked last time on libgen

>>10377561
I've seen Arnold recommended here before, but I doubt there's any difference.
>>10377574
Is anyone willing to generalize and show that the norm of a diagonal matrix is the largest value?

stat question from a noob. If i want to evaluate my t-score/ z-score and i already know its value how do i evaluate it without looking up a table?

I try to use matlab and its normcdf function but it requires samples but can't take more then one, which makes it pointless for double sample tests.

heeeelp.

>>10377822
If you have understood this argument >>10377162
then you can do it yourself. Moreover, the first statement here is probably as general as it gets

>>10377822
If you have understood this argument >>10377162 then you can do it yourself.
Also, the first statement given here >>10376798 is probably as general as it gets

>>10377867
anyone else thinks that stat is not a part of mathematics?

Fun new paradox: Suppose we have two points in a topological space, A and B, and a well ordered set I, beginning with A and ending with B, such that every element has a path leading it to its successor.
If the cardinality of I is sufficiently large, glueing the paths together into a path from A to B becomes impossible, but they're path connected nonetheless.

>>10378187
>glueing the paths together into a path from A to B becomes impossible
how so

>>10378187
Why would it be impossible

>>10378327
>>10378348
Suppose I has cardinality aleph-2. If you just stack the paths, you have a surjection from the unit interval to a set with aleph-2 cardinality.

>>10375982
Wait, is Mochizuki that old now?

>>10378187
"gluing every interval to its successor" doesn't really describe a path connected topology.
For example, the intervals 1..2..3.. are connected to each other, but then comes [math]\omega[/math] which doesn't have a predecessor, so it isn't connected in any way to the previous ones.
Well-ordered doesn't imply "every item has a previous one".

LOL

Taking my first course in functional analysis next semester. Anyone care to recommend some good books on an introductory level? Preferably ones with lots of examples and exercises, not just theorems and proofs.

>>10378906
>Preferably ones with lots of examples and exercises, not just theorems and proofs.
good luck with that. anyways, kyreyszig is a good intro text.

>do bad in an intro to Java course
>have complete programming existential crisis
>major in math because failed out computer science
>try to learn programming from c to lisp to haskell to python
>can't
>so mentally fucked that seeing a blank text editor or ide causes panic attack
>reading that same introductory drivel on how to make an if statement or what an integer is just reinforces self hate
>graduate math
>go to grad school
>try to avoid it but I'm expected to pick up minor programming for a masters
>talk with advisor tell him I really can't do this
>he says if course I can because :
>programming
>is
>literally
>math
>and I have such an intense interest in algebra and computer science theory
>take the class
>fail it
>fail it every year
>can't graduate have to stay longer
>talk with advisor
>he's confused but says they're willing to waive it considering all my other work in math
>graduate with math masters
>GPA is shit though
>enter massive depression that affects math skills
>don't do phd
>just live at home as a neet loser

>>10378931
>reading that same introductory drivel on how to make an if statement or what an integer is just reinforces self hate
and this is where you fucked up

>>10378948
Then how am I meant to learn a language

>>10378951
You just do it. Don't bother with the how and why. As long as it compiles, it's fine.

>>10378931
>be a CS poo in loo
>get angry at women's studies undergrads who rp as math phds on an assyrian fishing board for calling you a CS loser
>write up a post about how programming is actually really hard

>>10378960
But that kind of thinking is what causes my frustration
>>10378972
I wish

>>10378981
Boo hoo. You're learning a programming language, not stratified morse theory. Write code to be applied at first. Once you know the ins-and-outs through applied examples, you can then start to peel back the layers of abstraction down to the low level, etc.

Practicing for analysis midterm - good proof?

>>10377822
In general if [math]T\in B(\mathcal{H})[/math] on a separable Hilbert space [math]\mathcal{H}[/math] is Toeplitz then [math]|T| \leq |\operatorname{Spec}T(W)|[/math], where [math]W[/math] is the Wiener-Hopf symbol (math)/kernel (physics) of [math]T[/math] and the norm [math]|\cdot|[/math] here is the (virtual) Lebesgue measure. Equality is saturated by diagonalizable [math]T[/math], i.e. if [math]X = L^2(S^1,\mathbb{C})[/math] with a [math]L^1[/math] kernel [math]W[/math]: Fourier transform leads to [math]T[/math] being Toeplitz and unitary.
See Bottcher's book for a proof.

