/sci/ - Science & Math

I only study the day before the quiz/test.

Admissible Sets and Structures by Barwise.

>>9193069
Just got my honors algebra problem set, solved about 4/10 of them.

What is 23 x [6;33[?

>>9193155
Is this high school, freshman or sophomore?

Also if you want pick any of those (except 6 because the only proof I remember right now of the division algorithm is way too long for me to care to type it down) and I'll do it for you.

>>9193124

You are either a brainlet or a math Chad.

>>9193181
i don't think cross product is defined for integer and 1x2 matrices

>>9193069
I’m trying to learn about Cyclotomic Polynomials right now. Just for fun and bullshit.

>>9193069
I'm beginning to have doubts about the existence of even most rational "numbers".
Is this normal?

>>9193452
yeah for someone who just started to question how everything works

it's not a bad thing, we all kinda have a phase where we think everything is wrong, eventually you'll get to a point where you realize it's meaningless to try to redefine everything off some new standard if it doesn't even break way to any new discoveries, only to more limitations

go ahead and post in wildberger threads all you want, i'm not going to spend time trying to disprove a claim without evidence given
note: i'm not asking for evidence, just take it to the nearest wildberger thread

>>9193410
>math Chad
no such thing. everyone who's good at math studies or has studied extensively.

Reading Velleman's How to Prove It. Learning a lot coming from a physics background. Recommendations for future reading?

>>9193628
what're you planning on learning?
discrete math and linear algebra are both good for babbies first "with proofs" class

I've been reading Kosinski's Differential Manifolds following an anon's suggestion. Going through the introductory chapters right now to dust off some of that differential topology I forgot in the last year I've been out of school.

[eqn]7 \cdot a^{2} \cdot b^{2} = 11 \cdot c^{2} \cdot d^{2}[/eqn]

I'm trying to derive a contradiction from this. Any pointers? a, b, c, d are integers

>>9193637
How're you liking the book?

>>9193675
Pretty well. I like Kosinski's writing so far and I like the material. The exercises can be pretty involved and the book doesn't really hold your hand at all in terms of providing similar proofs or anything, but I haven't gotten too stuck on them. Pretty solid altogether.

>>9193721
That's great, if you need any other book suggestions I can probably recommend some other nice texts.

>>9193671
for all integers, a^2*b^2 = n^2 where n is another integer.
this means you can rewrite it to 7a^2=11b^2
for this to work, 11|7*a, meaning a must be an integer multiple of 11, meaning 77|a. also, 7|11*b, meaning 77|b.

no matter what you will always have a prime factorization of 7*11^a=11*7^b, which through division is equivalent to 11^(a-1)=7^(b-1), and for any integer a and b, they can never be equal since neither side will ever have a prime factorization of both 7 and 11.

rate my proof, i haven't taken a class with proofs yet

starting to read tao's notes on measure theory

https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf

is this "box" terminology typical? or it k-cell more common term?

>>9193792
>tao had notes on measure theory
didn't know about that
thanks

>>9193069
WHAT DO I DO LADS?!

I've got 2 days to learn Fourier series, Fourier transform and DFT & FFT.
So that I can teach it to a small group of people.

>>9194147
Fourier series is basically just using the orthogonality of sin and cos with respect to the integral inner product from 0 to L (or maybe it's -L to L, it's been a while)

>>9193634
>discrete math and linear algebra are both good
Wrong.

>>9194147
One day is more than enough for that stuff

>>9193671
>a, b, c, d are integers
>I'm trying to derive a contradiction from this
You're pretty much done. Arbitrary integers don't necessarily exist.

>>9194242
Explain

>>9193792
>>9194073

Tao's construction of the Lebesgue measure is godly. It is completely intuitive and very little out of left field proofs. It starts with a completely geometric notion of measure (measure of an interval is it's length, similarly his 'boxes' are n-volumes) and slowly adds in analytic machinery to get the class of Lebesgue measurable sets. It is so much better than the two other authors whose last names start with R that have written widely used measure theory books.

>>9193792

Anyone else feel like Tao's teaching materials are pedagogically really poorly made? It seems like he is trying to show how smart he is and not make people learn. It seems that all his blog posts and course notes are probably really great once you are a fields medalist, but I find them completely inaccessible, and I've taught myself very niche stuff like local field theory.

>>9194308
I can only speak for his intro measure theory book which I used in a grad analysis course. We used Royden in my undergrad measure theory class and Tao in my first semester grad analysis class. Maybe I just was inexperienced when I used Royden but I didn't really get measure theory until my grad class, and I thank Tao's book as well as a great instructor for that. I found every construction was specific and precise, and the exercises perfect complements and extensions of the material. I have never more than glanced at his blog or have read any of his other books.

>>9194303
yeah i started reading Tao's and got annoyed with how much exercise he just starts laying out. but the i went to Rudin and thought: wtf is this? So now im back on the Tao train

>>9194334
>>9194303
Rudin is a meme. Good work not falling for it.

i know that the complex numbers can be constructed as the quotient ring of the polynomial ring of real numbers by the ideal generated by (x^2+1)

is there any interesting structure to quotients of higher order ideals?
and what about polynomial rings of multiple variables?

one thing that comes to mind is [math] \displaystyle \frac{\mathbb{R}[x,y]}{x^3-x-y^2} [/math], or more generally, [math] \displaystyle \frac{\mathbb{R}[x,y]}{x^3+ax+b-y^2} [/math]
which is of particular interest because of the algebraic structures you can define on elliptic curves
what does this "look like"?
i feel like this might be approaching algebraic geometry territory, which i know nothing of
can i get a quick rundown on this?

>>9194336
It's called a coordinate ring, try skimming https://en.wikipedia.org/wiki/Affine_variety

Probability theory

>>9194356
And some algorithms textbook

Progressing slowly through Mendelson's Introduction to Topology while I live the wageslave life. I was an applied math major who didn't even take Real Analysis, so I probably should have done Baby Rudin or something first, but there is enough coverage of continuity and a little section that hammers epsilon/delta so we'll see how this goes.

Can you define the dot product for "continuous matrices" as as [eqn](A \cdot B)(x, y) = \int\limits_{-\infty}^{\infty}A(x,s)B(s,y)\ \text{d}s[/eqn]

>>9194375
>I was an applied math major
How can you major in something nonexistent?

