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Pastebin exists for a reason edition

Previously: >>13864113
Talk about mathematics
>>
Can anyone recommend a good textbook on proofs, calculus, linear algebra, set theory, abstract algebra, real analysis, complex analysis, topology, number theory, Galois theory, commutative algebra, algebraic topology, differential geometry, elliptic curves, algebraic geometry, or harmonic analysis?
>>
>>13888107
It's always greater, assuming i is positive
>>
>>13888120
Serge Lang wrote books about all those topics, including a book called Basic Mathematics which is about basic mathematics.
>>
>>13888120
https://openstax.org
free math books
>>
>>13888136
Basic and mathematicspilled
>>
>>13888107
Don't run crying to the old threads to dig the list up when you get 500 replies asking for book recommendations

And if you're so new that you don't even know to wait for the old thread to reach the bottom, then lurk and learn
>>
>>13888107
>implying that the complex numbers are able to be sensibly ordered
>>
>>13888551
Bibliographic order
>>
>>13888107
>>13888551
There are several mock orderings but no ordering that works as a drop-in extension to real inputs.
>>
>>13888107
Please extend the concept of "greater than" to the complex plane.
>>
>>13888120
Proofs:
>See below
Calculus:
>Who gives a shit. Just use Spivak, or Apostol or whatever else you like.
Linear algebra:
>Shilov
>Roman
Set theory:
>Halmos
>Jech
Abstract algebra:
>Basic Algebra I and II, by Jacobson
Real analysis and complex analysis:
>Rudin
Topology:
>Engelking
>Borisovich et al.
>Novikov
>Anything by Fomenko
Number theory:
>A Course in Arithmetic, by Serre
Galois theory:
>See abstract algebra
>Profinite Groups, by Wilson
>Galois Cohomology, by Serre
Commutative algebra:
>See abstract algebra
>Atiyah and Macdonald
>Eisenbud
Algebraic topology
>May
Differential geometry
>Lee
Elliptic curves
>Silverman
>Knapp
Algebraic geometry
>Shafarevich
>Hartshorne
Harmonic analysis
>Stein and Weiss.
>>
Yes it is greater because I said so stay mad stupid atheists.
>>
>Many Russian families have the tradition of giving hundreds of such problems to their children, and mine was no exception. The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sun rise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

I should just kill myself
>>
error: Class Complex does not implement Comparable -interface
>>
>>13888803
It's easy anon. Just write down the equations.
>>
What are some text books to really master geometry? I'm not just talking the bare minimum that's taugh in high-school, but really knowing it to the point I can formulate proofs entirely geometrically like was done back in newton's day.
>INB4 Euclid
I find it hard to believe that with all the progress since his time their aren't more complete and succinct books that the elements.
>>
>>13888803
Are you assuming they both are walking at the same speed? If so this is some very basic math mate, you can do it
>>
>>13888107
>!obkpuSWh5M
What's the point of using a trip if you're not going to use a name as well?
>>
>>13888936
Doing geometry only with pictures is difficult for a lot of people, that's why you usually see written proofs and no drawn proofs.
>>
>>13888952
I know, it takes real critical thinking and isn't just following a flow chart, which is why I'm asking if anyone knows of a really good geometry book.
>>
>>13888937
It is not assumed, in fact they aren't.
>>13888803
sunrise at t hours before noon
AB = v1*t+v2*t
4=v2*t/v1 -> v1*4=v2*t
9=v1*t/v2 -> v2*9=v1*t=v2*t2/4
t2=4*9=36
t=6
sunrise at 6 hours before noon
sunrise at 6 am
bonus:
AB=v1*6+v2*6
v1=v2*6/4=v2*3/2
AB=v2*9+v2*6=v2*15
v2=AB/15
v1=AB/10
so in fact the speeds are not the same
>>
>>13888551
>>13888619
>>13888672
>they haven't taken the lexicographic pill
NGMI
>>
>>13888107
>replace i with the y coordinate
>replace real numbers with x coordinate
>treat the points as vectors from (0,0)
-200i has a greater length than -1-i therefore it's greater
>>
Is there a reason for writing [math]\frac {\partial u} {\partial x}[/math] instead of [math] \frac {\partial (u)} {\partial (x)}[/math]? The first way of writing is more confusing for many of the undergraduates I TA for.
>>
>>13889343
writing dx^1,dx^2... for differential forms is the convention, saves a fair bit of time without brackets
idk how its confusing for them, dy/dx is the first thing they come across
>>
>>13888619
How doesn't the ordering defined by