>>10379384
CLEARLY
AM I BEING A GENIUS YET

>>10379471
Stop memeing retard

>>10379384
The proof should be two lines long, what definitions are you using?
>a closed set is equal to its closure, done
>>10379391
"Analysis of Toeplitz Operators" or "Toeplitz matrices, asymptotic linear algebra and functional equations"?

>>10375982
Is stats pure or applied math? Or somewhere between?

>>10379502
Lectures on Operator Theory.

>>10379516
That one isn't on libgen.io
Please rec something that is.

>>10379533
だが断る

>>10379538
>bullying the poor with JoJo references
You're the worst.

>>10379391
get the fuck out you fucking toy demon

>>10375982
This nigga is so fat lmao

>>10377793
based and underrated
>you will never be taught by a prof from /mg/

In asymptotic complexity of algorithms, why the base of algorithm is not necessary? I always saw O(logn), and base is never stated.

>>10377574
You don't know how to use commas.

>>10380441
The base just changes the constant factor, which big-O notation ignores. log_b(a)=log_c(a)/log_c(b).

>>10380326
Maybe you already have

>>10377793
Comfy, but you don't want to get too comfy or you'll never get anything done

>>10380866
>Comfy, but you don't want to get too comfy or you'll never get anything done
I'm never comfy, so that won't be a problem.

>>10380326
I doubt I will ever get that far.

>>10380441
The natural logarithm is just a constant multiple of logarithms of any other base.
The place where base matters is exponentials.

I-is topology interesting ? Do I have to take it if I want to get into any graduate schools in the future?

>>10381353
>is topology interesting ?
When you get far enough. The beginning is quite dry, as you'll be fucking with preimages, unions and intersections, but then you get some insight on why some things you've already encountered in analysis hold (without having to rely on stupid shit like infimum and supremum), which is quite nice. After that, you will get to the purer stuff like compactifications and Tychonoff's theorem etc., and then you will either see it as interesting or a re-emergence of autism. That is when you will know whether you want to go further that path.
>Do I have to take it if I want to get into any graduate schools in the future?
Probably not, but it will surely not hurt you.

any books similar to this with a focus on pure math that are easy to get into?

>>10381353
It's very interesting, but only once you get a bit into the material and even then it just might not be for you.
You should probably have taken analysis first.
I really think it's important to have topology if you want to go to grad school, it's foundational to pretty much anything.

>>10381353
>do I have to take topology
Point-set topology is a requirement for about 60% of everything, from Riemannian geometry to harmonic analysis and homological algebra. You absolutely should go study it.
But it's easy to pick up later. 90% definitions you use elsewhere, 10% Jordan's closed curve.

>taking differential equations class
>almost everything is "just use this specific trick that only works for this chapter and nowhere else"
I miss actually proving things before the professor would let us use it.

>Ind3 - Part 05 - Upper Bounds
https://www.youtube.com/watch?v=zEHmNP2wfZQ

How can I think of a projection with center L, a line?

So for example, in P^3 taking a projection with center the line being the z-axis (ie x=y=0) sends a point to its first two coordinates. So if we have a variety in P^3, how does this translate geometrically through this projection?

I understand projections with center being a point by a line tracing through a variety into a hyperplane, but I'm not seeing how the higher dimensional analogues work

Maybe one of you can give some insight here, I'm pretty desperate

>Given the polynomial is irreducible over the field Z5, find all zeros of p(x) in the field obtained by adjoining a zero of p(x) to F. There are 3 zeros.

>p(x) = x^3 + x^2 + 1

I feel like there's some trick here that I'm missing. All I can think to do (and I've gone pretty far down that road) is to explicitly calculate the polynomial for some arbitrary element of the extension field, which is this ghastly set of coefficients that doesn't give me any insight at all. I can (tediously) check that the zeros listed in the back of the book are in fact zeros this way. But all I can think in terms of strategy is to test out each possibility -- and there are 75 of them!)

this is a simple extension field, and I do understand why the added element is a zero.