>>9194336
>i know that the complex numbers can be constructed
Not really.

>>9194430
What do you mean?

>>9193750
here's a better proof: 7 is not a square
QED

Currently going through Aluffi Algebra Chapter 0, killed almost every section but I'm onto modules and fuck Im kind of plateauing. Any tips?

what math books have u actually read?

does anyone here go on irc?

>>9193628
depends on ur goals

>>9194384
sure, assuming appropriate regularity conditions

>>9194430

Is it possible to be bad at calculus but good at more advanced math? My math logic class comes really easily to me but calc 3 doesn't? What gives?

>>9195444
What exactly are you having trouble with?

>>9194336
In general, a ring R can be thought of as a ring of functions of the affine scheme Spec(R).


The schemes associated to the rings you listed are not particularly nice because the reals aren't algebraically closed.


For an alg. closed field k, and a prime ideal p of k[x1,...,xn]. The ring k[x1,...,xn]/p is isomorphic to the ring of regular functions on the variety V(p).


ex. If p = < y^2 - x^3 > then k[x,y]/p is the ring of regular functions on the affine curve y^2=x^3

>>9195638
>Is it possible to be bad at calculus but good at more advanced math?
I don't know anything about calculus, so probably.

>>9195638
I don't think it's terribly uncommon for that to be the case. I had a hard time with calc 3 because I'm pretty lazy with algebra so I had negative sign accumulation. I got an A, but I definitely worked for it.

There was a girl in our math department who failed Calc 3 but got a 40 on the Putnam exam one year and was allowed to teach graph theory as a senior.

>>9195638
If we accept the assumption that you're actually good at math the problem is that you're missing foundations.
In a logic course you start from scratch, but if there's concepts you didn't understand in the first two sections of calculus calc 3 is going to be difficult.

If you struggle with calc 3 while actually understanding calc 1+2 you are unfortunately a brainlet, there's nothing in that course that's more than a minor variation on stuff you already know.

>>9194147
>2 days
>2
>learn Fourier series
>learn DFT
>learn FFT
>learn it enough to teach people

I pity you and the people who will be confused and have such a beautiful topic ruined by an obvious mental retard.

Why are linear algebra and discrete mathematics (like combinatorics, set theory, graph theory, etc) all taught after calculus? there is no calculus prerequisite for any of this

are there more branches of math like this, where there aren't really prerequisites beyond geometry and algebra?

>>9195882
oh, probablility and statistics too

>>9195882
>are there more branches of math like this
Yes, basically anything even remotely interesting.

>>9195885
You actually use some calc in probability and stats such as applying the geometric telescoping series to certain problems.

How can math explain non numerical concepts like gravity?

I know you can calculate gravity with math but how does one use math to explain gravity?

>>9195872
WHY DO THEY CALL IT FAST FOURIER TRANSFORM IF I CAN'T LEARN IT IN LESS THAN 30 MINUTES

>>9195896
like????
???
>?
????

>>9195898
yeah but a lot of you don't use anything at all

>>9195905
a subfield known as physics

>>9195905
by giving physical things a symbolic representation, we can extend math into the real world
e.g. let s(x) be the displacement between two points in real space, s'(x)=v(x), or velocity

right now we're still in a math-like realm, so no real major assumptions have been made, this isn't true for most things higher than this. Pretty much everything is based on observations, or just fitting some math to the observations.

when it comes to gravity, almost all of it is based on observations (see Kepler's laws, all are based on observations)
all newton did was showed that F~m1m2/r^2, with there being a constant G for that, all based on data he had. this is good enough for every entry physics course and is pretty good for general purposes. Obviously this doesn't 100% hold true, otherwise general relativity wouldn't be a thing. There's also some discrepancies about how Mercury's path is like 50 arcseconds per century off or something.

In fact, Newton even tried to let F=Gm1m2/r^2 + Bm1m2/r^3 at one point.

so basically you just give axioms based on what you observe and hope they're right enough.

>>9195929
>yeah but a lot of you don't use anything at all
are you fucking serious?

Could a kind anon present https://link.springer.com/chapter/10.1007%2F978-3-642-02094-0_7

>>9195643
it's hard to say but it's the only section where i've had trouble almost every exercise

>>9193069
Brushing up on my Galois theory.
I'm trying to take a combinatorial approach.

If you've ever wondered how many irreducible monic polynomials of degree n there are over a finite field, here you go.

>>9195929
>yeah but a lot of you don't use anything at all
probability theory = measure theory with a bounded measure (and some weird-ass notation). integrals all over the place. if not, you're doing something wrong.

>>9195905
You could describe the 1/r^2 part pretty easily.

The surface area of a sphere is proportional to r^2.

If you are a distance r away from a point mass of mass m, you can imagine a sphere of radius r centered at the point mass (your position will be on the surface of the sphere). Now imagine the mass is radiating gravitational energy equally in all directions in proportion to m. If you calculate the flux density of this radiation at your point on the sphere, you will find it to be proportional to m/r^2.

>>9196156
what about fields of the form Z/p^n ?

>>9196190
>what about fields of the form Z/p^n ?
Those aren't fields (unless n=1).

>>9193069

I have an integral test coming, so can you guys post your nastiest integrals within the material of calc. 1 and 2.

>>9194147
>so that i can teach it to a small group of people
why are you teaching it if you dont even know it?

>>9196240
i guess he meant [math]\mathbb F_{p^n}[/math]

>>9196805
I think he thinks that [math] F_{p^n} [/math] is [math] \mathbb{Z} / p^n \mathbb{Z} [/math] .

>>9196156
what does this have to do with galois theory? pretty much all you need is the fact that the polynomial ring is a ufd. still a nice argument.

>>9196758
I had my integral test last week, turns out I diverge.

Get me into maths anons, I hate it.

>>9195628
So 10 is equal to 0.2 ?

>Proof of this theorem has been left as an exercise to the student

>>9196951
Then you prove it like you were solving any other problem, but no! It's the redditfrog retard who needs to have solutions in the back of the book instead of being able to proofread its (yes, it doesn't even deserve to be treated like a human bean) own proofs. Just give up, you will never become anything.