a + ib < x + iy if either a < x or (a = x and b < y)

extend the usual ordering of the real numbers?
>>
File: 1589184699900.jpg (231 KB, 564x860)
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>>13889386
the reals are an ordered field, which means order is in a sense ‘preserved’ over the field operations. i.e.
>i. [math]x+y<x+z[/math] if [math]y<z[/math]
>ii. [math]xy>0[/math] if [math]x>0[/math] and [math]y>0[/math]
bibliographic ordering is an order relation on the set of complex numbers, but order will not hold over operations. can you see why?
>>
>>13888935
>>13889001
I did it this way, but there was talk of a neater solution
This is how my friend did it:

t/4=9/t
t^2=36
t=6
(Copied from some dude on twitter cuz I can't post pictures)
>>13888937
Different
Otherwise at noon they would have met halfway and reached their destinations at the same time
>>
>>13888107
My lecturer uses Forster for his entire course in real analysis. Do you think it's any good? Should I work through a different book in my free time?
>>
why is there no information on the internet regarding partial limit? theres literally one question about it in quora with one reply and another page with a one sentence summary. what in the fuck?
>>
>>13889630
WTF is partiel limite?
>>
>>13889638
its a limit of a subsequence. this limit is called the partial limit of the original sequence
>>
>>13889656
Ha OK. You have The Bolzano Weierstrass theorem, if you're interested, but I thkink you already know it.
>>
>>13889001
>>13889528
Could be the same if you're autistic and take earths curvature into consideration, as well as removing the assumptions that they walk at a reasonable pace or that the day which they arrive is the same as the day they left/met.
>>
>>13888120
You'll never get to these but whatever here.
>Proofs
Hammock
>Calculus
Apostol
Hubbard and Hubbard
>Linear Algebra
Axler
>Set Theory
Schimmerling
>Abstract Algebra
Artin
>Real Analysis
Rudin
>Complex Analysis
Gamelin
>Topology
Dugundji
>Number Theory
Ireland and Rosen
>Galois
Don't know a specific book on this, but any algebra books should teach you the basics
>Commutative Algebra
Milne notes
>Algebraic Topology
Hatcher
>Differential Geometry
Beg: Shifrin notes
Adv: Tu
>Ellipitic Curves
Silverman and Tate
>Algebraic Geometry
Beg: Cox Little O'shea
Adv: Vakil Rising Sea
>Harmonic Analysis
Dunno
>>
>>13889660
yeah i do. i was looking for, say, a long page that covers the entirety of partial limits but i didnt find much.
i tried searching for other names like subsequential limit but theres not much information on that either. its weird because ive received a 7 page long pdf from my uni on it, and theres nothing extra that i could find online. maybe this is all there is, but i was hoping for at least some examples to study better. oh well.
>>
>>13889677
I can offer a nice fun fact about subsequences, maybe you know it already though, I'm not sure
>>
>>13889528
I guess, using v1/v2 as a constant, you have that it is equal to t/4 and also to 9/t, so these two are equal. But what matters is solving it, and modelling it correctly, understanding the problem. I prefer the more verbose way.
>>
>>13889677
Is it related to limit superior and limit inferior? I can't think of a reason you'd be so interested in a partial limit that you'd give a specific name to it.
>>
How did anon get good at analysis? In particular, doing inequalities.
>>
>>13889691
Inequalities suck ass. Just do a thousand of them.
>>
>>13889589
not rudin/10
>>
>>13889701
Do you know Amann/Escher? I skimmed it and it looks good but it has no solutions manual and I hate not knowing whether I did the problems correctly or not.
>>
>>13889691
So far, in all of my (advanced) undergrad math, I have honed my abilities by just doing the proofs I can, reading through the proofs I can't do, and absorbing the common general strategies.
>>
>>13889691
You just have to think about inequalities as bounds and then it becomes kind of second nature from there.

