/sci/ - Science & Math

>>11671193
tfw algebra 2

>>11671202

Why is there a meaningful difference between saying that two things are equal and that two things are the same? Why did the people who developed various theories to do with the fundamentals decide upon this? In the same way that you can do math with any axioms you like and see what you get, can you do set theory with the axiom that sameness and equality aren't two different things but are both 1 thing?

>>11671244
>two things are the same
Mathematically speaking this is meaningless.

>>11671248
Replace same with equivalent, or consider the possible versions of my post where I used the word isomorphic or some other related concept. Has anybody studied the axiomatic system where none of those concepts have any meaningful differences?

>>11671244
Typically, equal means set-theoretically equal and the same means isomorphic in the considered category.

>>11671244
>>11671287
>Why is there a meaningful difference between saying that two things are equal and that two things are the same?
I don't want to get into language games. Informally, you never properly capture what you talk about, but you always need an informal level to communicate. Sentences like
>Two sides of an equilateral triangle are the same
>The groups of two elements "(0,1) and addition mod "2, as well as "(1,-1) and multiplication" are the same.
Are two ways to start a discussion and distionction.

It would be easier to give you a satisfying answer if you pin your words down a bit, maybe even formalize things a bit. A lot can be said about equality and fields of math and philosphy are dedicated to it.
One must also take good care about mathematics "in-theory" statements, their meta-theoretical ramifications as well as those on the informal meta-level thereof.
Keywords are structural vs. material theories, extenionality and intentional equalities, substitution, isomorphism, explicit equivalence or apartness demands in constructive theory, etc., etc.
Here's some link relating to ways to approach the broader issue

https://ncatlab.org/nlab/show/structural+set+theory
https://en.wikipedia.org/wiki/Intuitionistic_type_theory
https://en.wikipedia.org/wiki/Confluence_(abstract_rewriting)
https://en.wikipedia.org/wiki/Apartness_relation

>Has anybody studied...
Yes, the philosphy of 40's category captures this indirectly - two objects found to be isomorphic (like the (0,1) and (1,-1) example, as modeled in a category of sets, found to be groupy objects) are understood of having invertible arrows between them such that one can translate the objects between any diagram expressing one and the same thing.
Martin Löf Type theory formalizes this, even.

>can you do set theory with...
Check out Bishop set theory for one view on it. Also afaik you get far even without extensionality in set theory, by considering equivalence classes.

Why are quotients of PIDs not also PIDs?

By the correspondence theorem, ideals [math]I[/math] of a PID [math]R[/math] containing some ideal [math](a)[/math] are in bijective correspondence with ideals [math]I+(a)[/math] in [math]R/(a)[/math]. But [math]I=(b)[/math] in [math]R[/math], so how or why does it not descend to [math](\pi (b))[/math], where [math]\pi[/math] is the projection.

>>11671314
Thanks

>>11671323
On second thought, they will not be domains unless of course, [math](a)[/math] is prime - however, the question stands - why are quotients of PIDs not principal ideal rings

>>11671331
Do the two SE threads (about primes, as you say) maybe help for the general case? Have you googled?
https://math.stackexchange.com/questions/2241356/quotient-of-a-pid-by-a-prime-ideal-is-a-pid-too/2241360
https://math.stackexchange.com/questions/32143/proving-the-quotient-of-a-principal-ideal-domain-by-a-prime-ideal-is-again-a-pri

>>11671327
Here's that referenced answer of Joel David Hamkins suggesting you can more or less savely drop Extensionality

https://mathoverflow.net/questions/168287/is-there-any-research-on-set-theory-without-extensionality-axiom

Also, so i can join the weebs and post anime girls with math books, try to understand what the Univalence Axiom wants
https://en.wikipedia.org/wiki/Homotopy_type_theory#The_univalence_axiom

(both of the two above wills will need some knowledge of the frameworks)

Also worth pointing out that iirc in the utterly failed New Math movement, numbers were informally defined as the class of all things of which there are those number of things.
(E.g. if there happen to be 31525264 apples in the world right now, the notion of apples is in the class of those things of which there are 31525264 many, and this class then "is" considered to be "the number" 31525264)
https://en.wikipedia.org/wiki/New_Math
I just mention that as a curiosity and because this idea has some history.

>>11671323
https://en.wikipedia.org/wiki/Principal_ideal_ring
>4. The quotient of any PIR is a PIR

>>11671331
>>11671323
But they are. Let R be a PID, S = R/I for some ideal I. If J is an ideal in S, its preimage in R is an ideal in R, so is generated by an element a. Then a+I generates J in S.

>>11671331
Do the two SE threads (about primes, as you say) maybe help for the general case? Have you googled?
https://math.stackexchange.com/questions/2241356/quotient-of-a-pid-by-a-prime-ideal-is-a-pid-too/2241360
https://math.stackexchange.com/questions/32143/proving-the-quotient-of-a-principal-ideal-domain-by-a-prime-ideal-is-again-a-pri

>>11671327
Here's that referenced answer of Joel David Hamkins suggesting you can more or less safely drop Extensionality
https://mathoverflow.net/questions/168287/is-there-any-research-on-set-theory-without-extensionality-axiom

Also, so i can join the weebs and post anime girls with math books, try to understand what the Univalence Axiom wants
https://en.wikipedia.org/wiki/Homotopy_type_theory#The_univalence_axiom

(both of the two above wills will need some knowledge of the frameworks)

Also worth pointing out that iirc in the utterly failed New Math movement, numbers were informally defined as the class of all things of which there are those number of things.
(E.g. if there happen to be 31525264 apples in the world right now, the notion of apples is in the class of those things of which there are 31525264 many, and this class then "is" considered to be "the number" 31525264)
https://en.wikipedia.org/wiki/New_Math
I just mention that as a curiosity and because this idea has some history

>>11671348
What the fuck happened with her hands?