>>9196977
pretty sure facebookfrog lad hasn't taken a proofs class (or any class involving proofs) yet, so it's kind of justified.

almost everyone struggles at first with proofs when you haven't done lots of practice with them before

>>9197046
>tumblr spacing

>>9197046
>everyone struggles with proofs when you haven't done lots of practice
>yet he complains about authors forcing him to practice
wew

>>9197055
>someone struggles with practice problems because they haven't done a lot of practice
yes it's a straight forward conclusion

i'm saying it's not like it's all hopeless because he struggled to prove a theorem, that it's just through lack of practice

>>9197057
the point is he's complaining about a book guiding him through the practice he needs, and that's fucking retarded. he needs to suck it up and do it

>>9196758
[eqn]\int_0^1 \log (x)\log(1-x) dx[/eqn]

>>9196190
>>9196240
refer to >>9196805
Sorry about the shit notation.
I'm pretty sure the counting argument still works if you replace p by p^N everywhere.

Counting the isomorphisms that fix the base field will be fun
>>9196863
If you start with a field F, you work with polynomials over F, namely, F[x].

To do a field extension you pick an irreducible polynomial p(x) in F[x].

Then you mod out by the ideal generated by p(x), <p(x)>. <p(x)> is just all polynomials having p(x) as a factor. You could write <p(x)> as (F[x])*p(x).

You get F[x]/<p(x)> is a field extension of F.
Addition works like you think it should (it inherits the structure from F). It is pretty much vectors with components in F.
Multiplication is done like in F[x] but then you reduce the result mod p(x)

To do iterated extensions, you let K=F[x]/<p(x)> and look at polynomials over K, namely, K[y].

You can pretty much represent the whole theory as multivariate polynomials satisfying the modularity rules defined when "modding out" by the polynomial ideals.

Where can a math major undergrad intern in a shitty country

>>9196758
[math]\int \sqrt{\tan{x}} \mathbb{d}x[/math]

have fun

>>9196758
[math]\int e^{e^x}dx[/math]

>>9196758
[math]\int e^{-x^2}dx[/math]

>>9197186
which country

>>9197227
pretty sure that you need to have studied integrals in more than one variable to solve it in a reasonable manner

>>9197246
New Zealand

>>9197205
[math] \int e^{e^x}dx [/math]. Let [math] u = e^x [/math]. Then [math] du = e^x dx = udx \implies dx = \frac{du}{u} [/math]. So the integral becomes [math] \int \frac{e^{u}}{u}du = Ei(u) + C = Ei(e^x) + C [/math]

Heh, that was trivial. You are like a little baby compared to me.

>>9197289
Integrate this
*unzips integral*
[math]e^{e^{e^{.....^{e^{e^{x}}}}}}[/math]

>>9197297
I am afraid not even the exponential integral can do against this BEAST. However, after quickly applying Tai's method I noticed it was trivial and therefore I leave it as an exercise to the reader.

quick how do I prove that the intersection of two subgroups of a group forms a subgroup

I can't see where to start with proving closure

>>9197361
https://en.wikipedia.org/wiki/Subgroup_test

>>9197251
u=x^2, du=(dx)2x=(dx)2u^(1/2)
Then know the gamma function.

>>9197361
for all [math]a,b\in H\cap K[/math], [math]H[/math] and [math]K[/math] being subgroups means [math] a*b\in H[/math] and [math]a*b\in K[/math], so [math]a*b\in H\cap K[/math]

>>9194336
For your question about higher order ideals: every ideal in k[x] is principal, and you can factor a generator f into irreducibles. Then k[x]/(f) is just a product of field extensions of k by the Chinese remainder theorem.

>>9197361
[math]a, b\in H\cap K \Rightarrow (a, b\in H\land a, b\in K) \Rightarrow (ab^{-1} \in H \land ab^{-1}\in K) \Rightarrow ab^{-1} \in H\cap K[/math]

>>9196938
the only numbers that are truly "real" and invariant are the primes. so no, 10 does not equal 2, but it is the product of 2 and 5.

>>9196925
read John Stillwell's elements of mathematics

>>9195534
BUMP

>>9197534
Not him but what's so good about it?

I have a problem that may seem like a pseudo problem, but its happened so many times and for so long that Im starting to think it's a cognitive disability. I will solve the problem entirely correctly on the scrap paper, but then transfer a slightly off answer onto the test sheet. Please help, its not dyslexia and Im not mentally retarded

I'm not studying but reading GMAT guide for brainlets because gmat techer is the only job I could get after a year of being a jobless msc graduate in maths.

>>9197784
>Math masters
For what purpose? Either do the PhD or do statistics

How would a brainlet go about learning physics, should i just get a textbook and start working or what?

>>9197683
I'm just really enjoying it. Providing such a high level overview is nice, and with that comes some historical context. But mostly it's the exploration and demonstration of the most obvious thing, like the greatest common divisor algoritm, pascals trianlge and binomial coefficients, etc.. I feel as though it's cultured me as a mathematician somewhat.

>>9197800
Are there other ways to learn?

I have a serious question for those who study math actively for a living: are you studying proofs?

only reason I ask is because this summer i was introduced to what math really is: proofs. I took real analysis as a chemistry major who almost failed DIFFERENTIAL EQUATIONS and this class shocked me beyond belief. There were 8 students in the class and it was allowed to continue. the proff is asian obv and was completely bursting with excitement every day on this subject. I didnt understand why until the last week

>>9197891
read a proofs book like Vellemens or Hammacks

>>9197800
smoke weed and stare at the stars, brainlet

Sorry if this isn't the right place to post this, but I was wondering if I could get some input. I'm planning what courses to take for fall of my second year, and I've never taken a rigorous math class, so I'm wondering if I'm biting off more than I can chew with this kind of course load:

Differential Equations I
Calculus III
Real Analysis I
Intro Probability (version that has calc II as prereq, there is a full brainlet probabilty class that doesn't need calculus but this isn't it)
Linear Algebra II (canonical forms, inner products, orthogonalization, spectral theory, etc)

Just looking at it it seems like it is probably too much and I should leave the linear algebra until later and take a computer science course (or something else completely unrelated), but I'm not sure. Any suggestions would be appreciated

>>9198049
if you've never taken a real rigorous math class, you might want to wait til next year to do real analysis, try speaking with your advisor
also
>diff eqs I
there's more than one diffy q class?