>>10375982

> n.j. sloane is an author of the paper
> can't stop reading it in his voice

FUCK

Injective means that if f(a) is not f(b), then a is not b. why is the contrapositive to this statement so worthless?

>>10383128
>if f(a) is not f(b), then a is not b
Nope. This is simply the fact that a function f: X -> Y sends every element in X to precisely one element in Y. If you had a = b but not f(a) = f(b), then f would send a to at least 2 elements in Y, and thus f would not be a function.

>>10381465
>After that, you will get to the purer stuff like compactifications and Tychonoff's theorem etc.,
I took topology years ago and this got me thinking, is the Stone-Cech compactification used for anything at all?

>>10381536
Diestel's graph theory

>>10382247
differential equations sucks the first time around, it's so boring
once you get into qualitative analysis of nonlinear equations which you can't solve, things get a lot more fun
you may do some of that in the first class, or you may have to wait until a second.

>>10383128
that's not what injective means
what you just said literally is equivalent to "if a = b then f(a) = f(b)" which is obviously true for any function
a function is injective if we have "if f(a) = f(b), then a = b" or equivalently, "if a is not equal to b, then f(a) is not equal to f(b)".
usually the former is easier to prove but the latter is easier to understand.

>>10383226
Think of it as a functor from the category of topological spaces to the category of compact spaces.

https://ilaba.wordpress.com/2018/01/21/as-you-do-unto-us/
>This post is for the men in mathematics who have been disturbed by the recent wave of disclosures and pushback against sexual harassment. You are horrified to learn that men have been doing such things, and you extend your sympathy to the victims, but you also need to know the possible implications for you. You’ve been asking us to clarify the rules: when you’re patting a woman on the back, where exactly do you have to stop before you get accused of grabbing her ass? Could we please draw red lines across our backs to demarcate the allowed from the unforgivable? You’ve been arguing about fairness, intentionality, proportionality, due process and reasonable doubt. You’ve been citing examples, both from the public sphere and from your own experience. I’ve never before seen so many men come to feminist discussions with well researched facts and cross-checked citations.

>That’s good. I’m very glad that you are doing this. I’ve been engaging in these discussions individually on social media as time permits, but I also want to post a few things here for those who might be interested.

>First, there’s a popular misconception that must be addressed, namely that such cases are only about the crossing of personal and sexual boundaries. No. Grabbing or exposing body parts at work is not just gross; it also derails and blocks our professional advancement and therefore our access to power in the society. Sadly, women at work are too often seen as primarily personal and sexual beings who should be satisfied with social popularity and possibly sexual gratification instead of seeking actual professional success. Our complaints about men who sabotage our careers are dismissed as “personal” disagreements...

>>10383226
> is the Stone-Cech compactification used for anything at all?
Well surprisingly, it can occur "naturally". I have just seen this used yesterday. If you take [math]R = \prod_{i \in I} F_i[/math] an infinite product of fields , then the prime ideals of R correspond naturally to the ultrafilters on your index set: if you have an ultrafilter U, then R maps to the ultraproduct of the F_i wrt U (which is a field by Los' theorem), hence the kernel gives a prime ideal; conversely, if you have a prime ideal [math]\mathfrak p[/math] of R and a subset S of I, then the "sequence" [math]x = (x_i)[/math] with [math]x_i = 1[/math] for i in S and 0 otherwise is an idempotent of R (ie. x^2 = x) and therefore either x or 1-x maps to 0 in [math]R/\mathfrak p[/math]. Hence, you can define an ultrafilter U by decreeing that S is in U iff the sequence x defined as above maps to 0 mod p.
Hence, set-theoretically it is the Stone-Cech compactification.
If, in addition you make some topological assumptions on the fields (I don't remember right now), then the set of prime ideals can be endowed with a topology that is homeo to the bone fide Stone-Cech compactification.

>>10383226
Stone-Cech compactification of natural numbers is useful
A cool example is Hindman's theorem, a combinatorial statement about integers which can be proved with ultrafilters and Stone-Cech compactification.