>>11671357
Donno, it's just a comic ya kno.

Anyway, I'm gonna read this paper today

https://arxiv.org/abs/0808.0754
>A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions

Who's interest, read it too and let's discussion

>>11671378
memetitles were a mistake

>>11671468
Admittedly.
Although I once made a list of some good ones

>Hodge's general conjecture is false for trivial reasons
>You Could Have Invented Spectral Sequences
>A minus sign that used to annoy me but now I know why it is there
>How not to prove the Poincare Conjecture
>Ramanujan's association with radicals in India
>Is the null-graph a pointless concept?
>Everybody knows what a Hopf algebra is
>On O_n
>Free rings and their relations
>Six standard deviations suffice.
>The importance of being straight
>The homotopy category is a homotopy category
>Division by three
>A Group of Order 8,315,553,613,086,720,000
>Holey Sheets
>On groups of order one.
>K-Theory and Reality
>The Joy of Sets.
>Applied Mathematics is Bad Mathematics
>Making the most out of zero branes and a weak background
>Can one hear the shape of a drum?
That one even has a Wikipedia article https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

also
http://www2.tcs.ifi.lmu.de/~jjohanns/cute.html

also, live in 15 minutes,
>MIT Categories LIVE - Mike Shulman: Conservativity of duals
Can't say if it's going to be understandable.

https://youtu.be/GwtChA9btgU

Any math that can be used to write malware ?

>>11671193
Have you guys ever met people with actual potential?
I had a calculus prof once that was a math major from my uni.
throughout his undergrad years, he had a shit ton of email correspondence with a field medalist prof from a very high tier uni that I wont name, said professor ended up taking him to his uni to do his phd under his tutelage as the only student being tutored by him.
My ex-prof has already published some new theorems and hes not even done with his phd.
needless to say he is the most brilliant man i've ever met and actually made a big influence on me, I wouldnt be surprised if he actually ends up winning a fields medal.
frankly I consider myself lucky to have not only met but been taught by someone that smart, it really changed the paradigm for me.

>>11671348
Fuck off tranny

>>11671947
>everyone who posts anime is a tranny
You're almost as bad as the trannies themselves.

>>11671947
>>11671954
>Everybody who posts anime images is a tranny
This is not true. There are plenty of losers who aren't trannies

>>11671974
creative

bros...

>>11671193
Can this problem be solved?
Define N.
O=42, P=43

3**(1+6x%o)=n
3**(4+6x%o)=n+1
3**(5+6x%o)=n+2

>>11671193
I'm pretty sure the one guy trying to make this book a meme has not actually read any of it
It's really fucking bad

>>11672426
This. Just read Jacobson.

>>11672426
IIRC the meme with it were the exercises.

Can someone suggest a book to self study algebraic geometry and algebraic topology this summer. So far I've taken some undergraduate/grad classes in abstract algebra and am familiar with some of the introductory concepts in alg geom like hilbert nullstellensatz and generally working in affine spaces. Some other background I have that may be useful is some complex analysis and analysis on manifolds. Not looking for anything too crazy just a good first text at the grad level.

>>11672463
>algebraic topology
Rotman.

>>11672426
>I'm pretty sure the one guy trying to make this book a meme has not actually read any of it
I'm not a "guy".

>I'm not a "guy".

How good are Milne's notes for learning algebraic geometry? And no, I'm not just going to read it and find out.

>>11672541
just read it and find out

Why is it so hard to prove conjectures
Why can't I just publish a paper where my argument is "it just feels right bro"

>>11672541
>How good are Milne's notes for learning algebraic geometry?
Why don't you read them and find out?

>>11672463
>Can someone suggest a book to self study algebraic geometry and algebraic topology this summer.

>>11672541
I'll have you read them and write me a report on them. Then I will read your report and tell you if the notes are good.

>>11671193
Last night someone tried to tell me countable infinity > uncountable infinity because "there are too many numbers man." Someone put me down, already.

>>11672585
How do you know uncountable infinity is bigger if you can't count the elements? Atheists 0 - Luciferian Cyberbuddhists 1

>>11672458
There are other issues (there's absolutely no attempt at organization, he just dumps random shit in random order even if it's literally backwards to how the theory should go), but the problems are probably the biggest ones, yes
The breakdown of your average end-of-chapter is something like
>2-ish problems that are literally just wrong (not unsolvable, wrong)
>another 3-4 problems that don't have a solution
>1-2 problems nicked from research papers
>3-4 hideous numeric problems that take up a page to several pages each
>10 3-liner trivialities
>5 decent tough problems
I read through a pretty decent chunk of the book(s) because I really like the idea of it (it is unique, and writing an algebra book that focuses on geometry and combinatorics instead of autistic structure theory is a nice idea), but it's just poorly executed.

>>11672594
That is a much better argument than what he gave. He was emotional and frantic.

>>11672600
>>2-ish problems that are literally just wrong (not unsolvable, wrong)
>>another 3-4 problems that don't have a solution
>>1-2 problems nicked from research papers
>>3-4 hideous numeric problems that take up a page to several pages each
>>10 3-liner trivialities
>>5 decent tough problems
What's the issue with this?

>>11672594
>>11672603
ZF only needs a minor tweak and then it's already consistent to postulate the subcountability of all uncountable sets. I think the intuition is that you postulate that all sets that provably exist have algorithms associated with it that enumerate their elements in a dovetailing procedure (although this computer science stuff isn't part of the formalism). Although I forgot what the coding of [math] {\mathbb R} [/math] is there.

>>11672692
You would need a much stronger argument than something you don't remember for me to suddenly think ZFC is close to having intervals be subcountable. The tweak would need to make the diagonal argument invalid.

>>11672730
>The tweak would need to make the diagonal argument invalid.
No, sets such as [math] {\mathbb R} [/math] would still be uncountable for the same reasons.