Book of proof or how to prove it?

>>9197794
Maybe he's yuropoor. Here it's standard to do msc before phd

>>9197800
Watch everything neil d tyson ever did

>>9198118
>you might want to wait til next year to do real analysis,
Hm I thought I'd be better off dropping something else instead if it was going to be an issue, since it seems like a pretty important course.

>there's more than one diffy q class?
Yeah, the second one additionally has a vector calculus prereq.

>>9198197
i mean real analysis is an important class, but can it wait until next semester?

best way for me to help you is to tell me what classes you want to/have to take next semester and what the prereqs are

>>9193452
Why do you doubt the existence of the rationals?

>>9198204
Nope not really, it would have to be the next year. Real analysis is only offered in fall, same for complex analysis and abstract algebra in winter.

There's nothing you have to take in a specific year and the math program as a lot of electives (unless you're doing an honors degree I guess), but obviously you want to get certain courses out of the way as soon as possible if they are prereqs for a lot of advanced shit.

>>9198204
>real analysis is an important class

>>9197800
Maybe ask on >>>/r/physics

I just got a copy of "A Concise Course in Algebraic Topology"

I know nothing pretty much nothing about Algebraic Topology but have a lot of experience with Cohomology.

Will I make it through?

>>9193069
Where does my soul go and will it come back?

>>9198209
They just seem too hacky to actually exist.

>>9198366
lmao bro just take the equivalence classes of Z^2 by the relation (a,b) ~ (x,y) iff ay = xb but only on elements where the second element in the pair is not zero because you cant divide by zero

you dont even need infinite sets or even negative numbers just a notion of multiplication of your numbers with themselves

>>9198402
This is exactly what I mean, you probably think of [math]``\mathbb{Z}"[/math] as a quotient of [math]``\mathbb{N} \times \mathbb{N}"[/math].
>you dont even need infinite sets
What is the reason for your usage of them if that's the case?
>a notion of multiplication of your numbers with themselves
Which need not be total. You need to actually prove that it is.

>>9197373
>Then know the gamma function
that's cheating

>>9198118
>there's more than one diffy q class?
In my uni there's ODE I, ODE II, PDE.

>>9193069
Differential Manifolds

>>9194147
It's just Linear Algebra

TURN ON CNN

A Proof There Exists Infinitely Many Primes with a Gap of Exactly 2
https://arxiv.org/abs/1709.09950

>>9197252
Move ASAP

Currently working on this problem, any tips/tricks on how to solve this? I'm not the best with coding theory

>>9198622
>A Proof There Exists Infinitely Many Primes
That can't be shown to be true though.

>>9198921
>That can't be shown to be true though.
but he did in the paper

>>9198829
The smallest field containing a 7th root of unity is [math] \mathcal{F}_{2^3} = \mathcal{F}_8 [/math] . so the minimal polynomial is some irreducible degree 3 polynomial.

>>9199665
correction: [math] \mathbb{F}_{2^3} [/math].

is there any intuition behind singular cohomology whatsoever ? I might have read somewhere that pic is related

>>9198622
>the references include a literal stachexchange question

Kek. The question was made 2 years ago so there is a small possibility that the author was thinking about this proof for 2 years ago and hit a wall and literally asked for a crucial part of his proof on the fucking internet. The absolute madman.

>>9198622
How does junk like this get on arxiv, I thought you had to have a sponsor to get something posted on there?

>>9199696

Has anyone torn it apart yet? It would be amazing to see the reaction of the established prime-gap people to a nobody proving the most juicy conjecture on arxiv using crowd-sourcing.

>>9198622
Does anybody know of a case where somebody formatted their paper in Word and actually turned out to not be a frothing nutjob?
I'm sure there must be counterexamples but I don't know any.

>>9200960
Recently, Thomas Royen.

I need to do a proof that Lp spaces are banach for Monday...hopefully gonna do some others over the weekend.
Gonna read through Stein and Shakarchi for my Complex Analysis class too. Then I've got some dynamical systems stuff to do but it's not bad, the book's really good.

>>9201181
What book are you using for dynamical systems?

>>9201189
It's an old-ass book by these guys Hirsch and Smale. My teacher used it in her grad school.
For a math book it's heavy on words and conveying concepts, and it does that pretty well.
Also it spends a lot of time covering Linear Algebra stuff I've already done in other classes so it's not difficult.

Wondering if you guys could help me with this problem. For some reason I find it hard to read or do mathematics without some kind of High BPM music playing, should I try to quit cold turkey in order to perhaps concentrate better?

>>9193069
I'm going to start learning classical mechanics, and probably go over some more precalculus.

>>9201410
>classical mechanics
Cool. Discuss it at >>>/r/physics/ though.

Hey /mg/ what is the background necessary to start any book on Algebraic Geometry?

>>9193155
Funny, I find 2 easier than 1. Why those * there?

>>9201696
commutative algebra

>>9201696
knowing a bit of commutative algebra could be useful, but you can also learn it along the road(or even postpone it), this way ag will give a geometric interpretation to the main theorems, and they won't be just a bunch of nonsensical lemmas and propositions
on the other hand i guess knowing bits of algebra and some topology is necessary
if you are learning by yourself give a try to Hulek

Are some of the exercises in this book really difficult or is it just me?

>>9197049
>"le reddit/tumblr spacing meme XD"
Only newfags get get mad over this

>>9202405
>le
>meme
>XD
see >>9197049

3x2 - 6x + 14

ax2 + bx + c = 0

a: 3
b: -6
c: 14

x = (-b ± sqrt(b2-4ac))/2a

+: x = (-(-6) + sqrt((-6)(-6) - 4(3)(14))/2(3)
x = ( 6 + sqrt(36 - 168))/6)
x = ( 6 + sqrt(-132))/6)
x = ( 6 + 11.49 ) / 6)
x = 17.49 / 6 = 2.915

3(2.9152) - 6(2.915) + 14
3(8.5) - 6(2.915) + 14
25.5 - 17.49 + 14
= 22.01

tell me why this don work

>>9201696
Go to the stacks project, download the entire document and start reading carefully from page 1.