>tfw you'll never be a mathematician in the 19th century making breakthroughs every day while drinking fancy liquor and fighting duels over women
why live? everyone's just a bunch of nerds now that spend their career proving tiny results no one really cares about

Good morning lads

I never applied myself in math class while in school, and as a result I probably have the math skills of a 5th grader. I think I'm of average intelligence so I could get better at math if I tried. Is Khan Academy a good way to improve my skills and get them up to college level? I'm starting college this summer.

>>10384125
>you'll never prove a major conjecture and publish it under a chuuni name for jokes
It hurts.

>>10384242
Yes, it is.

>>10384242
Yeah. Just keep in mind that math is very much a cumulatively learned subject, if you don't understand a topic and go on anyway you're going to get the wrecked. I think this is why so many people are terrible at math, they never got fractions or basic algebra or whatever down and their learning essentially stopped there. You can definitely get your math skills up to par before summer, you just have to make sure you're disciplined and take as much time as you need to really understand stuff, even if its elementary.

>>10380812
He used 1/2 commas correctly, that's pretty good

what gauge wire do i need for this diagram

injection mold heating control

>>10378356
It is entirely consistent to have a surjection from [0,1] to aleph_2.

>>10376049
I once saved a 10/10 one about Calabi-Yau manifolds but somehow deleted it :( have some Cirno instead

>>10384533
cirno a breeze

Could a reading group for Bishop's Constructive Analysis work?
>literally no requirements
>a reasonable enough topic to draw interest from oldfags and newfags
>fertile ground for shitposting
>super comfy

>>10384515

impossible to know without knowing what the volts/amps across each wire are.

you probably should aim for two different guages of wire: one for low voltage control circuitry and one for the mains/heating element

>>10384687

stfu and read the fucking book normie

>>10378931
Try assembler, nigger. All instructions you can do are basically

>move this number to there
>add this number to that number
>jump to that line

Once you understand assembler, you understand C. Once you understand C, every other programming language is child's play.

>>10384533
Not the same thing, but have this.

anybody here into physics? recommend a good book to build intuition? especially with those differential proofs they throw around all the time

>>10385612
Wen and Nakahara.

>>10385692

>>10385692
I was thinking something at more of a lower level to start with, like a good mechanics text book. or should i just bite the bullet and struggle through goldstein

>>10385797
https://arxiv.org/pdf/1511.02243.pdf
here you go boya
have some laughs

>>10385842
>Míra Rapčák
LOL i remember this guy from undergrad
wasn't stellar, to say the least
good to see he pulled his head out of his ass

>>10385926
keikao doori

>>10385972
Just out of curiosity, are you affiliated? Or was that just a random SYM paper you linked? Because it's not something people would know about, even among hep-th.

>>10383480
>comments are closed
disappointing
I was expecting some comedy gold down there

>>10386028
1 - 1
it's a match boys!

>>10385797
arnold

>>10384389
Not him but I've been postponing trigonometry for a while now, and that's the one thing that can cockblock me from pretty much all of college math. I don't know why I'm doing this with myself, maybe I'm scared of not getting it.
Sorry for the blogpost.

functional anal midterm soon

>>10377794
But Tenebaum is half the price though.

>>10378981
maybe try working top-down rather than bottom-up? Read SICP to understand the "why"

>>10384533
Was it tohou related? was it this?

>>10386101
what title?

i am so glad i am not in uni and have exams now

>>10383480
>You are horrified to learn that men have been doing such things, and you extend your sympathy to the victims,
what if I am not though?

Reminder to always make an even number of sign errors when doing arithmetic

>>10387226
Do you have a job? One of my worries is inevitably dropping math after I start wageslaving, I enjoy it but not enough to dedicate much of my wageslave time off to it.

I am getting a math associates degree because I can't justify getting into crazy debt for a math major. My plan is to learn the basics in community college ((calc 1-4), discreet, differential equations and linear algebra) and learn the rest off of books and the internet. What do you think of my plan /sci/? It's more for the love of it rather than for getting a job from it. I'd like to be able to make and/or read complicated proofs someday.