>ZFC
ZF
Since
https://en.wikipedia.org/wiki/Diaconescu%27s_theorem
spoils all the fun models

>>11672769
Thanks anon

>>11672806
There's some related yikes insights, such as the Specker sequence.

Of course it's relatively easy to define some uncomputable reals by case-wise definition on their decimal expansion. For any predicate P, let [math]a_n^P[math] be 1 is [math] P(n) [/math] is true and 0 otherwise. Since [math]\sum_{n=1}^\infty \dfrac{1}{2^n}=1[/math], we have that any [math]X^P:=\sum_{n=1}^\infty \dfrac{a_n^P}{2^n}[/math] is a real with [math] 0 \le X^P \le 1 [/math]. Now enumerate all textfiles by dovetailing over the number of characters resp. ascii characters from 1 to 256. All C++ programs will be among them. It's uncomputable whether a program in a Turing complete machine will halt. Let P be whether the n'th text in your enumeration is a valid C++ program that will halt and you got yourself a number _defined_ between 0 and 1 that has digits you can't compute.
Now Specker's insight finds you can't even escape such shitty incidents by restricting yourself to monotonically increasing definitions of numbers.

Of course, you can take LEM as axiom, and then a corollary of your theory is that
[math] \forall (n\in {\mathbb N}). \ (a_n=0) \lor (a_n=1) [math]
even though you know that there's infinitely many [math] k\in {\mathbb N} [/math] for which the value of a_k can really only be accessed by Plato in heaven.

The [math] n [/math]'s for which you can find the value of a_n is CE, since you can zic-zac-parallelize the C++ program execution in the same way you can run through all rational numbers and those programs that halt are in your set. But that set isn't all of [math] \mathbb N [/math] and you will never be able to know which n'th are wasted computation that will run forever.
That's what I mean with computer sciency dovetailing intution. Not all infinite subsets of [math] \mathbb N [/math] are in constructive bijection with [math]\mathbb N [/math], and there's your loophole where you can assets that all infinite uncoutable sets are actually of this type

>>11672806
There's some related yikes insights, such as the Specker sequence.

Of course it's relatively easy to define some uncomputable reals by case-wise definition on their decimal expansion. For any predicate P, let [math]a_n^P[math] be 1 is [math] P(n) [/math] is true and 0 otherwise. Since
[math]\sum_{n=1}^\infty \dfrac{1}{2^n}=1[/math],
we have that any
[math]X^P:=\sum_{n=1}^\infty \dfrac{a_n^P}{2^n}[/math]
is a real with [math] 0 \le X^P \le 1 [/math]. Now enumerate all textfiles by dovetailing over the number of characters resp. ascii characters from 1 to 256. All C++ programs will be among them. It's uncomputable whether a program in a Turing complete machine will halt. Let P be whether the n'th text in your enumeration is a valid C++ program that will halt and you got yourself a number _defined_ between 0 and 1 that has digits you can't compute.
Now Specker's insight finds you can't even escape such shitty incidents by restricting yourself to monotonically increasing definitions of numbers.

Of course, you can take LEM as axiom, and then a corollary of your theory is that
[math] \forall (n\in {\mathbb N}). \ (a_n=0) \lor (a_n=1) [/math]
even though you know that there's infinitely many [math] k\in {\mathbb N} [/math] for which the value of a_k can really only be accessed by Plato in heaven.

The [math] n [/math]'s for which you can find the value of a_n is CE, since you can zic-zac-parallelize the C++ program execution in the same way you can run through all rational numbers and those programs that halt are in your set. But that set isn't all of [math] \mathbb N [/math] and you will never be able to know which n'th are wasted computation that will run forever.
That's what I mean with computer sciency dovetailing intution. Not all infinite subsets of [math] \mathbb N [/math] are in constructive bijection with [math]\mathbb N [/math], and there's your loophole where you can assets that all infinite uncoutable sets are actually of this type

>>11672806
There's some related yikes insights, such as the Specker sequence.

Of course it's relatively easy to define some uncomputable reals by case-wise definition on their decimal expansion.
For any predicate P, let [math]a_n^P[/math] be 1 is [math] P(n) [/math] is true and 0 otherwise.
Since
[math]\sum_{n=1}^\infty \dfrac{1}{2^n}=1[/math],

we have that any
[math]X^P:=\sum_{n=1}^\infty \dfrac{a_n^P}{2^n}[/math]

is a real with [math] 0 \le X^P \le 1 [/math]. Now enumerate all textfiles by dovetailing over the number of characters resp. ascii characters from 1 to 256. All C++ programs will be among them. It's uncomputable whether a program in a Turing complete machine will halt. Let P be whether the n'th text in your enumeration is a valid C++ program that will halt and you got yourself a number _defined_ between 0 and 1 that has digits you can't compute.

Now Specker's insight finds you can't even escape such shitty incidents by restricting yourself to monotonically increasing definitions of numbers.

Of course, you can take LEM as axiom, and then a corollary of your theory is that
[math] \forall (n\in {\mathbb N}). \ (a_n=0) \lor (a_n=1) [/math]
even though you know that there's infinitely many [math] k\in {\mathbb N} [/math] for which the value of a_k can really only be accessed by Plato in heaven.

The [math] n [/math]'s for which you can find the value of a_n is CE, since you can zic-zac-parallelize the C++ program execution in the same way you can run through all rational numbers and those programs that halt are in your set. But that set isn't all of [math] \mathbb N [/math] and you will never be able to know which n'th are wasted computation that will run forever.
That's what I mean with computer sciency dovetailing intution. Not all infinite subsets of [math] \mathbb N [/math] are in constructive bijection with [math]\mathbb N [/math], and there's your loophole where you can assets that all infinite uncoutable sets are actually of this type

>>11672730
>>11672892
And mind you, Cantors argument is still valid regarding [math] [0,1]\subset {\mathbb R} [/math].
It says there's no bijective function
[math] b : {\mathbb N} \to [0,1] [/math].
Indeed, but for the subcountable model you can eventually (letting the procedure run forever) pop out all
[math] [0, 1]\mapsto [math]
in a countable fashion and say the reals is no more than what you witness like that.