>>9202526
>x = ( 6 + sqrt(-132))/6)
>x = ( 6 + 11.49 ) / 6)
the second line here is where you went wrong

>>9201410
>classical mechanics
Nice code word for symplectic geometry.

How would I solve this:
[math]\sum_{i=0}^{n}4^ii[/math]?

Could someone explain what a quotient ring is in brainlet terms?

>>9203062
You should really learn quotient groups first.

>>9203035
sum 4^i i =
4^1 + 4^2 + 4^2 + 4^3 + 4^3 + 4^3 + ... + n(4^n) =
(4^1 + 4^2 + 4^3 + ... + 4^n) + (4^2 + 4^3 + ... + 4^n) + .... + 4^n =
4(1 + 4 + 4^2 + ... + 4^(n-1)) + 4^2(1 + 4 + ... + 4^(n-2)) + .... + 4^n(1) =
sum_{i=1}^n (4^i sum_{j=0}^{n-i} 4^j) =
sum_{i=1}^n (4^i (4^(n-i+1)-1)/(4-1)) =
sum_{i=1}^n (4^i (4^(n-i+1)-1)/3) =
(1/3) sum_{i=1}^n 4^i (4^(n-i+1)-1) =
(1/3) sum_{i=1}^n 4^(n+1)-4^i =
(1/3) [sum_{i=1}^n 4^(n+1) - sum_{i=1}^n 4^i] =
(1/3) [n * 4^(n+1) - {(4^(n+1)-1)/(4-1)-1}] =
(1/3) [n * 4^(n+1) - {(4^(n+1)-1)/3-1}] =
(1/3) [n * 4^(n+1) - (4^(n+1)-4)/3]

>>9203035
Summation by parts turns it into a trivial problem

>>9203175
Thanks this was the method I was forgetting.
>>9203153
I think you're summing from 1 to n and also not taking to account that 4^i is getting multiplied by i.

>>9203182
>I think you're summing from 1 to n and also not taking to account that 4^i is getting multiplied by i.
it is multipled by i in my sum, read it again

also
http://www.wolframalpha.com/input/?i=sum+i*4%5Ei+from+i%3D0+to+n+%3D+(1%2F3)+%5Bn+*+4%5E(n%2B1)+-+(4%5E(n%2B1)-4)%2F3%5D
says it's right

>>9203062
>>9203137
this
Teaching Ring theory before Group theory is simply retarded.

>>9193069
Why does finding a GF feel like NP hard

>tfw independently discovered formula for pythagorean triples (n+1)^2=n^2 + b^2

>>9196758
http://www.mit.edu/~same/integrationbee.html

Check the qualifier tests.

>>9204130
>No girls
I didn't know MIT was so sexist wow

>>9203062
It's a generalization of the integers modulo n. In the same way that 4 and 9 are equivalent modulo 5, any two elements of a ring are equivalent if they differ by a multiple of an ideal.

So in the integers mod 5 example, the elements 4 and 9 could be written as 4 + 5Z = 9 + 5Z in the quotient ring Z/5Z, where 5Z is an ideal of Z. Basically, this means that there is some element a in Z such that 4 + a*5 = 9.

Well, similarly in an arbitrary ring R, for some ideal J of R, any two elements are equivalent in the quotient ring R/J if they differ by a multiple of the ideal J. So a + J = b + J implies that there is some r in R such that a = b + r*{finite sums of elements of J}

>>9203987
you are retarded

>>9194384
As long as it satisfies the inner product properties.

Hey guys I'm trying to understand this exercise. I must check if any of those functions are 3-linear function. I get the idea, but it should be 3-linear for any row/column, right? For example c), it's not 3-linear for row two, since D(ca2+a'2) =/= c*D(a2)+D(a'2) where a2 and a'2 are row two of an A matriz and c is an scalar, but it's linear for row 1 and 3.

>>9203987
Makes no sense. Some books, my college included, teaches Ring Theory before Group Theory.

What's the best linear algebra textbook?

>>9205467

Greub

>>9203062
the secret to quotients is to think of them in terms of relations. (these relations will then generate an ideal or their normal closure will generate a normal subgroup, but thats irrelevant for the moment so forget about all of that)

you introduce relations on your ring, like say you have a polynomial ring in 2 variables K[x,y], you could introduce the relations x=1 and y^2 = -1
x would simply vanish and y would become a squareroot of -1.

for a more concrete example, take the integers Z and two variables x and y. In the polynomial ring Z[x,y] introduce the relations x^2 = 2 and 2y = 1.

then you could view Z[x,y] modulo these relations (ie, modulo the ideal that these relations generate) as a subring of the real numbers, by identifying x with a square root of 2 and y with 1/2

>>9205496
more examples: you get the integers modulo 5 by introducing the relation 5=0 into the ring Z.
you can construct the complex numbers C from the reals R by identifying C with R[x] modulo the relation x^2=-1 (this is actually the standard algebraic way to do it)

using the relation x^2=0 would give you so-called "dual numbers" where you could think of x as an infinitesimal element
https://en.wikipedia.org/wiki/Dual_number#Algebraic_properties

>>9205407
Your college and those books are retarded.

>>9205544
>Your college and those books are retarded.
There's literally nothing wrong with Hungerford.

>>9205571
There isn't anything wrong with the older book, this retardation however doesn't even seem to cover modules.

>>9205581
It's an intro book, his other book is for graduate students.

>>9205584
An intro for retards perhaps, those other """graduate""" books are perfectly fine as an intro even for undergraduates.

>>9205467
Axler, but it might be problematic if you are a student and your professor goes heavy on determinants and computational bullcrap.

>>9205587
>those other """graduate""" books are perfectly fine as an intro even for undergraduates.
But the graduate book is specifically working under the assumption that the reader has already done some undergraduate course in algebra.

>>9193069

Pls someone give me a math subject to study, any subject is welcome. Pls give me interesting shit.

>>9205601
Category theory.

>>9205600
I wasn't talking about that specific book. There are plenty of better intro books, Algebra Chapter 0 for example.

>>9205601
cases where P > NP

>>9205601
Abstract harmonic analysis is god tier. Folland has a nice intro to it.