Most inequalities can be deduced by abusing the triangle inequalities in some way as well. Get very familiar with all the various triangle inequalities and you'll be skipping on through most inequality problems
>>
>>13888107
Re(-1-i)<Re(-200i)
Im(-1-i)>Im(-200i)
>>
>>13889696
>>13889708
>do problems
Of course I already am, just not getting much improvement. Can you recommend a good problem book?
>>13889713
I've heard of people eventually getting to the point where they're able to get an idea of how tight bounds actually are, which is where I'd like to be. Not quite sure how to get there though.
>>
>>13889684
He did it geometrically, via similar triangles. It's what I suspected would work but was too dumb to make work
> But what matters is solving it
I would agree generally but I was told there was a slicker way so I got stuck with that. Thank you anon.
>>
>>13889719
Did you read the other part of my post?
>>
Is apostol calculus any good for someone going for a engineering degree ?
>>
>>13889656
>>13889677
it is also termed a ‘subsequential limit’. the set [math]E[/math] of subsequential limits of a sequence [math]a_n[/math] has various properties; for example, [math]\sup E =\limsup a_n[/math] (supremum in the extended real sense) and similarly with [math]\inf E =\liminf a_n[/math]. see rudin PMA p.56.
>>13889705
no, i do not know amann/escher. only rudin. trust in the process.
>>
>>13889719
Bounds almost always fall from the triangle inequality which are always associated with some norm. Once you get an intuitive understanding of the commonly used norms (a.k.a the 1, 2, and infinite norm) it literally becomes like breathing unless you're working with a special norm.