>>11672730
>>11672892
And mind you, Cantors argument is still valid regarding [math] [0,1]\subset {\mathbb R} [/math].
It says there's no bijective function
[math] b : {\mathbb N} \to [0,1] [/math].
Indeed, but for the subcountable model you can eventually (letting the procedure run forever) pop out all
[math] [0, 1]\mapsto [/math]
in a countable fashion and say the reals is no more than what you witness like that.

>>11672980
[math] \zeta(x+i\,y) = \dfrac{2^{i\,y}}{2^{i\,y}-2^{1-x}}\cdot\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{n^x}\left[\cos(y\,\phi_n)-i\sin(y\,\phi_n)\right] [/math]

>>11672980
the GOAT

[math] \phi_n = \log(n) [/math]

>>11672980
>A wild challenger approaches

Is -0.1 with infinite ones equal to 0?

>>11673014
no, that's one over -9

>>11673027

0.9999999999999 = 1

0.9999999999999 - 1 = 1 - 1

0.111111111111111111111 = 0

>>11673075
>0.999 + 0.111 = 1

>>11673077
Yes :)

Let [math]R[/math] be a commutative ring with identity, and [math]\phi:R^m\to R^n[/math] a module homomorphism. If [math]\mathfrak m[/math] is a maximal ideal, then we have an induced vector space homomorphism [math]\phi_{\mathfrak m}: R^m/\mathfrak m R^m\to R^n/\mathfrak m R^n[/math].

In particular, this can be used to show that if [math]\phi[/math] is an isomorphism, then [math]m=n[/math], and if [math]\phi[/math] a surjection, then [math]m\geq n[/math], by basic dimension theory. However, the same argument cannot be used to show [math]m\leq n[/math] if [math]\phi[/math] is an injection. Why?

>>11673133
I expect one demonstrates some example where phi maps points not along single vector's span to points along a single vector's span

>>11673170
It is however true that an injection implies [math]m\leq n[/math], so I'm not sure if I could find that

Ive been working through naive set theory by halmos and had trouble with an exercise, so I looked up the solution
In the pic the guy asserts that the union of all natural numbers x that correspond to y in n must be an element of [math]\omega[/math]
But if the set of such x was infinite would that mean that the union of all such x is equal to [math]\omega[/math]
which would then make this an incorrect assumption?

>>11673133
I'd imagine that it's because an injection between the modules doesn't necessarily induce an injection between the vector spaces.
Consider, for example, [math]C[x][/math] as a module over itself, the injection given by multiplication by [math]x[/math], and the maximal ideal being the one generated by x.

>>11673184
It's assuming that omega is finite. Why would be the union of the subsets of a finite set be infinite?

>>11673177
wat, looks to me like you restated what you said before

>>11673184
This proof looks strangely long. Let me get this straight
You know that f is injective and it's codomain is finite but still need to argue that the domain of n is finite?
And also, you can't use that elements of w are finite?

>>11673177
wat, looks to me like you restated what you said before

>>11673184
This proof looks strangely long. Let me get this straight:

You know that f is injective and it's codomain n is finite but still need to argue that the domain of f is finite?
And also, you can't use that elements of w are finite?

if it isnt clear w is the set of natural numbers

>>11673231
>You know that f is injective and it's codomain is finite but still need to argue that the domain of n is finite?
yes figuring that out is the whole point of the exercise

>>11673228
finiteness has so far only been defined as a bijective function between some n in w and another set X existing without anything else proven so it should not be used to assert anything

>>11673133
Write [math]\pi_m,\pi_n[/math] for the maps onto the vector spaces. This argument is basically looking at the fact that [math]\phi_m \circ \pi_m = \pi_n \circ \phi[/math] . This can be used to get surjectivity of the vector space map (since compositions of surjections are surjective), and if phi is invertible then you can get two surjections in either direction, but you can't get injectivity out of this because [math]\pi_n[/math] will probably kill it off.

im taking grad algebra and real analysis next semester. what am i in for?

>>11673309
algebra and real analysis

>>11673309

>>11673270
>yes figuring that out is the whole point of the exercise
But the prove of that shouldn't rely on omega at all, it's true about every non-finite set.

Also, do you have Regularity (which rules out that w is an element of w)?

Can a physics Anon double check my math on this problem? It should be really simple but I haven't taken math or physics in 5 years and I'm worried I might have screwed something up figuring out the torque

>>11673291
>>11673227
>>11673170
I think the most appealing answer is that [math]\otimes_R R/\mathfrak m[/math] is right exact, so isomorphisms and surjections being preserved is immediate, while injectivity needs an actual argument.

>>11673380
dilate categorytranny

>>11673404
yikes

>>11673270
It's a proof by contradiction. It's assuming that omega is finite; therefore there exists a bijection from it to some natural number. Therefore, the number of elements of omega is finite, and since each member of omega is a natural number, one can define a bijection from it to itself, thus making those elements also finite. Finally, the union of these would also be finite if one were to combine the bijective functions into a single function.

>>11673380
Breddy nice.

how tf do you guys find a way to sit and study for so long? i want to learn more math but i cant sit down and read on my own

>>11673486
Find something you're interested in.
Train to be able to turn your phone off for 3 days without sperging.

>>11672917
Can you elaborate on why the procedure that eventually pops out all of
[math][0,1] \to {\mathbb N}[/math]
does not qualify as a function or at least let us make a bijection?
what are the functions here and how are they made?
are they TMs for Cauchy sequences?