>>9205601
see >>9205606 also check out type theory.

>>9205612
>any kind of analysis
Garbage which should be outlawed.

>>9205609
>Algebra Chapter 0
>doesn't even cover commutative algebra
>doesn't even cover representation theory

>>9205620
>Not seeing the beauty of analysis.
I can't fathom what it's like to have such shit taste. Do you know the absolute euphoria of using category theory in functional analysis?

>>9205623
>commutative algebra
No such thing.

>>9205616
>>9205606

Can someone recommend a book on category theory for a beginner who knows abstract algebra?

>>9205623
It's a complete meme to have a book with introductory sections to all of those subfields of algebra. It's better to get a book for group theory, a book for ring theory, a book for field theory, etc. Trying to fit too much into one book often leads to the material presented not bring very deep.

>>9205647
Use Reihl's text. It's free on her website.

>>9205647
Mac Lane, Awodey, Borceux, Leinster's intro. Try one of those, probably the first or the third, and maybe the fourth for a quick rundown.

>>9205663

thanks

>>9205661

also thanks

>>9205647
Awodey's "Category Theory" is pretty good.

>>9205851

got the pdf, will start reading tonight.

>>9205606

He said math subject not english subject.

If I have a 2x2 matrix A and am multiplying it by an unknown 2x2 matrix B to get a product matrix C, how would I find the values in B, given A and C? I'm sure this must be something simple I'm missing but I can't figure it out. The hint tells me you can turn it into a system of 4 equations with 4 variables but I'm not really sure how. Made me think of AX=B where you multiply the coefficient matrix by the variable matrix to get the constants, but that doesn't really work, does it? If X has to be a column matrix/vector in that case.

>>9206062
Also I tried to multiply C by the inverse matrix of A, but unless I'm a complete retard and just did it wrong, it seems this doesn't work the way it does for normal numerical values.

>>9206062
>>9206067
If [math]AB=C[/math], then [math]B=A^{-1}C[/math]. Are you sure you had the order of multiplication correct? Matrix multiplication is typically noncommutative.

>>9206090
Ah, yeah I had the wrong multiplication order, thanks. I know it's noncommutative, but what is the reason A inverse is on the left in this case? I guess it kind of makes sense intuitively but I'd prefer if there is a better explanation you could point me to. Seems to be the case in certain scenarios with multiplication of transposes too.

It's just this sort of algebra:

[math]A^{-1}C=A^{-1}AB=(A^{-1}A)B=IB=B[/math]
Whereas if we multiply by the other side, nothing really gets rid of that A on the left. (In fact we just change the basis of B)
[math]CA^{-1}=ABA^{-1}[/math]

>>9206118
>>9206125
I suppose I should also add that if A and B do commute, then we'd have [math] A^{-1}C=B[/math] and [math]CA^{-1}=B[/math] as in the latter case
[math]CA^{-1}=(AB)A^{-1}=(BA)A^{-1}=BI=B.[/math]

>>9206125
>>9206137
That makes perfect sense, thanks.

Currently trying to crack this little bugger of a problem

>>9206125
>>9206137
Hm I just realized this method doesn't work if the unknown matrix in the problem is the first one. Or am I missing something? Working backwards from what you did:

[math]A = IA = (B^{-1}B)A[/math]

Doesn't work because you can't define AB as C if the matrix is non-commutative, right? The question has no indication of whether it is, either.

>>9206197
Define BA as C, I meant

Tips/tricks/hints?

>>9206203
>>9206156
I'll tell you tomorrow.

>>9206205
Ohh c'mon, just a hint or 2, something to get me moving

>>9206197
>>9206199
In that case you would multiply on the right
[math]C=BA\implies CA^{-1}=B(AA^{1})=B[/math]
Are you saying that you wouldn't know which form C is in (i.e. you don't know if C=AB or C=BA)? In the question you said you have a matrix A and you multiply it by the matrix B. I would say that typically means that C=AB.

>>9206209
Yeah sorry for being unclear, it is fine when B is unknown. I'm saying I don't think that method works in other version of the problem where the unknown matrix is A. Since the matrices are defined as [math]AB = C[/math].

Then I would think [math]BA \neq C[/math] if they are not commutative, so that method doesn't work. Which is why I guess the hint suggests it can be solved another way.

>>9206203
For part b, [math]1+x^3+x^6+x^9+x^12=(1+x+x^2+x^3+x^4)(1-x+x^3-x^4+x^5-x^7+x^8)[/math]

>>9206226
Cheers champion, that's quite a handy little trick. Any suggestions for a)? I said alpha = [x] but that's probably silly.

>>9206222
Well when A is unknown and you know B, then you would just use B inverse (assuming that exists) in the same way. If B doesn't have an inverse then yeah you'd have to use other methods.

help

>>9206232
Oh for fucks sake I'm an idiot, kept looking at it in terms of [math]IA[/math], forgot that multiplication by identity matrix is always commutative. So you simply do [math]A=AI=A(BB^{-1})=(AB)B^{-1}=CB^{-1}[/math] right?

>>9206229
Not really, I'm a bit rusty on my field theory. This whole exercise looks like the construction of the splitting field of a polynomial from what I remember. [x] sounds familiarly correct (it'll provide the polynomial [math]1+[x]^2+[x]^3+[x]^4[/math], which you'll want to show is the same as [math][1+x+x^2+x^3+x^4]=0_f[/math]).

>>9206249
Yep, that looks good.

>>9205932
Nobody asked for your opinion.

>>9206246
Alright mang, look here. So [math] x^5-1 =(x-1)(x^4+x^3+x^2+x+1) [/math] so the 5th root of unity ([math]e^{2\pi i/5} [/math]) is a root of f(x). what does this root of unity look like in Z3? Plug that shit into f and mod 3 to make sure its zero. i used mathematica to check and it was

Hi guys, I'm studying for my upcoming discrete mathematics exam. I have no idea how to do the following problem.

(a) Use mathematical induction to prove that log2 n ≤ n for all integers n ≥ 1.

>>9206485
>prove the base case
>do the induction step

>>9206246
>finite field
Are you implying there is any other kind?

>>9206485
The answer is obvious I believe, unless I made a mistake, it is just that [math]n=(n-1) \cdot \frac{n}{(n-1)}[/math].

