/mg/ - math general

[Anonymous No.15003586]

Hilbert's tenth problem edition.
Formerly: >>14991073

Talk math.

[Anonymous No.15003705]

>>15003586
add an anime picture

[Anonymous No.15003715]

>>15003586
Thank you for not having an anime picture.

[Anonymous No.15003772]

>>15003705
>>15003715
After checking the previous threads, it turns out that you have strong feelings about anime pictures. If I had known ahead of time how much this meant to you I would have photoshopped an anime girl on top of the wikipedia screenshot.

[Anonymous No.15003808]

>>15003772
No one who matters actually cares, those are just the images I usually have on hand when making the new thread.

[Anonymous No.15003838]

Can anyone tell me what [math] \mathbb C [ \lambda][/math] means? My knowledge of abstract algebra is limited to axioms of of groups, rings and fields.

[Anonymous No.15003847]

>>15003838
It's just the ring of all polynomials in the variable lambda whose coefficients are complex numbers.

[Anonymous No.15003852]

>>15003847
Right, it's just what's written after it. I was being stupid. Thanks.

[Anonymous No.15003869]

>>15003808
that is an unreasonably long spout

[Anonymous No.15003939]

Doesn't the axiom of pairing imply the axiom of choice?

[Anonymous No.15003954]

Let G=U(20). Let H=<9> and K=<11> be subgroups of G. are G/H and G/K isomorphic?

[Anonymous No.15003960]

Does learning proofs and abstractions aid in developing an intuition for math?

I am an engineering undergrad but I wanna know if studying math from an abstract view (i.e., using math undergrad textbooks instead of just "mathematics of engineers" textbooks) will help me in understanding engineering mathematics.

[Anonymous No.15003962]

>>15003960
Yes

[Anonymous No.15003965]

>>15003960
what is an intuition for math? i've never understood anything intuitively in math

[Anonymous No.15003968]

>>15003954
How do we get undergrads to stop using funny notations literally only their professor uses instead of words?
I could only tell that [math]U(n)[/math] is the multiplicative group modulo n because of the definitions for H and K.

G/H and G/K don't even have the same order.

[Anonymous No.15003977]

>>15003968
says they dont have the same order
and
>>15003962
says they are isomorphic?

how do they not have the same order? i thought they both are just G to be honest

[Anonymous No.15003980]

>>15003586
Found picrel at my university library. Did Wildburger redpill them?

[Anonymous No.15003995]

>>15003977
>how do they not have the same order?
Because I calculated the order wrong in my head.
The correct way to solve this is decomposing [math]G[/math] as a direct product of [math]\mathbb{Z}_n[/math]s and then using that to calculate the modulos.
Have fun.

[Anonymous No.15003999]

>>15003962
Thanks, though I was hoping for an explanation.

>>15003965
I'd say intuition is when you "see" what's happening with the mathematics based on your previous knowledge, but can't rigorously prove it abstractly.

So, for example, there's a time-phase shifting property in Fourier transforms. You can prove it mathematically using the expression, but you can also intuitively "see" that shifting the function in time would result in a rotation in the frequency domain.
Then this kind of intuition is really helpful with quickly getting a 'feel' for themore advanced stuff like Hilbert transforms, where the mathematical proof is not so straightforward.

[Anonymous No.15004001]

>>15003980
ooh I've heard about this, isn't it the one where he proposes we should use squared distances and sin(x)^2 instead of angles?

[Anonymous No.15004003]

>>15003586
The order of a finite field is its number of elements, and is either a prime number or a prime power.

Does this mean there are no finite fields of order with a non-prime amount of elements?

[Anonymous No.15004022]

>>15003960
Maybe, but one thing that learning abstract math is going to help you in is discover new applications of math. It's hard to apply math in novel problems when you don't know the rigour. Fast Fourier transform for example. Even simple stuff like algorithms for solving linear equations is built on some heavy theory of matrices which often get left out in applied programmes.

[Anonymous No.15004023]

Time to learn lean anon

[Anonymous No.15004024]

>>15003939
How would it?
The axiom of pairing basically says if you have A and B, you can make C = {A, B}.

[Anonymous No.15004029]

>>15003995
decompose z20 = z2 + z2 + z5 and then find modulos of what?

[Anonymous No.15004031]

>>15004003
>Does this mean there are no finite fields of order with a non-prime amount of elements?
You just there are fields of prime power order, e.g. 4, 8, 9, etc.

[Anonymous No.15004032]

>>15004029
> z20 = z2 + z2 + z5
Not Z_20 dumbass, U(20).

[Anonymous No.15004035]

>>15004001
Yes, the very same

[Anonymous No.15004043]

>>15004029
Z20 = Z4 + Z5
But
Z4 isn't Z2 + Z2

[Anonymous No.15004051]

>>15003954
You first try to prove they are not isomorphic.
Do H and K have the same number of elements? If no, then G/H and G/K won't as well (in the case G is finite, which is what we have).

We check and we find that |H| = |K| = 2, so the quotients will have order |G|/|H| = 8/2 = 4. Shouldn't be hard to find an isomorphism if you quotient and look at the multiplication tables.

>>15004029
To use what the others are telling you, you will find that U(20) is isomorphic to <3>x<11> (so isomorphic to Z_4 + Z_2). So G/K is isomorphic to <3>, which is isomorphic to Z_4.
What about G/H? What will happen here?

[Anonymous No.15004052]

>>15004032
but i thought u(n) is isomorphic to z(n)

[Anonymous No.15004054]

>>15004052
Try something simpler:
Take n = 3.
What is the definition of U(3)?
How many elements does Z(3) have? How many does U(3) have?

[Anonymous No.15004100]

>>15004022
I see.
But balancing the coursework and self-learning abstract math would be brutal. I'll try and see if I am capable of it.

[Anonymous No.15004163]

>>15004031
Ah yes, sorry I meant only prime or prime powers. So then there is no finite field of order 6, for example? (6 is not prime nor is it a prime power)

[Anonymous No.15004244]

>>15004163
Correct.

The field elements under addition form a finite Abelian group of order n.
That means that adding n copies of any element will give 0, by Lagrange's Theorem.
Let k be the smallest (positive) natural number such that adding k copies of 1 gives 0.
We can think of any natural number N as an element of the field, where N is the sum of N copies of 1, i.e. N = N*1.

Now assume k is composite. Then k = cd, where c and d are greater than 1 and smaller than k.
Since they're smaller than k, and k is the smallest natural such that k*1 = 0, then c = c*1 and d = d*1 are not zero.
But cd = k = 0. That means that the field elements/natural numbers c and d nontrivial zero divisors.

So this is not a field, a contradiction. So k must be prime.

But what about the order of the field n? Sure, k must be prime, but what about n?

If n is not a prime nor a prime power, then it must have two different prime divisors, p and q.
Cauchy's theorem tells us that there must be some field elements x and y such that the additive order of x and y is p and q (respectively).
We know that kx = (k*1)x = 0x = 0. But p is the additive order of x, so p must divide k. But k is also prime, which implies k = p.
The same reasoning gives us k = q.
Now we have p = q. But we said they were different from the start, a contradiction.

[Anonymous No.15004285]

I don't know why but the notion of odd and even functions feels oddly (tongue sticking out) satisfying. Also, exercise 9 was fun, even if the hint was a big hint.

Hope I encounter these boys again in the future.

[Anonymous No.15004314]

>>15004285
Even if you didn't know the hint, just write down what things mean.

>"f is a sum of even and odd"
f(x) = g(x) + h(x)
g(-x) = g(x)
h(-x) = -h(x)

>We know how g and h behave when you replace x with -x. What about f(-x)?
f(-x) = g(-x)+h(-x) = g(x) - h(x)

Two equations in two unknowns:
f(x) = g(x) + h(x)
f(-x) = g(x) - h(x)

Solve.

[Anonymous No.15004478]

What was the most challenging textbook or paper you've successfully finished and understood after finishing undergrad?

[Anonymous No.15004484]

>>15004023
>Pentagon and Microsoft infiltrated forced math hot topics
PS
https://topos.site/blog/2022/11/statement-on-topos-and-the-ftx-foundation/

But okay, Americans gonna bomb, can't do nothing about that. The Pentagon issue aside, I like thos topics for the most part and Lean is probably nice.

[Anonymous No.15004502]

>>15004024
How does the axiom of pairing prevent you from taking arbritary elements of arbitrary sets as A, B?

[Anonymous No.15004514]

>>15004478
I don't think I've ever fully understood a single book or paper.

t. soon to finish MSc in maths

[Anonymous No.15004517]

>>15004502
What do you mean?

[Anonymous No.15004540]

>>15004517
The axiom of pairing:
>For all A, B there is a set C such that for any D, D is a member of C if and only if either D is equal to A or D is equal to B.
What's stopping me from taking arbitrary members of uncountable sets as A and B without using some kind of choice function?