>>11671885
I talked to a fields medalist for maybe 5 minutes one time. Really nice dude, and he was surprisingly normal. Felt exactly like talking to any other nobody prof at my campus.

>>11673553
Now I don't think you can use runtime-observed behaviour of "popping out" to turn the function around, thereby obtaining a bijection.
The set theories I had in mind definitely don't access definition/intentional aspects of procedures. I.e. they don't have a lambda calculus or something (again, the CS stuff is my intuition for the models.)

There are (were) various schools in those corners of math and various definitions of functions,
see e.g.
https://math.stackexchange.com/questions/176279/all-real-functions-are-continuous

Also want to highlight I also don't imagine the assertion that all sets are subcountable is constructive either.
I'm walking on thin ice here, though.

>>11673553
No I don't think you can use runtime-observed behaviour of "popping out" to turn the function around, thereby obtaining a bijection.
The set theories I had in mind definitely don't access definition/intentional aspects of procedures. I.e. they don't have a lambda calculus or something (again, the CS stuff is my intuition for the models.)

There are (were) various schools in those corners of math and various definitions of functions,
see e.g.
https://math.stackexchange.com/questions/176279/all-real-functions-are-continuous

Also want to highlight I also don't imagine the assertion that all sets are subcountable is constructive either.
I'm walking on thin ice here, though.

This also looks related

https://cs.stackexchange.com/questions/12664/where-am-i-wrong-countability-and-recursive-enumerability

>>11672486
interestingly enough, he taught at my school until he passed. wish i had the chance to meet him

>>11673629
On second thought, the CS-SE threads don't seem so helpful here, since they all assume a strong notion of countable (all subsets of N always seem to be taken as countable there).

Trying to do calculus in my final year of uni after not taking math forever, and I'm rusty.

I have been doing okay so far, but I have forgotten everything about radians and how they work.

Tried to look at khan academy for tutorials but they are poorly formatted or I dont know where to look. How would I figure out where to even begin here? I could solve this if it was normal numbers

>>11673868
I'm more looking for a title to google for tutorial rather than an explanation from you guys btw

>>11673868
>normal numbers
radians are "normal numbers"
they're some of the most "normal" ones
i don't know what to tell you

>>11672486
Why that over Hatcher? I ask because I have read Hatcher and not Rotman.

>>11673868
Select b to match the values at pi/2 (= 90 degrees) then select a to match the values at -pi/4 (= -45 degrees).

>>11673968
>Why that over Hatcher?
Hatcher is a meme.

>>11673975
Literally everyone in grad school suggested it, jesus.

>>11673974
>>11673974
Yeah I get how to do it and I eventually solved it but it took forever because i have no idea when to convert these radians / trig equations into their numbers like 1/sqrt(2) etc to actually proceed with solving.

Didn't help that when I plugged it into wolfram alpha the equation didn't solve correctly (probably due to bracket issue). Had to do the calculation piece by piece. There's probably some trig rule where sin/cos cancel out but I forgot it. Anyway, that took forever.

>>11674110
>There's probably some trig rule where sin/cos cancel out but I forgot it.
sin/cos is tan

>>11674119
Nice, yeah that would have made that last part a lot easier. I should have had identities open. It's not easy relearning years of math in a week. I forgot how good it feels to solve math problems

>>11673868
This task is incredibly simple, all you need are the basic trig values and being able to multiply/divide.
https://www.desmos.com/calculator/dj6pn3ymes

I want to do abstract algebra and the class has some vector stuff. Anyway whats a good book to refresh vector geometry quickly?

>>11671193
What the fuck is a "Ring"

I have an understand of sets, but rings? my dissertation is on robotics applications of convex sets

>>11674390
a ring is a set where you can add stuff and multiply stuff
e.g. the integers are a ring, polynomials are a ring, matrices are a ring

Is there an elegant way to simplify to e the limit definition of e without L'hopital?

>>11673968
I've read Rotman and can confirm it's good. I don't know if it beats Hatcher, since I haven't read his book.

>>11674630
and she does it again!! a totally worthless post

>>11674390
Ring is a set where you can add, SUBTRACT and multiply stuff.
>>11674405
>a ring is a set where you can add stuff and multiply stuff
(N,+,*) is not a ring

>>11673553
>>11673629
Via your remark, I spend some time digging yesterday and came to the conclusion that models with a computable "popping out" procedure for [math] [0,1] \subset {\mathbb R} [/math] was probably wishful thinking.

I refactored the following Wikipedia article quite a bit, it should now give a coherent intuition for set theories in which
[math] | 2^{ {\mathbb N} } | > | {\mathbb N} | [/math]
is independent (and in particular not provable). When trying to connect the computability talk with the reals, it's probably best to view [math] {\mathbb R} [/math] representable by monotone functions [math] {\mathbb Q}\to {0, 1} [/math] in the sense of Dedekind cuts, e.g.
[math] \sqrt{7} := r \mapsto \left( 0\ {\mathrm {if}}\ (r^2 < 7)\ {\mathrm {else}}\ 1 \right) [/math].

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

>>11673553
>>11673629
Via your remark, I spend some time digging yesterday and came to the conclusion that models with a computable "popping out" procedure for [math] [0,1] \subset {\mathbb R} [/math] was probably wishful thinking.

I refactored the following Wikipedia article quite a bit, it should now give a coherent intuition for set theories in which
[math] | 2^{ {\mathbb N} } | > | {\mathbb N} | [/math]
is independent (and in particular not provable), where the power on the left is understood as all there is to the function space.
When trying to connect the computability talk with the reals, it's probably best to view [math] {\mathbb R} [/math] representable by monotone functions in the sense of Dedekind cuts,
[math] {\mathbb Q}\to \{0, 1\} [/math]
e.g.
[math] \sqrt{7} \ :=\ r \ \mapsto\ \left( 0\ {\mathrm {if}}\ (r^2 < 7)\ {\mathrm {else}}\ 1 \right) [/math].