>>9194242
Something something burden of proof

What resources should I refer to if I want to learn about Hilbert's epsilon calculus? Or just first order logic and quantifier substition is all I need?

>>9206434
Cute

>>9207138
That I am, but it doesn't change the fact that math should look like a novel instead of a 10 year old practicing inequalities.

>>9206485
Let [math] n = 1 [/math]. Then [math] log_2 1 = 0 < 1 [/math].

Now suppose that [math] log_2 n < n [/math]. Then [math] log_2 (n+1) \leq log_2 (2n) = log_2 2 + log_2 n = 1 + log_2 n < 1 + n = n + 1 [/math]

Or you could tell your professor to stop being a fucking faggot and let you use the derivative to prove it for real numbers >1.

>>9206956
Technically yeah, just FOL. But if you want context, you need to be familiar with different deductive apparatuses as well as some basic proof theory.

I give up /math/
you bullied me to death
I thought i could become an academic and study math for fun
but your constant insulting and your constant "Maybe you shouldnt be in Math" finally got me and i quit. I am taking real analysis 2 and topology and i cant grasp the concepts. I am afraid of asking for questions because i fear they will tell me the same thing (you shouldnt be on math) even though i used to love math.
but fuck it plus the shitty money prospects looking after graduation and during grad school. I am tired of living like a poor person. I will take my job offer at this mobile app job and never look back at this general or math ever again. I loved it but the people ruined it for me. fuck you all i hope you are proud. and by the way you are all never gonna contribute anything significant ever. you ll die wrapped in a bag and tossed away in the trash and no one will think something is amiss. Fuck u all.

>>9207508
Math was never a social activity. You are retarded for coming here and asking questions when there are 500 books about every possible topic.

I come to this general and /sci/ in general to have mindless fun to relax from my studies. I Wildpost. I ask stupid questions on purpose. When faggots like you come asking questions I troll them to fuck with them. Why? Because /sci/ is not fucking stackexchange. This is supposed to be for mindless stupid fun. You are a retard for thinking that you should base your self-worth on what 4chan thinks of you. You are stupid. You shouldn't be in math. Neck yourself.

>>9207515
>math is not a social activity
>professor refuses to help anyone at class and encourages everyone to make friends to help each other
>most great mathematicians relied on others to help them reach their now famous results
>he thinks if he studies math by himself he will learn anything
not him but you are so wrong

>>9207521
There is a difference between a bunch of experts collaborating and an absolute retard coming to fucking 4chan to be spoonfed undergrad mathematics. This man may as well just drink bleach in hopes that the brain damage will glitch out and give him the solution to his analysis I homework.

>>9207526
why are you so bitter?
did chad take ur oneitis?

>>9207530
I wouldn't say I am bitter. As I said before, I come to 4chan to relax, which means that I take all my shit and let it out here because there are no consequences here. Today was really stressful because my girlfriend wanted to buy a jacket so I had to leave university early to take her there and buy it for her. And then when it was like 4 PM I remembered that today I needed to buy some electrical components for a physics class I'm taking as an elective so I had to rush the fuck there hoping they had what I needed and they did but it was all so rushed, it just feels bad. I am usually not this tight on my schedule but my girlfriend really fucked me up this time. Now I just came home and I'm having fun shitposting.

>>9207536
>he has a dumb gf instead of a smart bf

>>9207536
>he has fun shitposting
how low do you have to go to feel fun in such a brainlet activity

>>9207540
That's the thing, shitposting is not stupid as long as you do it [math] \mathbb{MATHEMATICALLY}[/math]

Then it technically counts as an intellectual endeavor.

>>9207543
Math isn't an intellectual activity. Or would you say drawing is one? You are either counting or making art.

>>9207543
>eating shit is not disgusting as long as you do it I̬̻͓̗̘̟̩͢͠͡ŗ͖͙̪̺̰̝͎̺̯̥o̵̡̦̟̫̩̖̹͎̣͖͡͡n̵̴̬̻͖̮͙i̢̢̠̬̣͓̘̩̖͍̗c̵̖͎̳̠͖̤̱à̵̟̦̱͖̪͍̙̣̕l̦̦̪̪̠̬̖͙͔͓͕͓͍͘l̴͎͎̦̘͙̱̕͝y̨͈͓͉̺͉̰̫̮̘͚͈͟
Delusion of grandeur is strong in this one

>>9207515
>reddit spacing

whats wrong with being spoonfed if i dont have time?

>>9207619
nothing

New to topology so maybe dumb question

can an operator be an element of a topological space?

like when considering the T1,T2,T3 topological spaces for example. and my two distinct points are operators, like a differential or something.

I finished doing school maths just when calculus started (17yo). I'm now in my late 20s. While I'm quick with basic arithmetic I don't know how to approach this problem (for a lotto type game played with friends)

I've got numbers 1 through 20. There are 10 of each number (so 10 1s, 10 2s, 10 20s) Two people play with each other. Each grab either two or three numbers.
I do I determine the probability of a tie? I don't want the smart ones to game a system that should just be a bit of fun between friends. I'm stupid, but at least smart enough to recognise that as numbers are removed people will be able to have a greater idea if it will be a different number or a tie between the two players.

>>9207791
>can an operator be an element of a topological space?
yes

>>9207828

thanks

Starting (homo/iso)morphisms but doing supplementary stuff about characteristic n and zero in rings. This is one of my proofs how bad is it

>>9207508
How sensitive can you be? Are you seriously quitting something because some legitimately autistic people trolled you to death on the most irrelevant board on a website only known for political radicalism and a completely fucked up user base?

What did wolframalpha mean by this

>>9208184
>What did wolframalpha mean by this
Probably that there exists no closed solution in terms of "standard" mathematical functions, but it is still possible to calculate a series approximating the solution.

>>9207467
>math should look like a novel instead of a 10 year old practicing inequalities
That it definitely should.

>>9207508
>I give up /math/
Stopped reading right there. Maybe you shouldn't be in Math.

>>9193155
>honors

>>9208171
Epic post "mate". Those completely fucked up radicals :D
I know about them from >>>/r/eddit/ I suppose I'm much like yourself in that regard.

>>9194308
Nah, it's just that you're a brainlet.