[Anonymous No.15004586]

>>15004540
The axiom of choice says that if you have a collection of sets, then there is a function that sends each set to some element in said set (i.e. f(S_i) is some element in S_i).
You're saying that the axiom of pairing, the ability to make {A, B} from A and B, implies that you can make a choice function.

The statement of the axiom of choice has you extracting elements from sets. But pairing makes you put (two) sets into a new set.
I don't see what you mean, because these are opposite things (extracting/putting).
Can you give a concrete example of what you mean?

[Anonymous No.15004741]

>>15004540
If you're about to apply the axiom of pairing, then you've already "chosen" A and B.
Pairing doesn't help with getting the sets you apply it to.

[Anonymous No.15004792]

>tfw literally no one in a 100 people class passed the abstract algebra exam

[Anonymous No.15004874]

>>15004792
What was the textbook?

[Anonymous No.15004924]

>>15003939
>>15004540
the axiom of choice is usually for infinitely many things, not just two things. you’re right that under the usual rules, if you’re given sets [math] X,Y[/math] and you know both are nonempty, then you can probably just assume you have [math]A\in X[/math] and [math]B \in Y[/math] and then form [math](A,B)[/math] as a kind of choice function. But there are two main problems when you look at the axiom of choice for infinitely many sets, say [math]X_1,X_2,\ldots[/math], namely:
>pairing is no longer enough, because would have to apply it infinitely many times, which is not allowed
>you have to explain how you came up with the elements [math]A_1,A_2,\ldots[/math]. “the sets are nonempty” is already a bit handwavey for just two sets, but for infinitely many it definitely isn’t good enough. It is a bit like constructive mathematics in this respect

[Anonymous No.15004935]

Retard here. I've never been good with numbers and maths, and get tripped up doing basic arithmetic in my head. I want to better my understanding of maths and ability to perform maths. I've been working on relearning math from scratch, starting with khanacademy basics so far.
Any fundamental tips/advice to view math differently or more efficient ways to think/approach maths for somebody starting from the bottom?

[Anonymous No.15005269]

>>15004244
very nice, thank you kind anon

[Anonymous No.15005440]

>>15004935
It's all a chain of true false statements.

[Anonymous No.15005442]

>>15004935
math is more about abstract reasoning skill and not as much of a focus on arithmetic as you may think if uninitiated.

[Anonymous No.15005464]

>>15003838
Wtf is with that hideous horizontal dotted line matrix notation?

[Anonymous No.15005472]

Can anyone offer me a resource for a good, concrete introduction to Lie Algebras? Every text I've tried goes 0-100 way too quick, and my professor for the course is completely useless.

[Anonymous No.15005482]

>>15003954
There is a more elementary way to solve this. Observe that every element x of G/H satisfies x^2 = 1 (1 is the identity in G/H), but the same is not true for every element in G/K. So, the two are not isomorphic.

[Anonymous No.15005503]

>>15005472
Lee.

[Anonymous No.15005554]

x

[Anonymous No.15005557]

trying to motivate myself to rework some vector calculus. cant seem to put the work in

[Anonymous No.15005643]

>>15004935
doing arithmetics in head is proportional to your working memory (which you can test using the digit span test). Doing actual math is more concerned with your reasoning and logical abilities. Both are different.

[Anonymous No.15005748]

Is there some kind of "wiki" that has more rigorous definitions for a lot of physics concepts, kind of like planetmath but physics instead of math?

[Anonymous No.15005758]

Is there any reason ZFC is preferred over NBG?

[Anonymous No.15005762]

Is there some kind of "wiki" that has more rigorous definitions for a lot of physics concepts, kind of like planetmath but physics instead of math?

[Anonymous No.15005789]

>>15005758
What do you mean preferred? NBG is an extension of ZFC, when you use classes, you use NBG implicitly, if not you stay in the realm of ZFC.

[Anonymous No.15005876]

>complete chapter 1 of bogachev's measure theory book
>113 problems
lol. Any other books that do that?

[Anonymous No.15005930]

What's the best way to progress through a chapter that you find difficult to understand? Should I power through the whole chapter first even if there are hazy spots so I can get the "big picture", or should I slow down and keep looking at every page until everything is clear?

[Anonymous No.15005944]

>>15005876
This is why books should break it up in between sections.

[Anonymous No.15005951]

I'm not going to move on until I've figured this word problem out on my own.

[Anonymous No.15005953]

>>15005643
Yeah my education actually involves neurosci and I'm painfully aware of how my poor memory and almost lack of STM interferes within my daily life, especially keeping track of numbers in my head.
Thanks for the replies, anons. I'm still working through baby stuff but I also want to 100% everything before moving on to the next thing.

[Anonymous No.15005955]

>>15005951
I think that's a waste of time. Why not just read an explanation of how to solve such problems in the general case, and then you would know how?

[Anonymous No.15005956]

>>15005955
Because I am prideful.

[Anonymous No.15006080]

>>15005944
I don't mind it too much since there were tons of examples inbetween so it's not like I have no clue how to approach these exercises. Still the largest number I've ever seen at the end of a chapter, let alone first chapter.

[Anonymous No.15006084]

>>15005951
You can try to derive everything on your own if you want to. If you, on your deathbed, discover the Pythagorean Theorem while everyone else is already past the Riemann Hypothesis, then that's entirely your choice.

[Anonymous No.15006090]

>Convex uniform 5-polytopes gem even though trannies love them
coalson spotted
TND Aryan white gem isogonal isotopic polytope victory
also Disdyakis rhombic prismatic honeycomb coal post touhou in cover of thread chud

[Anonymous No.15006094]

>>15006090
>Got runcic snub 244 honeycomp.png from the 'wiki even thoughie its kinda gemmy
its over 6D 2,0 superconformal field theory hitlerian fregearyans

[Anonymous No.15006125]

>>15005930
>Should I power through the whole chapter first even if there are hazy spots so I can get the "big picture"
Yeah, if you get the big picture vaguely, it might be easier to understand the details after that.
>or should I slow down and keep looking at every page until everything is clear?
Yeah, after you know what they're motivated by.

[Anonymous No.15006184]

>>15004874
Fraleigh

[Anonymous No.15006224]

>>15003715
this

I was skipping the old thread without even considering the context because of the image, this one got me though

[Anonymous No.15006343]

>>15004514
by that constraint I wanted to avoid stuff like "this 300-page state-of-the-art algebraic geometry paper that made no sense to me". By successfully finishing I mean mostly that you got the main line of reasoning.

[Anonymous No.15006350]

>>15006224
I have been absolutely filtered by a Game Theory problem I thought would be easy enough for my dumb ass to model.
Do any of you think you could tackle a game theory problem? I may post a thread if not.

[Anonymous No.15006355]

>15006224
Did not mean to reply to this post here >>15006350
oops

[Anonymous No.15006437]

>>15004023
coq proof of false

[Anonymous No.15006544]

does a number system based on prime numbers exist?
not sure if it's clear what i'm trying to say

[Anonymous No.15006557]

>>15006544
Yeah, binary

[Anonymous No.15006633]

>>15004478
Bump, I'm interested.
Or are there only undergrads itt?

[Anonymous No.15006665]

>>15004478
this one tbqhimo

[Anonymous No.15006682]

>>15006665
Yeah I can imagine that being tough to work through.

[Anonymous No.15006686]

>>15006670
This is my thread. I wish I took maths in Highschool anons. Even Game Theory filters me. To be able to solve problems such as the one in this thread what would you reccomend I start learning?

[Anonymous No.15007292]

hello. what is the best elementary algebra textbook

[Anonymous No.15007604]

>>15007292
update: im reading the openstax one

[Anonymous No.15007911]

John Burgess' Set Theory a good refresher/introduction to the subject? In particular chapter 6 and beyond

[Anonymous No.15007931]

basic number theory is kind of recking me
should I just learn humanities or engineering instead?
it's going to get a lot harder, isn't it?

[Anonymous No.15007938]

>>15007931
I wouldn't be too concerned, Number Theory filters many people. You either grasp the subject or you don't. And yes, it will get much, much harder.

[Anonymous No.15007964]

>>15007938
Doesn't that go with any other learnable thing that exists?

[Anonymous No.15007966]

>>15007938
>>15007964
to be clear, my main issue is applying the theory in practice
I don't know why but I have a much harder time applying the theory I learned
any task I haven't spent a lot of time practicing seems impossible to solve on my own

[Anonymous No.15007967]

After every study session should this be happening in order to get better? "Purposeful practice should take you out of your comfort zone. It should feel painful to do. As I’ve mentioned earlier, playing a piece of music you already know how to play — for instance, during a musical performance — does not count, and writing a program that utilizes techniques you already know to use also does not count for the purpose of mastery. Practice that makes you better should take maximal effort, and thus feel terribly unpleasant to do."