>>10387561
>calc 1-4, discreet
>community college debt
The first one suggests you're a monkey brazillian. The second that you're mutt.
Which is it?

Honestly, if you can learn from books in the first place, you might as well not waste time at community college.
>>10387117
>midterms in february
North hemisphere niggas baka.

i need an actual good book about coding theory that isn't obsessed with applications

>>10387651
who the fuck cares about codes in and of themselves?

>>10387703

I didn't until I learned that the degree k boolean functions (refering to the A.N.F. representation) are reed-muller codes. So coding theory is useful for studying boolean functions.

Is there any merit to Coq or GAP?

>>10376018
a+b = c

just finish this current and next one and grab the next one over and over

>>10387945
Have no idea about Coq, but GAP is pretty good for calculations with groups. Can do pretty much everything and has huge libraries for finite groups. It is also reasonably well equipped to do representation theory of finite groups.

In my opinion it beats MAGMA, because its open source and has thus less bugs/incorrect data, although it lacks some of MAGMAs functionality if you need some very specific things, like homological algebra stuffs etc.

SAGE is also not bad at providing a nice all around computational toolset, but it is not as focused as GAP. I think you can access some of GAP's shit with SAGE, so that is a plus I guess.

any recommendations for a good intro to PDE's book?

>>10388008
Is Gap hard to pick up

>>10384592
And so is elementary arithmetic, fittingly.
>>10387216
It's this one, thanks <3
>>10387561
Sounds like you want to take math as a hobby. If you won't even pursue a related, perhaps applied field such as engineering, then it's better to not pay for college at all. When you pay for a college or university you are NOT paying for the lectures —you might even be able to audit the lectures for free unless they're up their own ass or something—, but rather to certificate your expertise in a given field of knowledge. You're paying for the grading, your title and maybe some small services if the institution offers them. Your idea of "learning the basics" for the love of it is noble, but the way you propose to execute it amounts to a big waste of money that'll only benefit the shoddy college you're planning to attend.
Given that you can access the Internet you pretty much have no excuses to avoid self-studying: readily available books, notes, recorded lectures and community forums given that you know how to search for them.

>>10388511
I disagree. He should get that associated

how do we colonize numbers

>>10388415
Evans

Where do i go after Linear Algebra and Calc 2? I want to end up in topology and eventually understand mochizuki somewhat

>>10388596
heh

>>10388641
>end up in topology
>eventually understand mochizuki
Haha what?

>>10388646
thats my life goal but i guess im retarded, where should i go next though if i want to be a memethmatician

>>10388641
I'll say it so no-one else has to
Even if mochizuki is right, YOU will never understand anything he has written down. To understand the first paragraph you must be on track to completing a PhD in arithmetic geometry.

>>10388513
An associate's in math is pretty much worthless on it's own, that's why I said that unless he plans to continue his studies on a related field, it's a waste of time.

>>10388641
Real and complexanalysis (unless you already did proper proofs in calc) and abstract algebra.
or you can just take topology directly, it has little prerequesites. if you are into arithmetric geometry, start with elementary number theory.

honestly, studying stuff to understand mochizuki (who is probably wrong on abc) instead of doing stuff you actually enjoy sounds like a horrible plan.

>>10388660
It 100% isn't are you crazy

>>10388641
Mochizuki is a meme. You'll waste 10 years of your life trying to understand his proof of abc only to realize that it's wrong.

mathlet here, any help with this question? It seems to me it would depend on what the function was right? even though the integrals use all the same values.

>>10388696
look at the area you integrate over and match them to the dz dy dx. you can commute the order of integration but not the area each variable is integrated over

okay thanks so the only answer that matches up is b

>>10388709
yup

>>10388653
if you have to ask you'll never know

If U is a subspace of a vector space V with a complementary subspace W_1, then if for any other complementary subspace W_2, how do I show dim(W_1) = dim(W_2)?

What do you guys think of this one?

Answer each question about the surface area S on a surface given by a positive function z= f(x,y) over a region R in the xy-plane. Explain each answer.

(a) Is it possible for S to equal the area of R?


(b) Can S be greater than the area of R?