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

>>11675276
(and then elements of [0,1] not being "poppable" due to Rice's theorem, https://en.wikipedia.org/wiki/Rice%27s_theorem))

https://www.math.purdue.edu/~ppatzt/Stability2020/
Stability in Topology, Arithmetic, and Representation Theory

>>11674825
Aww!

bros....

Threadly reminder to sell your soul to physicists.

Is this the most autistic version of the nullstellensatz?

>>11676408
What's special about that one?

How to verify this identity?
cscx + sinx = (2-cos^2x)/sinx

>>11676565
what have you tried?

>>11676565
Start with the definitions and use sin^2 cos^2 = 1.

why am i so shit?

>>11676603
because you are faggot

>>11676603
god is punishing you for your homosexuality
repent sinner

>>11676612
>>11676614
is wanting to get fucked by a cute girl with a strapon really makes me gay and thus shit?

>>11676631
go get fucked by our resident tranny

>>11676672
>our resident tranny
you're gonna need to be a bit more specific than that

>>11676684
this slut >>11676316

>>11676699
>tfw when yukarifag is our resident tranny anymore
Why live?

>>11676408
>no sheaves
>no category theory
That's not even more autistic than the Ruckert Nullstellensatz for complex spaces.
Absolutely weak sauce.

>>11673868
For f to be continuous, for any a in its domain, we must have that lim_(x->a) f(x) = f(a). Set a = pi/2. If the limit exists, then so do the left and right-hand limits, so lim_(x+->pi/2) bsin(x) = pi/2 = b*lim_(x+->pi/2) sin(x) = b*1 (I can never remember whether its + or - here, stupid fucking notation). Then b = pi/2. Do something similar with acos(x)

Is there any greater honor than this?

>>11674379
>vector geometry
You don't need this for abstract algebra.

Where can I find a nerdy pure math major BF?

>>11676797
Are you a biological female?

>>11676837
Yes

So I found these on wikipedia
>Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.[4]
>There exist five three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b.
And it sounds interesting but I've just started linear algebra. I appreciate that it's gonna take a while to get there but what topic in math is this a part of? On seeing it my immediate response is to ask that if we have found working three dimensional algebras why do we need the quaternions for three dimensional math, but I recognise that I don't know enough math yet to understand whatever the answer to that question is.
Once I reach mathematical maturity, what topic should I be looking into to study the applications or lack thereof of these things?

>>11676797
What standards are you looking for? Is good personal hygiene, some motivation in household tasks, disinterest in media targeted to kids, being ok to boring in conversation, lower middle class wealth and average attractiveness enough? I'm in Colorado and looking for somebody I can do better than just tolerate the company of.

>>11676893
>what topic in math is this a part of
Abstract algebra.
>if we have found working three dimensional algebras
The quaternions are not three dimensional over the complex numbers. There is the relation ij = k, so we need only 1 and j to generate the quaternions as a complex algebra = a ring that is also a vector space over the complex numbers.
>what topic should I be looking into to study the applications or lack thereof of these things
No idea about real life applications, but mathsy stuff that has "algebraic" in the name is a good place to start.

>>11676699
>>11676672
Rude, fampai!

>>11676859
post pusy

>>11676944
i don't want to get banned, but if you post peepee then i will

>>11676914
Let's meet up sometime

>>11676893
>but what topic in math is this a part of?
You could say it's it's own subfield of abstract algebra, namely the one studying associative algebras. It's quite close to linear algebra.

>>unital associative algebras over the field of complex numbers
>On seeing it my immediate response is to ask that if we have found working three dimensional algebras why do we need the quaternions for three dimensional math
The quaternion algebras are 4 dimensional over the reals. What you're thinking of is the Hamiltonian quaternions restricted to unit norm (one equation, thus bounding it to 3 dimensions), which induce a representation covering the rotation group.
In that representation, the tangent space is one of quaternions with unit direction set to zero, another condition. It's relevant. But it's also not the only rep.
>You best get used to associative algebras, you're already in one
Over C, the quaternions are 2 dimensional.
Let me shill those notes on the matter
https://youtu.be/tkS_6xY132g
https://youtu.be/iotcTNB5Q8g
https://youtu.be/gBMdTSXhYsY

I also saw a book pdf (with more of a cohomology and number theory flavour I think) being handed around here 2 threads ago, that seemed related.

>what topic should I be looking into to study the applications or lack thereof of these things?
I deal with quaternions at my job 5 days a week, so... mechanics is one direction (Aerial navigation, Robotics, Computer Graphics). Here's a good sort of practical (probability theory oriented) writeup slash formula collection (only concerned with the Hamiltonian quaternions)
http://www.iri.upc.edu/people/jsola/JoanSola/objectes/notes/kinematics.pdf

Here's the kind of application this relates to
https://youtu.be/LblxKvbfEoo

Why was I taught differentiation as my intro to calculus as part of my Engineering Maths module instead of limits? Limits are usually the first thing introduced in calculus afaik

>>11676991
Basic differention is easier for brainlets than basic limits so your class was probably for brainlets

hello bros i want to give online live tutoring to a friend but i dont really know what tools should i use to show math on screen, drawing equations in paint/gimp doesn't seem like a good idea same with LaTeX as im not really well versed in it and it usually takes a little bit for it to compile to something readable so bros do you know of any tools that would allow me to show good looking math in a efficient way?

>>11677314
point a webcam at a piece of paper

>>11677332
sadly i dont really have a good enough webcam to do it

>>11677336
do you not own a cellphone? even your phone camera would be sufficient to see a sheet of paper

>>11677314
>>11677336

>>11677314
Do you have a drawing pad? I used to teach my bf stuff using that. No need for LaTeX, but you still get pretty looking text even with Paint.

https://www.youtube.com/watch?v=Z3kci1dE3yc

bros she has the euler hat on

>>11671193
hello I want to learn more about stats and stochastic processes
do you guys have any recommendations?