I know that practice is different than studying but does this still apply to studying math?

[Anonymous No.15007976]

>An étale morphism is a scheme morphism which is formally étale and locally of finite presentation. Equivalently a morphism is étale if and only if it is:
>Flat and unramified;
>Smooth and unramified;
>Smooth and of relative dimension 0;
>Smooth and locally a quasi-finite morphism.
What the fuck? Why is this basically the same thing as being a local homeo/diffeomorphism? Is there a good way I should I be thinking about the étale topology? Also, any good sources for this that explain it to retards?

[Anonymous No.15007993]

>>15007976
>>Smooth and of relative dimension 0;
You know how to visualize a local homeomorphism, right? [math]\pi: \mathbb{R} \to S^1[/math]?

[Anonymous No.15008019]

>>15007993
Yeah, sure, but it seems difficult to understand what exactly is going on in precise terms.

[Anonymous No.15008134]

Bros?

[Anonymous No.15008145]

>>15008134
Have you tried calculating the integral?

[Anonymous No.15008183]

>>15007967
This phrasing is stupid. Effort isn't "painful" unless you're a lazy NEET slug
Yeah, you need to be challenging yourself in order to learn anything, but if you actually enjoy what you're doing, then challenging yourself is fun

[Anonymous No.15008193]

How I can make a mathematical model of an idle game?
If I try to model it as an infinite one-player game in the game theoretic sense, will that work as a way for exploring strategies?

[Anonymous No.15008203]

>>15007931
Assuming you already know propositional logic, try Calvin Long's book, it's a much gentler book. If you're still struggling, study something else. I also used to struggle a lot with it, but after I came back to it from Analysis, it became a lot easier.

[Anonymous No.15008206]

>>15006437
It has the name "coq". What can you expect.

[Anonymous No.15008221]

>>15006557
i meant a system in which the only whole numbers are prime numbers

[Anonymous No.15008238]

>>15005464
The person was too lazy to look up how to tex non-horizontal dots
>>15005557
> vector calculus
not mathematics

[Anonymous No.15008255]

>>15008183
I agree that the word "painful" might not be the most appropriate, but if you're learning something that for some reason feels difficult and/or frustrating then depending on your personality you might want to stop studying with every new line on every page. It's not always pleasant when you're living it, and I
I think your tolerance for this kind of situations probably correlates with whether or not you end up as one who "has always sucked at math".

[Anonymous No.15008279]

>>15006544
Set of prime numbers, with operation a*b = multiples of all primes less than and including max(a,b), plus 1. Clearly closed under this operation, commutative, identity is 2, no inverses. Pretty useless it seems.

[Anonymous No.15008298]

>>15008279
I don't get it.
What would 3*7 be?

[Anonymous No.15008327]

>>15008193
Model each round of the endless game as a short length of time, the minimum unit of time between each action the player can do, where actions are things like thinking, clicking the clicker, buying an upgrade etc

[Anonymous No.15008337]

wow 4chan has a thread full of failed mathematicians.
4chan makes me feel better about myself
so many loosers in just one forum.

[Anonymous No.15008386]

>>15008298
2*7=3*7=5*7=7*7=7*5=7*3=7*2

All of them, according to the post, result in {2, 3, 4, 5, 7} +1, whatever that is.

[Anonymous No.15008463]

Why don't undergraduate Linear Algebra books talk about cross product?

[Anonymous No.15008499]

>>15008463
Because it has more to do with geometry or physics than linear algebra.

[Anonymous No.15008502]

>>15008463
But they do. Most of them have a chapter about multilinear algebra in which the cross product is discussed as a special example of an alternating multilinearform.
The cross product itself is just not very important since it only exist in [math]\mathbb{R}^3[/math].

[Anonymous No.15008509]

>>15008386
>>15008279
It doesn't make any sense.
How is this closed? You get a set at the end. And how is 2 an identity?

[Anonymous No.15008520]

>>15008499
Then what if I want to learn Physics or Geometry?
>>15008502
Undergraduate.

[Anonymous No.15008528]

>>15008463
discussion on a generalized version of the cross product are usually withheld until later classes that use these books.
https://en.wikipedia.org/wiki/Exterior_algebra
you can read about it here a bit.

>>15008502
also this

[Anonymous No.15008679]

>>15008463
i just checked, my intro to linear algebra book (not english) mentioned it halfway through. with that said, it's not a crazy complicated subject, and a mechanics class might just explain it if the prof realizes the students don't know the fucking cross product

[Anonymous No.15008720]

>>15008509
Binary operations that end in sets can be closed, consider addition of Von Neumann ordinals

[Anonymous No.15009024]

>>15008720
Consider giving simple example

[Anonymous No.15009222]

I want to stop being a retard and get good at math. Do you guys have a "reading guide" like /lit/ does?

[Anonymous No.15009411]

>>15009222
We don't teach reading here

[Anonymous No.15009524]

>>15008238
>> vector calculus
>not mathematics
i dont really know what to call it. i could describe it kinda. using a sort of derivatives of fourier transforms to find one thats smooth. if that makes any sense. i call it vector calculus because i have to use arrays of complex numbers and the array math gets messy

it is math. maybe not "your" kind of math whatever that might be

[Anonymous No.15009600]

>>15008528
>that use these books.
Use what books?

[Anonymous No.15009606]

>>15008238
>vector calculus
>not mathematics
congrats you got me to rage for about 0.01 seconds, have a (you)

[Anonymous No.15009942]

>>15008183
I got it from an article, too lazy to send link.

[Anonymous No.15009962]

Simple example/s of using a "mathematical tool" to solve a problem, including an abstract one, that would otherwise be too hard or unclear how to solve?

[Anonymous No.15010001]

how should and ESL switch to studying math in english?
I will have to bite the bullet eventually, so I'd like to make the change asap.
is there anything I can do besides just studying with english books?

people who made the switch, did you revisit topics that you had already studied or just started learning new topics in english immediately?

[Anonymous No.15010060]

the math subreddit is a literal joke and this thread is even worse than that.
4chan is such a dumpster fire.

[Anonymous No.15010093]

>>15009962
Mechanics.

[Anonymous No.15010108]

>>15010093
Yes, I meant more a problem that would be considered to be in mathematics. An abstract, mathematical problem.

[Anonymous No.15010115]

>>15010060
Try mathchan

[Anonymous No.15010124]

>>15006544
How do you imagine you use your number system? Perhaps the set of finite fields, whose orders can onky be powers of a prime, satisfies your needs for certain (re)definitions of the usual arithmetic.

[Anonymous No.15010125]

>>15010115
>5. You may not solicit anything from Mathchan users
>>a. Do not ask for homework help, whether to provide review, hint part of a solution or full solution with any of your assignments

[Anonymous No.15010126]

I've made some progress on approximating sums with integrals (related to Euler Maclaurin summation).
What I wanted to find is an nxn matrix, M, such that:
[math]x^{m} = {1 \over n} tr(\int\limits_{xI_n - M}^{xI_n + M}y^{m}dy) [/math]
for m=0,1,...,2n-1.
I wanted to solve for the eigenvalues but doing it directly was a pain.
It took a little effort but I managed to get a way to find the characteristic polynomial of M, p(x) = det(xI-M).
Let q(x) = (-x/n)^n * p(-n/x). Let even(x) and odd(x) be the even and odd parts of q.
It turns out that odd(x)/even(x) = tanh(x/2) + O(x^(2n+1)).
You basically get q from the pade expansion of tanh.
You could rearrange this to get:
odd(x)*cosh(x/2) - even(x)*sinh(x/2) = O(x^(2n+1))
then differentiate twice to get a recursive formula for q in terms of the q of order n-1.
It turns out there is an even easier way to compute q using the continued fraction formula for tan(x).
https://en.wikipedia.org/wiki/Trigonometric_functions#Continued_fraction_expansion
For n = 1, q(x) = 1+x/2 which gives p(x) = x - 1/2
For n = 2, q(x) = 1 + x/2 + (x^2)/12 which gives p(x) = x^2 - x + 1/3
For n = 3, q(x) = 1 + x/2 + (x^2)/10 + (x^3)/120 which gives p(x) = x^3 - (x^2)(3/2) + x(9/10) - 9/40

This is some black magic and I never expected to see a Pade show up.

[Anonymous No.15010127]

>>15010125
What's it matter, when there's only like a post per week?