A lot of inequalities can be broken up into cases as well, and for some it's easier to look at it that way.
>>
>>13889736
I'm an eng student as well. I used it, it was weird. In my case, the rigour surpassed what was expected of the class. There weren't enough drilling exercises probably. And the order wasn't the same: Apostol begins with integration, and classes usually start with differentiation. You have to jump around a lot on other subjects as well.
>>
>>13889723
>>13889750
I didn't realize the triangle inequality could refer to something other than the one you get from the euclidean norm, so wasn't entirely sure what you meant. In hindsight it was kind of obvious. Your suggestion is in line with what I've noticed from watching people tackle Olympiad inequality problems, so I guess I'll check out what resources their community uses to learn and prepare for them.
>>
2-dimensional turing machine simulation
https://www.youtube.com/watch?v=GLc52sC8woI
>>
Post things you've came up with only to find that they were already know. Pic related.
At the begining of the year i was playing around with infinite equations and discovered the middle one formula. I was very happy to actually find something interesting by myself, but I got kinda down when i had found that my formula has been known for centuries.
>>
>>13889943
Nice pic
>>
File: 1622778860182.jpg (1.51 MB, 4000x2250)
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1.51 MB JPG
>>13889943
i forgot the pic
>>
>>13889949
O Kurwa
>>
File: 1636243715155.jpg (1.92 MB, 4000x2250)
1.92 MB
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the rest
>>
>>13889973
and the last one: the last formula from previous pic in case where the first sum doesn't generate infinitely many terms
>>
File: 1627843537180.jpg (1.52 MB, 3088x2096)
1.52 MB
1.52 MB JPG
>>13890041
fuck, i keep forgeting to choose files
>>
>>13890046
your u's and n's and n's and 1's are confusing
>>
>>13890064
i know my u's and n's look pretty simmilar, sorry. Which part do you find unreadable? But what is wrong with my 1's?
>>
>>13890076
the ones look like n's
>>
File: 1610365193607.png (7 KB, 448x384)
7 KB
7 KB PNG
>>13890085
>>
>>13890115
In what godforsaken universe is that an N? Honestly, you desperately need to fix that.
>>
File: nnu1.png (4 KB, 303x519)
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>>13890115
wat
its weird because the rest of your semi-cursive looks nice
post full alphabet, i want to see your z
>>
File: 1619951857129.jpg (1.45 MB, 4000x2250)
1.45 MB
1.45 MB JPG
>>13890155
>>
>>13890281
Your n is awful.
>>
>>13890287
His i is awful as well
>>
>>13890281
was expecting polish crossed z's desu
anyway sort out your h's m's n's and q's
they are terrifying
>>
Why is the pushout of two inclusions [math]S^{n-1}\to D^{n}[/math] homeomorphic to S^n? I've been it cited frequently as obvious. I can see that, for n=2, the visual representation of this is having two disks glued with their boundaries forming an equator. I probably can write down an explicit bijection and check that it is open, but considering that the result is considered so 'obvious', I think I'm missing something that would let me deduce that identification on D^n+D^n is homeomorph to S^ right away. Is constructing an explicing map really the canonical way of checking homeomorphism?
>>
>>13890342
>I've been it cited
I've seen
>>
>>13890340
no q's in my alphabet R comes after P
>>
>>13890350
then your R is pretty borked
>>
>>13888107
There's not an order relation defined for complex numbers. Order is defined on the modulus.
>>
>>13890342
In an n-sphere x_0^2 + ... x_n^2 = 1, both the upper half x_0>=0 and the lower half x_0 <= are homeomorphic to n-disks by orthogonal projection onto the x_0=0 plane. Their intersection is the n-1 sphere.
>>
>>13890395
Well, seems to be the answer. Thank you
>>
File: 1609832563409.png (29 KB, 712x468)
29 KB
29 KB PNG
>>13890046
ok, i did a lot of print screen and paint but i have made a computer font version
>>
>>13890457
anon wtf have some self-respect
>>
I wish I could be like my heroes Ted Kaczynski and Grigori Perelman
>>
the fact that
[math](a_n)_{n \in \mathbb N}, a_n < b \Rightarrow \lim_{n \rightarrow \infty} a_n \leq b[/math]
is in my opion the most important fact in analysis
actually it might be the essence of analysis itself
>>
>>13890616
>Oswald Teichmüller missing
[math]\text{s}\text{oyman detected}[/math]
>>
>page 3×3
What are some exciting facts about number 9?
>>
>>13892347
it's the smallest odd number which can be represented as the [math]a^b[/math] where a and b are both primes and a=/=b
>>
>>13890457
You should learn latex/mathtex
>>
>>13890992
i'm stupid, please elaborate
>>
Are there any real world applications of spectral graph theory?
>>
>>13890992
That's true in literally every ordered set with the order topology, those can get a lot wilder than R
>>
>>13890380
Is this true? Can anyone confirm?
>>
>>13890992
Yes the order properties of R are the most essential, and the fact that it's a field is almost accessory to them.

https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals
>>
>>13889677
pickup a book on convergence spaces
>>
File: haww.jpg (81 KB, 600x854)
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The annual inflation rate R_m is published every month and expresses the price p_m increase of good to 12 months prior.
R_m = p_m / p_{m-12}

Say on month m=0, the price is p=1 and the rate equals some R_0 (given value).
Say over the span of one year, the rate R_m increases to R_12 (given value, determining how p_m went up in the first 12 months) and then stays constant.