(c) Can S be less than the area of R?

im getting contradictory answers from the internet on this

>>10378834
What book is this?

>>10388877
Never mind, found it myself.

https://books.google.co.jp/books?id=YSe4hUBM7uEC&pg=PA160&lpg=PA160&dq=%22Cauchy+is+crazy,+and+there+is+no%22&source=bl&ots=qoP2reyk0K&sig=ACfU3U2RW7MeVUbY2uLFuLUEMFdjbtM_ow&hl=en&sa=X&ved=2ahUKEwiixOzm_rzgAhXGi7wKHeC8AxMQ6AEwAHoECAUQAQ#v=onepage&q=%22Cauchy%20is%20crazy%2C%20and%20there%20is%20no%22&f=false

>>10388804
[math]U \oplus W_1 \cong V \cong U \oplus W_2[/math], so [math]W_1 \cong V/U \cong W_2[/math].

>>10388874
>(a)
True, take f as a constant function

>(b)
True, pick literally any continuous function.

>(c)
False, but I'm too lazy to write down the proof. Try doing it with simple functions and then take limits using the monotone convergence theorem.

>>10388874
Yes, yes, no.

It's the 2D version of whether the length of the curve defined by y=f(x) over [a,b] can be equal to, greater than, or less than b-a.

Essentially, you're comparing the integral of 1/cos(a) (where a is the angle between the line/surface and the horizontal) to the integral of 1, both over the same region. And 1/cos(a)>=1.

>>10388507
It is about as hard as any other mathematical program, such as Mathematica, MAGMA, etc. If you have a basic understaning of any other classical programming language, you should be fine. There is plenty of documentation, about how to use GAP and all its functions, with lots of examples. See
https://www.gap-system.org/Doc/manuals.html

All in all I'd say it is very similar to python.

>>10375982
/engi/ student here, anybody know of any good resources for control systems? Been doing diff EQs for several years now but it still hasn't clicked, and now i'm stepping into analysis and shit's getting out of control

>>10388415
Strauss, if undergrad. If grad, well... I guess Evans.
>>10388628
Oh my lord, how could you suggest something like that with no context?

>>10388804
If you haven't done quotient spaces, do this >>10388930 but take dimensions instead of quotienting by U. Remember the dimension of a direct sum is the sum of the dimensions.

>>10389170
>control systems
not maths

>>10388804
this is essentially the rank-nullity theorem (consider the projection onto W_1 wrt U and prove that it is an isomorphism in restriction to W_2)

>>10388804
You can also do it in a more non-algebraic way by considering a basis of U and extending it to a basis of V.

>>10388874
>>10388874
This is basically asking if the vertical projection of the surface has more or less surface area.

(a) Any constant function will do. Even easier is z=0.

(b) While almost every other continuous function will do, I think a more general way to do it that doesn't involving hand-waving is to take a small open subset inside R and create a partition-of-unity-type function on z=0.

(c) While intuitively this is correct, it's not entirely obvious. I think the proof can be handled in a couple stages:

(1) Consider any P be any plane in 3D space, and let R' be any bounded subset and R its projection onto the xy plane. Then Area(R')cos(t)=Area(R) where t is the angle the plane P makes with the xy plane. In particular, the area in the xy plane is smaller or equal to the area in the plane P.

(2) Any (almost everywhere) continuous function (or Riemann integrable) can be subdivided into small enough rectangles, such that for any epsilon, the discrepancy between the surface area and the sum of the area of the rectangles is less than epsilon.

(3) Take any bounded region S of the surface and let R be the vertical projection. The area of the sum of the rectangles covering S is bounded below by Area(R) by (1)

(4) Since epsilon was arbitrary, you can conclude that Area(S) is bigger or equal to Area(R).

HELP! How do I construct an isomorphism of vector spaces [math] U\cap W \rightarrow \text{ker}(A)[/math] for subspaces [math] U,W[/math] and [math] A(u,w) := u+w [/math] for [math] u \in U, w \in W[/math]

>>10389680
[math]u \rightarrow (u, -u)[/math]

>>10389697
THAT'S WHAT I THOUGHT WHY THE FUCK IS LINEAR ALGEBRA SO GAY
thank you animeposter

Easy brain teaser to start off your day:

Is R^R T4 space in the box topology?