[Anonymous No.15010131]

>>15008509
Brother have you ever constructed the natural numbers
Every number is a set

[Anonymous No.15010133]

>>15010127
In an (absurd) ideal world, the majority of /sci/'s (non-funny) qualityposting would be anons collaborating and throwing shit at the wall while trying to solve open problems in mathematics and science.
In actuality, the majority of /sci/'s qualityposting is people spoonfeeding anons and answering homework questions (inb4 this question isn't ACTUAL homework because I'm self-studying this book by myself with no teacher and need help).

[Anonymous No.15010182]

This book is so fucking bad holy shit. Why does this get shilled so hard. Written for brainlets.

[Anonymous No.15010198]

I keep hearing a lot of praise for Dover books on Mathematics on /mg/. Is there a list of the good ones? Which ones to buy?

[Anonymous No.15010211]

>>15010182
check out Kallenberg. It's my go-to probability book

[Anonymous No.15010236]

>>15010133
Do you have any thoughts on the notion of statistical evidence being used in doing mathematics?

[Anonymous No.15010272]

>>15010131
Then say "union" instead of "plus" and make it clear you're using the set-theoretic construction of the naturals.

[Anonymous No.15010273]

>>15010211
I don't care about measure theory (at the moment)

[Anonymous No.15010275]

>>15010131
>>15010272
It's also literally just the max operation, so why didn't you just say that?

[Anonymous No.15010286]

Sup guys, I have a question regarding time-series analysis.
I have a time series and I performed a continuous wavelet transform to obtain the time-frequency spectra. Each one of the spectral components will be a time series in itself that is correlated with the adjacent spectral components, right?
Now, my question is: can I perform some tests on the spectral components (that are usually performed to the non-decomposed time series) such as stationarity tests or am I violating some assumption?

[Anonymous No.15010903]

Calc 1 CS retard coming through,
I'm working on L'Hopitals rule and other log derivatives right now, can someone frame this in a real world context for me? I can work with the laws and formulas, but I find that the concepts of logs and ln aren't very intuitive to me. I understand acceleration and velocity are derivatives of position, and the concept of derivatives/limits, but beyond that I'm struggling to make sense of the problems and what I'm working with. I just dont really understand why and when to take the log or to apply different rules for the problem at hand.

>pic rel
like what is the best way to look at this and go about approaching it? I had a retarded pre-calculus teacher so my log/ln knowledge is a little lacking, some clarity or advice on how to think about something like this would be appreciated bros.

[Anonymous No.15010931]

>>15010903
For that problem I wouldn't use l'hospital. Just apply the log rule

[eqn] \log(a) - \log(b) = \log \left( \frac{a}{b} \right)[/eqn] and remove the common factor from both polynomials [math](x-1)[/math].

[Anonymous No.15010970]

>>15010133
lambdaplusjs doesn't have any ban on homework help

[Anonymous No.15011003]

>>15010060
Math Redditardation stories? I'll start

>middle schooler asks about rationalizing a denominator
>top post is a redditard going into long-winded essay about muh Galois conjugates
>looked like he was just trying to flex on a middle schooler
>doesn't even help the kid with his question

>discussion in mathhelp about a calc 1 limit problem
>replies say to asker "you have to put 'lim' in every step of a limit problem"
>respond with saying it's okay to do algebraic manipulations on an expression before taking the limit
>get dislikes

>write something like x = -(-x) as part of a response
>some guy replies with a rage post
>says some weird retarded shit about "but then you can do -(-(-(...))) infinitely which is not allowed because of prime factorization"
>discussion was related to reals to begin with
>the guy actually had a PhD in CS

[Anonymous No.15011054]

>>15010286
Sounds plausible. What's the statement of the conjecture?

[Anonymous No.15011367]

I've been stuck on this question for days
I need some help. I don't know if it's the wording that fucks with me
[math] I_a [/math] is the largest interval containing [math] a [/math] such that the restriction of [math] f(x) = cos(x) [/math] to [math] I_a [/math] is injective

[Anonymous No.15011387]

>>15011367
(Using x instead of a to avoid confusion with grammar)
Assuming you mean open interval, by looking at the graph of cos, I_x is just the interval (B, C), where B is the largest multiple of pi that is less x, and C is the smallest multiple of pi that is greater than x.
(If x is a multiple of pi, then no such interval exists)

Take a look at the picture and tell me what N(1) is. (actually give your answer so we can see if you understood)
Then compute the rest.

[Anonymous No.15011399]

>>15011387
It's actually a closed interval, I forgot to mention that
And I still don't really get it. N(a) is defined as the "numbers of integer points contained in I_a", I just don't understand what that fundamentally even means
If you can give me an explanation for a random N(a), then maybe I can understand it more intuitively

[Anonymous No.15011406]

>>15011399
Okay, so look at the graph in >>15011387 and look at 8.
The red dots represent multiples of pi.
8 is between 2pi and 3pi.
You can see that between this two numbers, cos is injective, because it's strictly decreasing (same goes if we used a number different than 8 and cos was strictly increasing)
You can't get a larger interval containing 8 where cos is injective, because then you will go over the max/min, and the values will be repeated in the image.
So I_8 = [2pi, 3pi].

Okay we have the interval [2pi, 3pi].
The question just asks "how many integer points are there in this interval?"
You can see you have 7, 8, and 9 are the only integers in this interval. These are 3 integers, so N(8) = 3.

[Anonymous No.15011405]

>>15011399
Do you know what [math]I_1[/math] is?

[Anonymous No.15011413]

>>15011406
holy shit thank you
>>15011405
[I_a] would be [0, pi]

[Anonymous No.15011417]

>>15011413
[math]I_1[/math]**
typo

[Anonymous No.15011420]

>>15011413
By the way, do you know now what it was about the question that confused you originally?

[Anonymous No.15011422]

>>15011413
Note that I_x may not be defined if x is a max/min.
For example, what is I_0?
Both [-pi, 0] and [0, pi] work as a largest interval.

[Anonymous No.15011426]

>>15011420
The wording, it just didn't click in my head that it was asking about how many integers there was in between

[Anonymous No.15011431]

>>15011387
Wait, also doesn't this mean that every N(a) = 3? Except for the mins and maxes

[Anonymous No.15011446]

>>15011431
No. Because on rare occasions you get 4 numbers. Those are the only two possibilities: 3 and 4.

For example: there are 4 integers between 7pi and 8pi.
This is because 7pi is very close to and slightly less than 22 (pi ~ 22/7), so you have {22, 23, 24, 25} in the interval.

[Anonymous No.15011485]

>>15011446
That actually goes directly into the next question.
The max is 4, and the min is 3.
But how do I actually prove this? I assume I just need [math] n - 1 [/math] comparisons since it's the max, right?

[Anonymous No.15011505]

>>15011485
Assume that the max is bigger than 4.
Then there are at least 5 consecutive integer points.
Take these 5 consecutive points and look at the distance between the biggest one and the smallest one. How far apart are they?

[Anonymous No.15011561]

>>15011505
>Then there are at least 5 consecutive integer points.
What do you mean here?

[Anonymous No.15011585]

>>15011561
Let's say the max is bigger than 4, so at least 5.
Let's say the max is 20.
That means there is some x such that I_x contains 20 integer points.
Let's say something like 41, 42, 43, ..., 60 are in I_x.
That also means it contains 5 points. Like 53, 54, 55, 56, 57.
They are 5 consecutive integers.
How far apart are 53 and 57?
What does this tell us when we also consider the length of I_x?

[Anonymous No.15011625]

>>15003586
How often is new maths invented?

[Anonymous No.15011820]

When you understand every step in a proof (or at least think so), but it still doesn't give that satisfactory "so obvious!" feeling that many of them do after you finish them what should you do? Revisit it after one month? Read it once every day pondering it? Let it go and only work at it again when you next encounter it mentioned somewhere?

[Anonymous No.15011931]

>>15011820
Well, some proofs are just shit.
What you could try is trying to reproduce the proof from memory, that usually makes it easier to see which steps are important, and which are just routine calculation. The hope is that you can memorize just one step, and then reconstruct the rest of the proof from there.

[Anonymous No.15011956]

>>15011820
definitely don't just let it go (when I did that, I would eventually just forget the proof or proof techniques used, sometimes even part of the theorem, and ultimately have wasted my time. Imo, either study it in full or don't). Studying proofs, from my experience, also is *the* way to get better at math.
I'd suggest trying to break down the proof into as many sub-proofs as possible. That way, you can work out which parts exactly are unclear. Oftentimes, small but very important steps are skipped, and once you realize what those are, it suddenly all makes sense (at least this has been the case for me).

[Anonymous No.15012001]

What is a Weil Algebra?

[Anonymous No.15012143]

>>15003586
>try to prove that [math]I \cdot J = \{i\cdot j | i \in I, j \in J\}[/math] is an ideal if I,J are ideals
>seems easy
>fail almost an hour
>read the problem again
>[math]I \cdot J[/math] is actually the ideal generated by [math] \{i\cdot j | i \in I, j \in J\}[/math]
how do you guys deal with suicidal thoughts?