What's the price p_m analytically?
What is p_{2*12} for R_0=0.02 and R_12=6.2?
>>
>>13890992
It's an important fact, yes. But the most important is that integration of x^a depends on if a<1 or a>1
>>
>>13892931
what does p_{2*12} mean?
>>
>>13893198
p_24
>>
>>13892720
Yes, it's true. There is no ordering on C that respects multiplication.
>>
>>13892931
>writing LaTeX without [maths] [/maths] (remove s)
please kys
>>
File: invariants.pdf (289 KB, PDF)
289 KB
289 KB PDF
bros, i'm an undergrad taking part in a seminar on invariant theory - every student is assigned a topic and is supposed to deliver an expository "lecture" on it to the others. the topic i got assigned is invariants of the conjugation action on (complex) matrices. this is the first piece of expository writing i've ever produced, so if anybody feels like reading this and critiquing it (on pretty much anything from content to style), i'd be very glad to hear them
cheers
>>
>>13893375
Do you have something else than a PDF? I don't download binaries coming from 4chan.
>>
>>13893439
wdym "something else", do you want the tex file? i also uploaded this to r/math, here's a link i used there https://www.reddit.com/r/math/comments/r2t8se/anybody_want_to_offer_constructive_criticism_on/ (4chan isn't letting me post the wetransfer link)
>>
>>13893500
Something that I can read without compromising security of my computer.
>>
>>13893517
dude, i don't know what to say to you, all i have is the pdf (which i ran through some site that strips metadata) and the tex original
>>
>>13893375
>>13893439
>>13893517
here, i pastebin'd the latex code
>>
>>13893545
forgot the fucking link, lol
https://pastebin.com/FAWUCP12
>>
>let
>>
>>13888107
See here for how to represent gaussian integers in the base (-1 + i).

https://www.math.uwaterloo.ca/~wgilbert/Research/ArithCxBases.pdf

Related project Euler problem.

https://projecteuler.net/problem=508

Your picture may have a legit answer assuming you didn't cut off important shit like a retard.
>>
>>13893548
Can't you screenshot each page? I would read it, I like the subject. But TeX is Chinese for me.
>>
>>13893736
i've worked harder getting this to you than i have on the damn paper, lol
here: https://imgur.com/a/eGIiYmT
>>
How can you not love complex analysis and Fourier analysis ?
>>
>>13892810
any recommendations? the only thing i find are a few books by a szymon dolecki
>>13893375
keep your equations austere; i am not familiar with the content of your seminar, but page 4 (and 3) look nasty.
>>
I'm trying to find the de Rham cohomology of the twice punctured plane. There's one part that I'm not sure about. Is it true that [math] \mathbb{R}^2 / \mathbb{R} \cong \mathbb{R}[/math]?
>>
>>13893796
>https://imgur.com/a/eGIiYmT
Thx
>>
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>>13893851
>keep your equations austere; i am not familiar with the content of your seminar, but page 4 (and 3) look nasty.
i've since modified the formatting a bit, does this look better? if by "nasty" you mean they're long (like the example 3.2), i'm afraid that's kind of unavoidable
>>
>>13893855
>Is it true that \mathbb{R}^2 / \mathbb{R} \cong \mathbb{R}?
do you mean as vector spaces or topologically? the former's a definite yes, i think so is the latter
>>
>>13893796
What is the due date?
>>
>>13893915
i'm supposed to present some time in December/January, but i'm pretty much done, just adding/modifying things here and there
why do you ask?
>>
>>13893924
Right now, I'm reading it by skipping the computations (I hate computations), but maybe I will reread it more carefully (the subject is interesting).
>>
>>13893934
thanks for the interest, although i don't expect you'll find anything fundamentally (i.e. excepting typos) wrong with the computations, since none of this is original work
>>
>>13893943
I know, I'm doing a first reading to understand about what it is.

In fact you just want to prove fundamental theorem 3.8 ?
>>
>>13893949
in a nutshell - yes, that's the salient part the professor wants me to cover. i've also written a bit about the closures of conjugacy classes at the end. i also wanted to start with a motivating discussion on the quotient map for n=2,3 and maybe talk a bit more about the closures of conjugacy classes, but i'm afraid all of this won't fit into the 1.5 hours i'm allotted (for the sake of the other students i wanted to be really thorough with the results i use, hence i prove most of the stuff, don't just quote it)
>>
>>13894007
dont be afraid to replace some of the chugging working with 'it can be shown', then just writing equality (assuming you are presenting in person on a blackboard)
dont spend too long on them, your audience knows how to use matrices and manipulate simple summations/products, expose subtleties and repeat the main ideas
>>
>>13894007
Ok. My first remarks (it's my opinion, I can be wrong, I can be a dick, it's your call):

Why not introduce fundamental thm 3.8 into the introduction. It was not clear that the finite generators were those si functions, you define later. I know that it's not easy to explain it without a lot of details but...