>>11677314
>>11677348
So here's what's probably the minimal, 10 line html solution. Save it as a .html file and then you should be able to drag it into a web browser. Edit it in a text editor and just refresh the page to re-render. Needs internet connection.

https://gist.github.com/Nikolaj-K/00e425a2857cd4a0991f504d14276c30

>>11677423
>doxing yourself on /sci/
i shiggy diglett

germans ...

>>11677449
>>doxing yourself on /sci/
>i shiggy diglett
What do you mean?

>>11677343
i do own one but dont really have a good way to setup it up
>>11677357
>Do you have a drawing pad?
if i had one i wouldnt be asking this question
>>11677423
thank you very much seems like a nice solution to my problem

>>11677462
are you asking what shiggy diglett means or are you still missing the fact that you just posted a bunch of personal info on 4chan?

>>11677462
https://math.stackexchange.com/users/18993/nikolaj-k

>>11677470
>if i had one i wouldnt be asking this question
This would have been obvious maybe 10 years ago, but with the trend of people online becoming more and more incapable of using their brains... you get what I mean.

>>11677462
Respect for having Nick Land in a thumbnail.

Nighty-night, /mg/.

>>11677472
That's not me you responding to there.

Anyway, 4chan is among the 1000 most visited websites on the web, I don't think there's much to it.

>>11677491
gnmg

>>11677503
>Anyway, 4chan is among the 1000 most visited websites on the web
That's the point. There are a lot of people here, and all of them are anonymous. Some of them are raging assholes. All it takes is one guy who thinks you might enjoy 800 emails a day about Chinese penis enlargement herbs to make posting your email here a stupid idea

Is the following correct?

> Let [math]\phi:R\to S[/math] be a finite homomorphism of rings. Show that for any prime ideal [math]p\subset R[/math] there are only finitely many prime ideals [math]q\subset S[/math] such that [math]q\cap R=p[/math].

Let [math]T:=\phi(R-p)[/math]. If [math]\phi[/math] is finite, then the induced morphism [math]\phi':R_p\to S_T[/math] is also finite. We also have another induced finite morphism [math]k:= R_p/pR_p\to A:= S_T/\phi'(p)S_T[/math]. The field [math]k[/math] has a unique proper ideal corresponding to [math]p[/math], and [math]A[/math] is a finite [math]k[/math]-algebra, so in particular, it has a finite amount of maximal ideals, which correspond precisely to all the primes [math]q[/math] that pull back to [math]p[/math].

>>11677523
I occasionally engage with Indians sending me their General Relativity alternatives. If someone goes as far as to finding my inquiry mail and writing a spam bot with supplement ads, I'll respect the man.

Will we ever get an explicit well-ordering for R or is this task impossible?

>>11677546
https://math.stackexchange.com/questions/6501/is-there-a-known-well-ordering-of-the-reals/6504#6504
"no"

>>11677546
Without AoC it does not exist
With AoC is does exist but is nonconstructable

>>11677449
Nikolaj already posts his videos here every now and then, anon.
https://www.youtube.com/channel/UCcrSMnEYhIPX_p127jI23qw

>>11677314
mathanim

how long does it take for amphetamines to start charging it's toll|? this seems too good to be true

>>11677755
anywhere from a few months to never. all depends on how hard you're going at them
if you're popping them like candy all day long you're going to run into issues pretty fast, if you use them for 1 week every semester before finals you're never going to encounter any tolerance issues

How do i show that Bousfield localization preserves fibers?

>>11677586
I wish I was at my stuff as good as he is at his.

>>11678481
Suppose you have fibration sequences [math]F \xrightarrow{i} E \xrightarrow{p} B[/math] and [math]F' \xrightarrow{j} E' \xrightarrow{q} B'[/math]. There is a result* saying that if you have maps [math]f\colon F\to F', e\colon E\to E', b\colon B\to B'[/math] such that [math]e\circ i = j\circ f, b\circ p = q\circ e[/math] and any two of the three are localisations, then so is the third one. Now, suppose [math]e, b[/math] are precisely that (we don't assume that the map between the fibres exists, we don't know it yet!), and consider the composite [math]q\circ e\circ i = b\circ p\circ i \simeq *[/math] ([math]p\circ i \simeq *[/math]). It follows that we may factor [math]e\circ i[/math] through the homotopy fibre of [math]q[/math], and so we have [math]f\colon F\to F'[/math] such that [math]j\circ f = e\circ i[/math]. Now, the conditions for the referenced result are satisfied, and so we may conclude that [math]f\colon F\to F'[/math] is a localisation. Let's then think a bit. We have shown that the localisation of the fibre we started with is a fibre for the localised version of the fibration we started with, and that is what we were trying to do. More precisely, any localisation of the original fibre is homotopy equivalent to any fibre of the localised fibration.

>>11678496
And obviously I forgot the reference.
* 2.3.12 https://web.math.rochester.edu/people/faculty/jnei/exalgmethod.pdf

>>11678497
https://www.youtube.com/watch?v=bRIVTEBaLkA
Aren't they just cute, btw?

>>11678498
>2 million views

>>11678496
Ok thanks animu bitch

>>11678502
Could definitely have more.

>>11678512
You're welcome iff you formulate a similar thing for cofibrations.

>prerequisites: basic number theory, at the level of Weil's book.

>>11678539

stay mad analysisboi

>>11677419
pick up any book on the subject and start reading. don't get caught in the trap of looking for the perfect textbook.

>>11678539
>>prerequisites: basic number theory, at the level of Weil's book.
Name one (1) such case.