[Anonymous No.15012206]

>>15010903
Simplify the log to what he says >>15010931 however there is no common factor anon is wrong. But you can now use L'Hopitals rule.

At this stage in math it can be difficult to find a real world equivalent. A log basically squishes an exponential growth function into a linear space. It is the inverse of exponential growth consider the relation of the functions graphed in this picture notice the symmetry.

[Anonymous No.15012220]

>>15012206
[eqn]x^7 - 1 = (x-1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) \\
x^5 - 1 = (x-1)(x^4 + x^3 + x^2 + x + 1)
[/eqn]
How is there not a common factor?

[Anonymous No.15012227]

>>15012220
Okay, you right. But that's kinda cursed desu.

[Anonymous No.15012248]

>>15012227
(x-1) is always a factor of (x^n - 1) because 1 is a root.
The other factor will be 1+x+x^2+x^3+...+x^(n-1)

[Anonymous No.15012649]

So what's a *typical* PhD like in the US?

Do you take courses? Or is it only research?
How many years is it?
Do you have to teach?
What's the financial situation like? Do they pay you or what? Do they house you?
Are there "entrance exams" or something like that?

I wanna get a PhD, but I basically know nothing.

[Anonymous No.15012654]

>>15012227
it's pretty common, brother

[Anonymous No.15012695]

>>15012649
>" I want to be a mathematician "
Win a (hard) STEAM olympiad as teenager first, or else, any tesis you will be capable doing will be just an ad hoc argument for a very particular problem without any potential for generalizations and hence, will not provide any fundamental breakthroughs in the conceptual understanding of highly educated people (including yourself, a phD). But whos talking is a channer and with no formal education on the field, i am just a curious person which knows that be capable of doing routine textbooks exercises is not enought to pursue STEAM career. If youre not talented from birth its pitifull trying anything seriously. So ok, try a phD, but dont overreact when do you realize that your tesis is fated to oblivium.

[Anonymous No.15012712]

Since there is no physics general right now and it is pointless to create one, I'm going to ask my physics question here and hope that some of you know the basics of electrodynamics:
In multipole expansion, how can I visualize a quadrupole or higher order terms? I have my quadrupole tensor, but what would the 4-point charge distributions corresponding to that tensor actually look like?

[Anonymous No.15012743]

Can someone explain me why wikipedia got so fucking shit? I was just looking for the formula for the geometric series on
https://en.wikipedia.org/wiki/Geometric_series
and no where on this side is a simple
[eqn]
\sum_{k=0}^{n} p^k = \frac{1-p^{k+1}}{1-p}.
[/eqn]
All I see are some completely useless graphics and random concrete values of p.
This is just an example. There are many mathematical articles on wikipedia with this completely irrelevant information.

[Anonymous No.15012814]

>>15012743
Based blind retard

[Anonymous No.15012825]

What's the most important thing I should know/review before going into (ordinary) Differential Equations?

[Anonymous No.15012829]

>>15012814
This is not in the form i wrote it. I' m not screening for an expression written like that if Im in a hurry.
Why not write it with Sigma? Why isnt it at the top of the article?

[Anonymous No.15012839]

>>15012829
>>15012814
Also wikipedia is supposted to be an encyclopaedia. Students shouldnt be allowed to fill up entire articles with useless information just because their teacher decided it would be a nice idea for a project.

[Anonymous No.15012859]

>>15012829
>Why isnt it at the top of the article?
Because you opened the page for the geometric SERIES, which has infinitely-many terms, when you want the sum of finitely-many terms, retard.

[Anonymous No.15012879]

>>15012859
you sound sharp

[Anonymous No.15012895]

>>15012859
The geometric SERIES is not at the top of the article either you autist. The top of the page is only filled with useless information. Some easily recognizable equation of the geometric series is nowhere to be found either. My point is that even though the information is somewhere it is obscured by useless information.

[Anonymous No.15012906]

>>15012895
It's literally at the top of the first section.
Stop being a retard.

Stop opening the wrong pages, and stop trying to read "in a hurry".
That's retard behavior.

[Anonymous No.15012954]

>>15012906
How does this even relate to my point? Also, try broadening your vocabulary.

[Anonymous No.15012965]

I’ve been having trouble finding the definition of cotorsion of a sheaf and could only find one random website defining it. Can you tell me if this definition is standard.
Let F be a sheaf, then there is a natural map phi from F to the double dual of F.
If phi is surjective, we say F is cotorsion free. Otherwise, we say F has cotorsion.

[Anonymous No.15013002]

>>15012829
>Why not write it with Sigma?
But the form with the sigma is right there in anon's pic you replied to!?

[Anonymous No.15013032]

>>15013002
1) It's not written as an equation its interrupted by text for whatever reason.
2) It's not at the head of the article its written in a subsection that is called "Coefficient a". I also have no explanation why a subsection named like this is needed for in an article about the gemetric series.

I don't exactly get what you guys are arguing for. Do you honestly believe this article, or in general math articles, are well written? That these articles convey information as fast as an encyclopedia is supposed to?

[Anonymous No.15013034]

>>15012712
Gradient? vector field?

[Anonymous No.15013045]

>>15012954
Stop being a speed-reading retard.
I will keep talking down to you in simple language until you understand you are a retard and that the problem is you, not others.

[Anonymous No.15013049]

>>15013045
You're such an insufferable dense cunt.

[Anonymous No.15013054]

>>15013049
I was thinking the same thing about you.
No one ever has a problem with math Wikipedia.
Only you.
You're the problem, retard.

Why are you in a hurry instead of scrolling and reading through normally? Behind on your deadlines because you're a retard?

[Anonymous No.15013083]

stop fighting

[Anonymous No.15013127]

>>15012743
This is because the particular formula you were looking for was actually the sum of a finite geometric series, and that is just one detail of a large concept.

[Anonymous No.15013174]

>>15013127
The geometric series is not a large concept and its wikipedia article should reflect that. A section "Coefficient a" is not needed. Neither is a section "Historic insights" with subsections about some Greeks. Deriving it myself took me less time than actually reading through all that mess.

What is needed, is a simple equation of the most common uses.

This is how a mathematical wikipedia entry should look like
https://en.wikipedia.org/wiki/Minakshisundaram%E2%80%93Pleijel_zeta_function
https://en.wikipedia.org/wiki/Conformal_geometry

I sincerely doubt that this geometric series article is of use to anyone.

[Anonymous No.15013179]

>>15013032
>Do you honestly believe this article, or in general math articles, are well written?
Not really

>It's not written as an equation its interrupted by text for whatever reason.
It's a piecewise case distinction.
You'll learn about it in your first semester.

[Anonymous No.15013190]

>>15013179
>It's a piecewise case distinction.
>>15012906
?

[Anonymous No.15013199]

>>15013190
If you click though the chain of links upwards, you'll end up at >>15012814

Remember, the image of which you (mistakenly) asserted that it doesn't involve a sigma.

I think the conversation should have been over with the post of the first anon who pointed out that, if you're looking for a formula for the finite sum, you shouldn't expect it at the top of the geometric series (i.e. infinite sum) Wikipedia page.
The second anon screencapped the section in the geometric series article where, despite the article not being about the sum you're interested in, it actually is also presented there. He drew a red arrow to where it is in his screencap.

[Anonymous No.15013207]

>>15013199
1) I agree the conversation should be over but
2) >>15012906 involves no case distinction. I have no idea what you are talking about.
3) The formula of the geometric sum does not involve a Sigma
4) I know that the information is in there. My point all along was that is obscured by useless information. Even the beforementioned formula is obscured by a completely useless statement between the sum and its closed form.

[Anonymous No.15013214]

>>15010182
Why do you think it's bad?
I think it's good

[Anonymous No.15013219]

>>15013207
2) There is a distinction in the FINITE sum case. How are you supposed to divide by 1-r when r = 1?
There is no distinction in the INFINITE sum case. Stop confusing two different things.
3) Sigma is just a notation. You can write the sum with a + or you can use a sigma, it's still the same thing. Why are you hung up on this?
4) The information that you call "useless" is the stuff you keep skipping on. That's why you don't understand the difference between "finite sum" and "infinite sum". Stop being a retard.

[Anonymous No.15013242]

>>15013219
Jesus Christ, learn to read you fucking spastic. You are replying to some warped version of this "conversation".

[Anonymous No.15013247]

>>15013242
The only one that needs to read is the retard speed-reading in a "hurry".

[Anonymous No.15013271]

>>15013247
You type like someone who learned English by reading fansubbed manga. Don't excuse yourself by being an ESL.