>>10389972
Isn't R^Z already not T4 since you can make two sequences that converge to the same value, but one from below and the other from above?

If [math] a = t_1^{g_1} t_2^{g_2} \ldots t_v^{g_v}[/math] and [math] b = t_1^{h_1} t_2^{h_2} \ldots t_v^{h_v}[/math], and if [math] d = \gcd(a,b) = t_1^{c_1} t_2^{c_2} \ldots t_v^{c_v}[/math], how the f do I show that [math] c_i = \text{min}(h_i, g_i)[/math] for all [math] i[/math]???

>>10390149
This question is so stupid that I'm not even gonna give an ironic response. My best advice to check that you're not actually mentally retarded would be to try it out with some numbers to see why

>>10390190
It is fucking stupid mate, but it's one of four questions on our homework.

>>10390149
Induct on the number of ts.

>>10390149
that's actually an open problem

>>10390222
Well then gimme my money fag

>>10390149
how do i practice at reading and getting good at this notation? just read textbooks?

>>10390149
Assuming that the t_i are all coprime, it's so obvious I wouldn't have a clue what a proof would look like. If they aren't required to be coprime, you're in trouble.

>>10390149
use the definition

>>10390312
What notation?

>>10390338
If they aren't coprime it isn't true.
g=2^1 4^2, h=2^2 4^1

https://www.youtube.com/watch?v=7UwzTdbhHzc

>>10390688
>>10390338
>Nitpicking what is obviously what the author meant
What if it's a non-commutative ring without unique factorisation? Huh? Thought of that, retards?

>>10390312
To "get good" at any sort of mathematics you need to actually DO mathematics, not just read them passively. I know you don't literally mean to just read the book, but this is an important thing to note, nonetheless. The following quote from Paul Halmos encapsulates my point well:

>The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method, the method in which the teacher plays the role of an omniscient but largely uncommunicative referee between the learner and the facts. (...)

>The right way to read mathematics is first to read the definitions of the concepts and the statements of the theorems, and then, putting the book aside, to try to discover the appropriate proofs. If the theorems are not trivial, the attempt might fail, but it is likely to be instructive just the same. To the passive reader a routine computation and a miracle of ingenuity come with equal ease, and later, when he must depend on himself, he will find that they went just as easily as they came. The active reader, who has found out what does not work, is in a much better position to understand the reason for the success of the author's method, and, later, to find answers that are not in books.

As for the "notation" you speak of, it's just a pretty standard way to write, as generically as possible, a number as the product of its prime factors. for instance, instead of [math]a = t_1^{g_1} t_2^{g_2} \ldots t_v^{g_v}[/math] we could've written [math]24=2^3\cdot3[/math], but this is merely one particular case: here, [math]a=24[/math] and [math]v=2[/math]. A good book if you want to study this, in particular, is David Burton's Elementary Number Theory.

>>10390787
there's no such thing as a "non"-commutative "ring."
there are dumb garbage "structures" which morons drool over to make themselves feel adequate, and there are rings (which are equipped with a commutative multiplication operation and unity).

>>10391824
>rings must have unity

>>10391824
>there's no such thing as a "non"-commutative "ring."

>>10391824
I'm not sure if you're just meme-ing or what, but you're probably familiar with matrices, which for a noncommutative algebra. All of quantum mechanics is based on (possibly) noncommutative operators. Without them you would not have Heisenberg uncertainty, for instance.

As far as I am aware (keep in mind I am a geometer not an algebraist) rings without unit are mostly useful as a technical trick when working with augmented rings (which are widespread in algebra and geometry, c.f. deformation theory in algebraic geometry which studies augmented commutative algebras or stacky generalizations thereof). Simply: given a map R -> k of k-algebras the kernel will be a nonunital algebra. You might also frequently run into them in the theory of Hochschild homology, etc.

Is Khan academy recommended for a mathlet? I pretty much stopped paying attention around pre-calculus in school.

>>10392025
Yeah, and you're pretty much the target audience. Go for it.