>>9208311
I upboated (I am so random LAWL) your post my fellow 9gagger.

>>9194308
>how smart he is
He's pretty much a retard when it comes to actual mathematics.

Let [math]\varphi : A \to B[/math], [math]\psi : C \to A[/math] be morphisms in an additive category.
How do I show that from [math]\varphi \circ \psi = 0 \implies \psi = 0[/math] it follows that [math]\varphi[/math] is a monomorphism? It seems like I'm missing something really obvious.

>>9208706
phi psi_1 = phi psi_2 => phi (psi_1-psi_2)=0 => psi_1=psi_2 => phi is a monomorphism

Got a 52 on a precalculus midterm due to questions over concepts. Brainet indeed.

>>9208706
The proof goes like >>9208717 posted. For intuition, you should consider that ψ a generalized element in your category. Then your equation would be φ(ψ)=0, and (for abelian groups) this means ψ is in the kernel of φ. If this implies ψ=0, then (in the case of abelian groups) you conclude that φ is injective. Analogously, this implies φ is a monomorphism in an additive category.

>>9207477
please use \implies or \Rightarrow when applicable
it gets a bit hard to read

>>9208789
but there's no implications there, just a chain of inequalities

>>9208794
yeah i realized that like 2 seconds after i pressed submit

If you solved a Millenium Problem would you refuse the money just like Perelman did?

I need to cram everything from Algebra 2 to calculus 1 in a few months. So I can continue my education because I'm a retard.

>>9208800
I guess that's why people say that autism speaks.

>>9207515
>>9208171
>>9208307
>>9208311

You think i meant you are the ones that discouragef me??? no it was people irl
proffesors refuse to help me and tell me to make friends
none of them want to give me help because they are always busy
i never have TA s in my classes
No one outside math can sympathize my struggle
If i ask help to my autistic peers either they are dumber than me or much smarter than me and ignore me.
i never have time to catch up
whats the point of goin to grad school if you arent an egotistical faggot that think he will contribute to math?
People look weird at me when i ask math questions
i cant do this math. also no money and only scholarship money to live like a beggar. if i didnt know things such as having fun, money , and other things outside and math and i was an autist that only finds joy in math. i ll probably make the stupid decision in goin to math for grad school..
good bye /math/ applied and actuarials are not real mathematicians but pure maths is full autism. you cant win here

>>9208846
>You think i meant you are the ones that discouragef me??? no it was people irl
>proffesors refuse to help me and tell me to make friends
>none of them want to give me help because they are always busy
>i never have TA s in my classes
>No one outside math can sympathize my struggle
>If i ask help to my autistic peers either they are dumber than me or much smarter than me and ignore me.
>i never have time to catch up
>whats the point of goin to grad school if you arent an egotistical faggot that think he will contribute to math?
>People look weird at me when i ask math questions
>i cant do this math. also no money and only scholarship money to live like a beggar. if i didnt know things such as having fun, money , and other things outside and math and i was an autist that only finds joy in math. i ll probably make the stupid decision in goin to math for grad school..
>good bye /math/ applied and actuarials are not real mathematicians but pure maths is full autism. you cant win here
cringe

>>9208846
Never post in these threads again, you weak fuck!!!

>>9207508
>but fuck it plus the shitty money prospects looking after graduation and during grad school.
but there's tons of scholarships/grants to apply for in grad school, and well paying jobs after graduation like cryptographers, actuaries, etc.

>>9208846
also i am 25. i tried to ge to college asap after tons of financial set backs but i got here when i was 23 and now i see 19 year olds doin shit like nothin. i wann kys desu. time was against me. it always was

>>9208717
Thanks, I got it.
>>9208734
>For intuition, you should consider that ψ a generalized element in your category.
Thanks for reminding me of this.

>>9208802
>If you solved a Millenium Problem
Not "if", "when".

>>9208877
No problem. I recommend you check out the embedding theorem by Mitchell. It allows you to think of objects in a small abelian category as left modules over a unitary ring, and that makes things a lot more intuitive. Of course, assuming you are to restrict your additive categories to abelian ones.

>>9208904
I've only heard about it but I'm planning to learn about it pretty soon.

Threadly reminder to work with physicists.

Fuck off with the anime girls reaction faces, faggots.

>>9208927
>faggots
Why the homophobia?

>>9208932
>>9208925
>redditry and physishitism from the same retard
Why am I not surprised?

>>9208927
make me

>>9208927
You might want to switch to another website.

>>9208935
Not quite.

>>9208927
Sure, I'll drop the anime girls.

>>9208925
Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat, because theirs' are as good as mincemeat anyway.

>>9208950
>Physishits should be held down on the ground and pommeled with a straight edge over the head until their brain is mincemeat, because theirs' are as good as mincemeat anyway.
cringe

>>9208945
You and your subhuman friend aren't really different "people".

>>9208953
>You and your subhuman friend aren't really different "people".
We're all friends here.

>>9208927
anime website

>mfw fags here think they are smarter than avg people lmao

>>9208957
I don't believe humans and physishits can truly be "friends".

>>9208966
>lmao
see >>9208959 for food.

I'm studying Real Analysis. I'm always studying reals. It's taking up my life but I don't mind it's kinda cool.
It definitely gives me a huge nerdcock

>>9209006
> he thinks real is hard
hope you are a non math major taking this class. Otherwise I got some bad news for you

Would it be correct to say that ML-based prediction algorithms are concerned not with the past and future, but only the present because of the law of large numbers?

>plan on sharing about recent uni classes on variation of parameters for particular solutions of linear differential equations
>subjects i still havent seen pop up
>hate for physicists
>charitable amount of smug anime girls
i like maths and my major is the closest ill get to pure maths, but why the hate tho
pls no bully

>>9209381
What is your post even trying to say?

>>9208927
Grad Set Theory, one of our hw is to prove the transfinite recursion theorem. I saw the proof in my undergrad set theory course and have it in my notes, trying hard not to look.

>>9209587
avatarfagging is against the rules faggot

>>9209726
>reaction images are avatars
R*ddit vermin isn't welcome here. Feel free to fuck off:
>>>/r/eddit/

Linear Programming

>tfw applied math master race

>>9209898
>applied math
No such thing.