[Anonymous No.15013663]

>>15013214
You must be a midwit then. It shies a lot away from difficult topics. It give several examples never letting you ever think for yourself. It's bloated with shitty repetitive exercises. Out of the 700 exercises it has, maybe like 200 are worth doing. Read Feller for an actual non-measure probability written for thinking individuals.

[Anonymous No.15013694]

Guys I went to the Wikipedia page of Integral Domains and they didn't give the definition of a Topology at the top of the page?!?!?!?!
Why do Wikipedia pages suck so much?!?!

[Anonymous No.15013836]

>>15013207
I explicitly pointed you to the post with the screen cap with the right pic (namely, again, this: >>15012814, I linked to it in the post you just are replying to too) from which the current reply chain originates. It contains both a sigma and a case distinction.

[Anonymous No.15013844]

>>15013836
Since this already failed twice, let me maybe be more explicit. The post with the sigma, the red arrow and the case distinction is >>15012814

So what the chains of posts here is NOT referring to, but you come back to is the following post: >>15012906
(=NOT)

In other words, we have the following list:
* >>15012814
* >>15012906

Consider the first of the two, not the second.
I.e. the post ending with a "4"

Hint: It has a red arrow in it. And it ends with a 4.
Explicitly speaking, it's this one:
>>15012814

[Anonymous No.15013852]

>>15013836
>>15013844
Dude, he's just a retard. Make fun of him and laugh.

[Anonymous No.15013916]

>>15013844
>>15012814
>>15012743
Compare those two again formulas, ask yourself which one you would like to see in your lecture notes and than ask yourself if you actually understood what I talking about.
I know you will nitpick again so prelude the equation with a p neq 1

[Anonymous No.15013931]

>>15013844
>>15013836
See what I mean? >>15013916 >>15013852

[Anonymous No.15013933]

>go to wrong page
>bitch about not finding the formula

[Anonymous No.15013973]

>A ring is an abelian group with some additional structure.
>Need the notion of a ring to define a module.
>An abelian group is just a [math] \mathbb{Z} [/math]-module.
Thank you for coming to my ted talk.

[Anonymous No.15014012]

>>15013174
>The geometric series is not a large concept
It's actually a cool topic with many directions

https://en.wikipedia.org/wiki/Neumann_series

In functional analysis.

For just numbers, some angles on the series and the sum which come in handy are

[math]\dfrac{z^s}{(1-z)^{s+1}} = \sum_{k=s}^\infty {k\choose s} z^{k}[/math]

[math]\dfrac{1}{(1-z)^s} = \sum_{k,m \geq 0} \left[\begin{matrix} k \\ m \end{matrix} \right] \dfrac{1}{k!} s^m z^k[/math]

[math] \dfrac{1}{1-(z+w)} = \sum_{k,m \geq 0} \binom{k+m}{k}\, w^m\, z^i [/math]

[math]\sum_{k=0}^\infty f(z^k)\,z^k=\dfrac{1}{1-z}\cdot\int_0^1 f(s)\,{\mathrm d}_zs[/math]
(see q-integral)

[math]\prod_{k=0}^{n-1} \frac{1}{(1-q^kt)}=\sum_{k=0}^\infty {n+k-1\choose{}k}_q t^k[/math]

There's more on the "neighboring" product formulas

[math]\dfrac{1}{1-x}=\exp(-\log(1-x))=\exp(\sum_n\frac{x^n}{n})=\prod_n\exp(\frac{x^n}{n})[/math]

which is the crucial relation in some Weierstrass factorizations of more general functions.
You got the factoring out of (a-b) in

[math]a^{n+1}-b^{n+1} = (a-b)\sum_{k=0}^n a^k\,b^{n-k}[/math]

(and similarly in)

[math]a^n-b^n = (a-b)\prod_{k=1}^{n-1} (a-b\cdot{\mathrm e}^{2\pi i\frac{k}{n}}) [/math]

The interplay of sum and product comes back in integral equation theory, where for an operator A on a Banach space you can define the resolvent, basically

R(z, A) = -1/(1-A/z)

and the difference of two such geometric series for A and B still has a factor (A-B).
See pic related.

The map from a to 1/(1-a) with a in a ring is generalized to Star semirings, and that's a whole other can of nutritious worms

https://en.wikipedia.org/wiki/Semiring#Star_semirings

[Anonymous No.15014028]

The resolvent formula being the statement that

[math] \dfrac{1}{A-z} - \dfrac{1}{B-z} = \dfrac{1}{A-z}\cdot (B-A)\cdot \dfrac{1}{B-z} [/math]

in this order, still survives if A is an operator in a general normed vector space

[Anonymous No.15014031]

>Study geometry
>Expect to learn about cool shapes and wacky spaces, drawing lots of different cool pictures.
>Instead study functors and keep drawing the same diagram of a pull-back, and hope to one day parse whatever the fuck is an [math] \infty [/math]-category.
What the fuck? Everything's just a functor?

[Anonymous No.15014079]

>>15014031
>Everything's just a functor?
To the extent that categories are a general context for associative operations and functors are homomorphisms of such contexts, it shouldn't be too surprising that it pops up.

Functors in topology (I mean sheaf theory) just code continuous function spaces effectively.
But not everything is associative, and so you cane come up with even algebraic things where the native formulation usually doesn't smell too much of category theory (at least if you restrict yourself to sane diagrams). The commutator C=[A,B] as a map of (A,B) being such a non-associative thing.
Here's another, much smaller one: The function which takes two rock-paper-scissors players A,B who have chosen as move, and returns the winner C of the game.

PS the nLab doesn't contain the finite sum formula, time to REEE

[Anonymous No.15014203]

>>15013663
Why are you so angry?
I don't think it fulfills the purpose you want it to; it is a fantastic textbook to get an intuitive understanding of statistics and elementary probability. If you want a 'proper' treatment, what reason could you possibly have to skip measure theory?

[Anonymous No.15014208]

Is there any simple explaination how to calculate lyapunov spectrum for 4D system of ODE?
From what I have found it doen't look as difficult task, but people just don't explain their code and what they do.

[Anonymous No.15014228]

>>15003705
There should be an anime picture at every math related wikipedia page.

[Anonymous No.15014391]

>>15014228

[Anonymous No.15014396]

>>15014012
This is actually another good example of how a mathematics article on wikipedia should look like. Even though the setting is more general the information is much more concise and accessible.
The fraction exponential formula is also really neat.

I should get into the habit of looking for more general articles even when I'm only interested in special cases. They appear to be less prone to turning into convoluted messes due to normies being normies.

[Anonymous No.15014402]

>>15014396
You couldn't even handle a factor of "a".
RETARD

[Anonymous No.15014405]

>>15004478
Matsumura's Commutative Ring Theory
I dedicated almost an entire year to read it fully. It was mainly to fill the details for Hartshorne's Algebraic Geometry, but I wanted to understand it from beginning to end anyways

[Anonymous No.15014416]

How is the second equality in equation 14 true?
[eqn] E(Z_x) = \sum_{k=1}^{\infty} k \cdot \left[ F_k(x) - F_{k-1}(x) \right] = x \sum_{k=1}^{\infty} k \cdot (-\log x)^{k-1} / (k-1)! [\eqn] where [math] F_0(x) = 0, \quad \forall x [/math] and [eqn] F_n(x) = x \sum_{k=0}^{n-1} \frac{(-\log x)^k}{k!} \quad \forall n \in \mathbb{Z}_{\ge 1}. [/eqn]
When I tried to work this out, I got [eqn] E(Z_x) = x \sum_{k=1}^{\infty} k \cdot (-\log x)^{k-1} / (k-1)! \sum_{i=0}^{k-2} \frac{(-\log x)^i}{i!} [/math]

[Anonymous No.15014425]

>>15014416
Trying to enter math on this site is suffering.
[eqn] E(Z_x) = \sum_{k=1}^{\infty} k \cdot \left[ F_k(x) - F_{k-1}(x) \right] = x \sum_{k=1}^{\infty} k \cdot (-\log x)^{k-1} / (k-1)! [/eqn] where [math] F_0(x) = 0, \quad \forall x [/math] and [eqn] F_n(x) = x \sum_{k=0}^{n-1} \frac{(-\log x)^k}{k!} \quad \forall n \in \mathbb{Z}_{\ge 1}. [/eqn]
When I tried to work this out, I got [eqn] E(Z_x) = x \sum_{k=1}^{\infty} k \cdot (-\log x)^{k-1} / (k-1)! \sum_{i=0}^{k-2} \frac{(-\log x)^i}{i!} [/eqn]

[Anonymous No.15014430]

>>15014425
Not an answer to your question, but do you know how to algebraically manipulate equations like this of arbitrary length/complexity?