>Some remarks about the contents

* Introduction
f(g.A)=f(A) and not f(g.A)=A

* Definition 3.11
Aren't everyone aware that there is only one topology for a normed vector space of finite dimension? Why do you precise?

* Lemma 3.12
Is it necessary? It's a basic theorem about continuous extension of a function (I don't know how to say it in English).

* 4 the proof
The first introduction phrase, is not necessary. Just say that you'll use the following U. Don't need to say that it was a contest between a lot of candidates.

* Definition 4.2
Calling the set of companion matrix K confused me. K is your field. I made the confusion during Corollary 4.5, but you're coherent.

About the style

"We're dealing with"... I'm not sure, but I don't think that it's the kind of expression we want to read in a paper. English is not my first language, but I think that "to deal with" is too informal.

>"tricks such as averaging?"
"Tricks"!!! Where do you think you are? In the ghetto? No, don't talk like that, the one reading your paper isn't your friend. I am not your pal, bro.

>It is a standard result
That's your opinion. I don't think it's good practice to talk about your opinions.

>It is no coincident two of these functions look familiar
It looks like you're animating a TV show. If you're talking to students, yes, you can say that. But a paper isn't a TV show.

>require rudiments of field theory
It's your opinion, just state that it requires some results of field theory not cover by this paper. Of course, when talking you can say that.

By the way, thank for the paper, I knew that there was such a theorem, but that's the first time that I read it explicitly.
>>
>>13894087
corrected

not all of the students have taken a topology class and while they've definitely covered this in analysis i thought it germane to mention this briefly. perhaps i'll omit it in the lecture itself, i just thought it'd be good to have in the written version

you mean "Let A" etc? fair enough

i wanted to call the set C, but i use that for the conjugacy classes, so i settled for K. maybe i'll rename it to R since it's the image of \rho(\cc^n)

it's not like i'm submitting this to a journal, this is basically for my fellow students, but fair enough

i'm pretty sure this is standard nomenclature, e.g. "Weyl's unitarian trick" or "Rabinowitsch trick"

i get your meaning, but you can't convince me the fact that similar matrices have equal characteristic polynomials isn't standard

that's exactly what i'm doing

i suppose i could rephrase that

>By the way, thank for the paper, I knew that there was such a theorem, but that's the first time that I read it explicitly.
thank you for reading and taking the time to comment, this has been helpful. if you learned something from this, then this means i'm on the right track
>>
>>13894120
>i'm pretty sure this is standard nomenclature, e.g. "Weyl's unitarian trick" or "Rabinowitsch trick"
Why not. I never read the equivalent in French, but if you're sure, go ahead.
>>
>>13894120
>similar matrices have equal characteristic polynomials
isnt this direct from jordan canonical form?
>>
>>13894131
ah, so you're a frog? merci beaucoup pour reading
i always wonder, how do you guys don't get cognitive dissonance from the conflicting definitions (i.e. yours and that of others), e.g. when you call numbers \geq0 positive and so on
>>13894138
kek no, that's determinants 101
[math]\det(tI-PAP^{-1})=\det(P(tI-A)P{-1})=\det(tI-A)[/math]
>>
>>13894147
>when you call numbers \geq0 positive and so on
That's your conventions that are crazy, not ours :D How could you call the natural numbers the non-negative integers? That's insane!
>>
>>13892791
Tarski was retarded
>for all X, Y ⊆ R, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.
Let X = (0,1), Y = [1,2]
>>
>>13894499
z=1?
>>
>>13894506
wtf that doesnt align with my one line gotcha moment for a well studied set of axioms
will burger told me the reals were broken everywhere and all the time!
>>
>>13888134
i = 0.000003
>>
>>13889674
Heyyy shifrin and tu, very rare good taste
>>
>>13888551
Order them within a Hilbert curve
>>
>>13894956
Hilbert curves have many self-intersections
>>
Are there any interesting extensions to the riemann integral that don't agree with the lebesgue integral?
>>
My GPA is going to be officially less than 3.0 now and I'm pretty screwed as far as graduate school.





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