[Anonymous No.15014470]

>>15009962
AAAAaaah.... I guess the Königsberg bridge problem could be a good example? What do you think?

[Anonymous No.15014752]

>>15014470
I was thinking of something more like a device for a problem that is already abstractified. Maybe some kind of function that makes it almost visually clear and easy to see.

[Anonymous No.15014853]

>>15013034
I don't really know what you want to tell or ask me. Do you mean that I should look at the gradient/electrical field and use that to come up with the charge distribution the quadrupole resembles? Because I have tried that, I just don't understand it since for me it does not look like the field of just a bunch of point charges

[Anonymous No.15014893]

>>15014425
recompute [math]F_k(x)-F_{k-1}(x)[/math] carefully, you replaced this expression with the formula for [math]F_{k-2}(x)[/math] which is not correct. you should find that [math]F_k, F_{k-1}[/math] differ by just one term

[Anonymous No.15015263]

Vakil’s notes are very good but my god the writing style is infuriating

[Anonymous No.15015435]

Terry?

http://shanghikid.50megs.com/Otakudom/ContReal/contReal.htm

[Anonymous No.15015461]

>>15003586
Wouldn't it be hilarious if you could just visualize all of this shit?

[Anonymous No.15015704]

What are some good books with problems that are interesting to solve, but don't require any formal education in mathematics? Basically like basic geometric stuff that doesn't require me to know tons pf definitions. I just want some riddles to sharpen my mind.
Bonus points for small books that I can easily carry with me all the time to have something to do when waiting

[Anonymous No.15015705]

>>15015263
There is room for informal math but this guy goes too far

>>15014405
>Ring Theory
I wish there was a ring theory text that's not mathematicians-oriented, in the sense that it spends a bulk of the text on polynomial equation who's numerical solution is trivial.
I'd like to see a ring theory text, commutative or not, that generalizes some theorems true for important rings and fields, like formal power series and the reals or at least the rationals or some Lie algebras over subfields of R with characteristic.

[Anonymous No.15015708]

>>15015704
I think "how to prove it" is a standard rec for your situation

https://users.metu.edu.tr/home205/serge/wwwhome/courses/111-2011/textbook-math111.pdf

[Anonymous No.15015717]

>>15015708
I heard of it, isn't it more like a textbook with some exercises though?
I don't really want to read something, just have someproblems to think about occasionally

[Anonymous No.15015727]

>>15015704
>>15015717
Mathematics via problems
Problem solving strategies

[Anonymous No.15015733]

>>15015717
I guess so - but why do you explicitly want to avoid learning some concepts to express problems you can't otherwise deal with?

[Anonymous No.15015757]

How do you isolate this X here? No logs, please, and no, it's not homework, I'm just experimenting.

[Anonymous No.15015767]

>>15015757
It is homework and you need log for that.

[Anonymous No.15015777]

>>15015767
it's not homework, and I know I need log. What i'm asking is there a way without log?

[Anonymous No.15015781]

>>15015777
There's no way to isolate [math]x[/math] in [math]a^x = b[/math] without logarithms.

[Anonymous No.15015815]

>>15015777
Do you not understand what need means dumb esl?

[Anonymous No.15016161]

>>15015727
Thank you
>>15015733
Because I just do this for fun, and I enjoy thinking about simple to understand but hard to solve problems more than reading definitions and formalisms.

[Anonymous No.15016182]

taking a logic class at my local community college and I feel like I'm Ben Shapiro facts and logic mastermind. I would like to learn more about logic at my leisure but I can't wait to learn to prove simple mathematical statements. I feel like I would be able to reason about questions that nag me all the time about math when I learn it. Cheers to maths

[Anonymous No.15016209]

What does it take to prove 2+2=4?

[Anonymous No.15016219]

>>15016209
Using the definition you have for "2", "+", and "4".

[Anonymous No.15016226]

>>15016219
Okay that's a good start thanks. I know you guys could probably do this but it's amazing to think how many people can't.

[Anonymous No.15016230]

>>15016226
It's not a matter of ability, you stupid retard.

[Anonymous No.15016246]

>>15016209
Not much
.. + .. = ....
+ is a fusion sign so you just take it away and have
.... = ....

[Anonymous No.15016600]

>13 math courses, 42 credits in math
>10 upper level math courses, 30 UL credits

>3 physics courses, 11 credits
>4 CS courses, 12 credits

>only 65/120 credits in shit that actually matters
I can't believe I am at the finish line and I realize now that just about half my time here just went into the garbage, for shit like western culture, survey of american history, english 101, personal finance, etc. Saving grace is my total debt will be zero dollars.

[Anonymous No.15016701]

I dropped out of calc 2 to be a forklift driver and have been mostly lurking mg, posting once a week or so about something the board has taught me, usually something in combinatorics where I can get by with stubborness instead of brain cells, for 3 years. I recently wanted to actually understand large ordinals to a better extent than the fucking Vsauce video so I got a copy of Hrbacek's Introduction to Set Theory and braced myself for a couple months of self study.
I got filtered at the third content page and cannot understand what the fuck parameters are supposed to be. Don't try to explain them to me, instead please answer this question:
>I don't want to give up on this entirely. Should I keep attempting Hrbacek on pen and paper, drawing out truth tables for whatever the fucking terms are until I fucking get it or instead try reading How to Prove It?

[Anonymous No.15016720]

>>15016209
The same amount of effort it takes, if not more,
to prove that 1+1=2. Of course, the result was
said to be "occasionally useful".

[Anonymous No.15016894]

Wtf does Latex have warnings for? Not like it's an actual programming language

[Anonymous No.15016900]

>>15016894
Latex is turing complete. You can code Doom in it.

[Anonymous No.15016903]

>>15016894
Wrong. Latex is a programming language. It generates input for Tex to process.

[Anonymous No.15016967]

>>15016900
>>15016903
>ACKSHULLY

[Anonymous No.15017018]

>>15016967
bro you’re on ackshually the board, the website.

[Anonymous No.15017133]

>>15016967
You expect a math thread not to be full of pedants?

[Anonymous No.15017466]

>>15016701
Just take it slowly. Analyze each proof in the book carefully. Try to write out every single step explicitly. Imo, that's the only way to self-study: by studying proofs and examples extensively, then applying that knowledge in exercises.

[Anonymous No.15017978]

https://link.springer.com/book/10.1007/978-1-4419-8616-0
https://link.springer.com/book/10.1007/978-0-387-71568-1
https://link.springer.com/book/10.1007/978-3-319-23488-5

Which one should I get from the Springer sale?

[Anonymous No.15018034]

>>15003586
infinity + 104 + miscellaneous

[Anonymous No.15018043]

>>15016894
I also know people who implemented Runge Kutta solvers in LaTex.
(I'm not the ackshually guy)

Btw. there's the concept of "Pac-Man complete", to get more out of such discussions

[Anonymous No.15018247]

>>15017978
Get the first one, its behind a paywall. The other two are free if you are associated with a university in any way.

https://files.catbox.moe/ldx58v.pdf

https://files.catbox.moe/v9bjmk.pdf

[Anonymous No.15018369]

you guys have any good sites I can use for exercises and practice? I just want something besides Khan academy and my current resources are a bit barebones.

[Anonymous No.15018379]

>>15018369
All those sites will just use textbook problems. If you need step by step solution assistance, find textbooks with teacher companion guides.

[Anonymous No.15018382]

>>15018379
good point, I will likely focus better if I have it right in front of me as well, anytime I do work on khan academy or otherwise on the computer I zone out too much and lose focus. I'll find some good precalc books to download until I can get physical copies.

[Anonymous No.15018390]

>>15018382
If you're in precalc, just use stewarts. Massive, very thorough book. You can get lang's basic math if you want to be an autist, but stewart's worked really well for me.

[None !X3333333n. No.15018443]

hello, I have a question (not homework, no time pressure) about operator spectra
Say we have the infinite dimensional matrix
[math]
\begin{equation*}
M: {\ell}^2(V \rightarrow W) =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0\\
0 & \frac{1}{4} & 0 & 0 & 0 \\
0 & 0 & \frac{1}{9} & 0 & 0 \\
0 & 0 & 0 & \frac{1}{16} & 0 \\
0 & 0 & 0 & 0 & \ddots
\end{pmatrix}
\end{equation*}
[/math]
which takes a vector from [math]{\ell}^2 \rightarrow {\ell}^2[/math].

I think:
this matrix is injective so it should have a left inverse, but fails to surject so it doesn't have a right inverse and [math]0[/math] should be in its spectrum.
So my question is:
what's an example of a vector that this matrix won't hit (thus not having a right inverse)?
I can't think of any, but it seems like there should be an easy one.
thanks!

[Anonymous No.15018462]

>>15018443
>>>/sci/sqt
[math]a_n = 1/2^{n/2}[/math]

[None !OhXsuiCIDE No.15018470]

>>15018462
oh... right, lmao
thank you anon!! <3
>sqt
ha yea, sry

[Anonymous No.15018487]

>>15018443
Just take a_n = 1/n^2 (the diagonal entries)
Its preimage is (1,1,1,1,1,...), which isn't in l^2.

[None !X3333333n. No.15018500]

>>15018487
wouldn't the input space also being [math]\ell^2[/math] mean that vector wouldn't be in [math]V[/math] tho?

[Anonymous No.15018512]

>>15018500
Yes, I said (1,1,1,1,...) isn't in l^2.

[Anonymous No.15018521]

i ordered some new vector calculus pants. these destroy programmer socks in terms of increasing abilities to hack difficult to unsolvable math problems that make you rich

[None !X3333333n. No.15018536]

>>15018512
oh no anon you're completely right
idk why I was so hung up on (1,1,1,1,..) not being something to consider
baka thank you very much sorry for being dense lol

[Anonymous No.15018553]

>>15018390
Im on precalc but my algebra and trig can certainly use some brushing up, but I imagine it'll correct itself as I go through this book. Thanks anon ill be trying my best

[Anonymous No.15018574]

>>15018553
I personally didn't like Stewart's for self teaching, try Sheldon Axler's precalculus if you feel the same way.
Obviously they teach the same shit but Axler provides mostly thorough, complete solutions with his thought process for all the odd numbered problems.

[Anonymous No.15018604]

>>15018521
Looks like those pants could destroy some virgins too.

[Anonymous No.15018712]

>>15018574
Yeha I just did a short session with Stewarts, not a fan though its good to keep around I suppose. Im giving Sheldon a shot right now and this feels far more intuitive for self study, thanks again

[Anonymous No.15018881]

>>15006184
That is the worst book I have read in my entire academic career, barring my community college world religion textbook

[Anonymous No.15018946]

why is the institute for advanced study so incompetent that they can't record lectures without them being 10 frames per second or so quiet you need a fucking 1000$ sound system to boost the volume to hear.

[Anonymous No.15018981]

>>15010001
It's really easy, just whenever you see a word you don't know, write it down in a document along with its translation. Then you have your own custom-made dictionary you can consult whenever you need to. Grammar doesn't matter at all for math, so don't worry about it.

[Anonymous No.15019875]

>>15018946
>At the stage in your life where you are doing actual math
>Can't figure out how to get and install a program that fixes the sound for you
Should a couple classes in basic internet competency be required to attend grad school?

[Anonymous No.15020790]

>>15003586
how is this unsolved? lmao

[Anonymous No.15020801]

>homework problem from this week:
>prove a number is rational if and only if its digits are eventually periodic
God this problem makes me feel retarded.

[Anonymous No.15020858]

>>15020801
Isn't this quite easy, you can always present periodic part as ratio of two natural numbers.

[Anonymous No.15020864]

>>15020801
Which direction do you have trouble with?
Either way you will have to use the geometric series formula in the form

[eqn]\sum_{k=1}^\infty 10^{-kn} = \frac{10^{-n}}{1 - 10^{-n}} = \frac{1}{10^n - 1}[/eqn]
at some point.

If you also have to prove it for bases other than the decimal base then just replace 10 with b.

[Anonymous No.15020865]

Nand gate in quintic solutions is all. Convex uniform, duoprism.
nitrogen electron distribution is the main interest

[Anonymous No.15020885]

>>15020865
How do you get a duoprism from a quintic? I would have thought that any trigonometric 3-solids need a sextic to define.

[Anonymous No.15020930]

Let U be the number out of 10 days a stock goes up and D be the number of ways it goes down. The solution says U and D are positively correlated which makes sense but if I write D=10-U
[eqn]cov(U,D)= cov(U,10-U)=E(-U^2+10U) - E(U)E(10-U) = -E(U^2)+E(U)^2[/eqn]
isn't the right-hand side negative due to Jensens inequality?

[Anonymous No.15020938]

>>15020930
>positively correlated
Aren't they negatively correlated?

[Anonymous No.15020940]

>>15020858
>>15020864
The tough part is that everything about series was cut from our curriculum (because of high dropout rates).

Our professor showed us a weird proof for the existence of decimal expansion without series. And that's the only clue we get.

Our books still contain a chapter about series which I read through and used to show that periodic implies rational. Tomorrow I'll try it the other way around.

[Anonymous No.15020944]

>>15020790
why shouldn't it be?

[Anonymous No.15020947]

>>15020940
>Our professor showed us a weird proof for the existence of decimal expansion without series
Post it

[Anonymous No.15020952]

>>15020930
>>15020938
Just to elaborate:
When U is small, D is large. When U is large, D is small.
So take U = 0, you get D = 10. Take U = 10, you get D = 0.
That's negative correlation.

[Anonymous No.15020959]

>>15020930
>>15020952
The solution says positive. But if you think so too its probably just an error. Thanks.

[Anonymous No.15021659]

>>15003586
What’s the answer? HELP!

[Anonymous No.15021672]

>>15021659
Retard

[Anonymous No.15021683]

>>15021672
I’m not in the top 20%

[Anonymous No.15021690]

>>15021659
>divide 50 by half
Does it mean 50 by 2
or 50 by 0.5

[Anonymous No.15021722]

>>15021690
That’s why it’s a trick question.
Probably why writing out an equation is shit.

[Anonymous No.15021902]

>>15021722
There is also scaling element to it which is
50 / 0.5 scaled is 1 / 0.01 or 100 / 1
>if you forget left->right

[Anonymous No.15021924]

>>15021902
and the same for 2 it can be either 25 / 1 or 1 / 0.04
so
1 / 0.01 + 20 x 2
100 / 1 + 20 x 2
25 / 1 + 20 x 2
1 / 0.04 + 20 x 2

so 65 90 140 or 240
>4 answers

[Anonymous No.15022111]

>>15021659
90.
Order of ops is irrelevant for stuff like that, just perform the actions in the order you're told to.

[Anonymous No.15022895]

X (number of uni graduates) = 30% of n
Z (number of proficient uni graduates) = 1.2% of n
Y (percentage of Z from X) = 4%

How could I do this equation without knowing n?

[Anonymous No.15022954]

soundtrack: https://soundcloud.com/user-223534166/house-music-anthems-synonymous-with-1993

[Anonymous No.15022966]

>>15022954

[Anonymous No.15022983]

>>15022966
Note: when using LyX, commas after formulas should be placed inside the formula when the formula would otherwise end in a symbolic character in order to avoid wide kerning.

[Anonymous No.15022990]

My computational skills are worse than the ability to grasp the course concepts and do proofs it seems.
I want to grind out some strictly computational problems every day, that are about fractions, powers, simplifying stuff and things like that (anything one needs to be able to do the computational side of real analysis problems).
I was thinking of doing the art of problem solving books, do you guys have other recommendations or are they decent enough that i shouldnt think too much about a better solution?

[Anonymous No.15023000]

Was talking to my professor and a few other students about Wolfram's books, and I got funny looks and told he was a crank, but no real explanation why.
Is this true? Everything he's written seems spot on, and he doesn't claim any wild theories.

[Anonymous No.15023005]

>>15022983

[Anonymous No.15023012]

>>15023000
honestly go to /biz/ or /g/ if you want to talk about Wolfram
his organization makes scientific software
he is relevant to scientific technology
that's what he's known for
he is in the business of selling scientific software
go find people who care
don't try to make us care about shit

[Anonymous No.15023023]

>>15022990
https://archive.org/details/spivak-m.-calculus-2008
https://archive.org/details/spivak-m.-calculus-2008/page/n15/mode/2up
Spend some time with it.
Almost certainly it's worth printing out some of the pages, especially the ones with problems.

[Anonymous No.15023101]

>>15023012
So is the problem that he made software and h as money? Maybe I should ask in a physics thread instead. Same hostility I experienced today in office hours. Bizarre. Did he kill your dog?

[Anonymous No.15023115]

>>15022895
Easy...you don't have to know n.

[eqn] {Z \over X} = {{1.2 \over 100}n \over {30 \over 100}n} = {1.2 \over 30} = 0.04 = {4 \over 100}= Y [/eqn]

When two different percentages are parts of the
same group or population (n), the ratio of these
percentages is constant--a percentage of two
percentages.

[Anonymous No.15023129]

>>15022895
You read chapter 7.2 of your textbook (devore) before attempting homework problems.

[Anonymous No.15023295]

What is the key conceptual difference between set-theoretic foundations and category/type-theoretic foundations?
I reckon that set approaches do rules of inference first and constructions second while category/type approaches do the other way around, but I feel like a pseud even saying this.

[noo No.15023353]

>>15023351