I'll have a Hodge Structure. Tate, not mixed.

why does math trigger my schizophrenia?

>>11676859

post a pic of your hand (or tits) with a timestamp?

>>11679259
>schizophrenia
>>>/x/

Pill me on Fuchsian groups.

>>11679295
s/[math]\chi [/math]/izopfrenia.

>>11679328
Consider a model for the hyperbolic plane, then take its Möbius group. It can be shown that the Möbius group is a topological group, and then you can consider its discrete subgroups. Then you get all sorts of nice things like how every commutative Fuchsian group is cyclic, and Dirichlet fundamental domains etc. This one's pretty short, so go ahead and give it a read: http://www.f.waseda.jp/ykomori/umemoto-master.pdf

>>11678496
>at his stuff
It's not so clear what that is, to me.

>>11677491
The pic related it where Land prose meets the sciences.

In other news, today I came across a nice paper giving (and indeed providing the code implementaton for)
concise computable bijections between [math] { \mathbb N } [/math] and
[math] \bigcup_n { \mathbb N }^n [/math],
i.e. explicit enumerations of all finite lists of [math] { \mathbb N } [/math].
In fact, permutations on [math] \bigcup_n { \mathbb N }^n [/math] are given that (bijecively) pass through [math] { \mathbb N } [/math]:

https://arxiv.org/abs/1301.0129
https://arxiv.org/pdf/1301.0129.pdf

>>11679518
>It's not so clear what that is, to me.
Basically, he wrote it in a funny way, but the two out of three thingy is this diagram.

>>11679518
>It's not so clear what that is, to me.
it means she's been stroking her cock to your pictures

pure math is formalism
applied math is platonism

>>11679589
Platonism is nothing

>>11679518
>It's not so clear what that is, to me.
I don't know. It could be the fact that I don't know too much about set theory and formal logic etc., and thus you may appear like an omniscient wizard to me. Regardless of that, I think you are pretty cool.
>The pic related it where Land prose meets the sciences.
Something to watch at some point then. Grazie schön for the rec.

>>11679518
>We argue
Why don't string theorists just prove things instead of arguing?

>>11679646
I don't think I could be this much of a fag if I was doing it on purpose

>>11679684
>fag
[math]\sum_{j\in J}\mathbb{Z}_j [/math] or perhaps [math]\sum\limits_{i=1}^n \mathbb{Z}/m_i \mathbb{Z}[/math]?

Let p be prime, n >= 1 be an integer, and F a field of order p^n.

Show that for any d | p^n -1 (where | means divides), the multiplicative group (F \ {0}, *) contains an element of order d

>>11679697
what have you tried?

>>11679697
just count the elements bro

>>11679702

Nothing, man. I know that p must be the order of the characteristic of F, but I have absolutely no idea how to proceed or even approach the problem.
I've come here out of desperation

>>11679749
>I've tried nothing
then all you get is a hint. [math]d | p^n-1 \rightarrow x^d-1 | x^{p^n-1}-1[/math]

>>11679646
I learn set theory for "practical purposes", i.e. for other math, from books and papers, never sat in on a lecture about it, and can't really use it in my day job. So that's not the stuff.

Regarding your pic related, I have a collegue who made this, I think it's really good:
https://youtu.be/lrOVKHg_PJQ

>>11679766
Much obliged, friend.

>>11679787
That's a good video. I used to go run on a treadmill before going to the office, and I would watch stuff like that at the same time. Perfect for that, no idea why.

>>11671193
Does anyone know of a good source to practice the skill of rearranging algebraic equations. I'm fascinated by it, but I don't have much practice at it. :(

>>11680019
cardanos method cubyc solutyon

do generators of a convex set map to generators of the projection of the set? I think the answer is no. just do the parallel projection
[math] mathbb{R}^2 \to \mathbb{R}[\math]

I get obsessed with problems but I don't have the IQ to approach and tinker it with different strategy and angles to actually solve it, I just end up wasting hours repeating the same try hoping it will work

>>11680390
What the fuck do you mean with generators, extreme points?

What is the deal with the definition of injective functions? This image is from Wikipedia, clearly every element in X, maps to a different element in Y. 'Distict element X maps to distinct element Y'
But then you scroll down to the definition, and it claims that for f(x) to be injective, then for every element in X if f(x1) = (fx2) then x1 = x2 and somehow this defines an injective function, which would imply that multiple elements in X map to the same Y if f(x1) = (fx2). ???wat

do you guys think they could solve th Bertrand's box paradox?

https://www.youtube.com/watch?v=Ingk207JdJY

>>11680440
in vector spaces, it would be the spanning set of whose elements the linear combinations of form the space.

>>11680445
wait I figured it out, the notation should be f(x) = f(y)

I'm deciding modules for third year undergrad. I'm not sure whether to do Galois theory or a module in "Rings and Modules".

I've done algebra up until sylow groups and basic ring theory. Anybody who can shed light onto these two subjects would be great

>>11680445
injective just means 1-to-1,
it's just trying to say f is 1-to-1 and if x1 = x2, f(x1) = f(x2)

and if x1 != x2, then f(x1) != f(x2)

How long has COVID set back mathematics?

>>11680549
69 days

>>11680485
If you're interested in pursuing math it doesn't really matter much because those are both things you'll have to take soon anyway.
I would probably go with the modules course, personally. Galois theory is a very fun class, but unless you intend to pursue specific branches of number theory, it's honestly kind of a dead end. Modules on the other hand are extremely useful to know about early. You'll see them all the time basically no matter what you decide to do unless you're some caveman graph theorist or something.

https://vixra.org/pdf/2004.0065v1.pdf
https://vixra.org/pdf/1502.0038v1.pdf

>rationalizing the denominator in the current millenia
why do people still do this?

>>11680617
because their high school teachers told them to

>>11671193
Am I the only one here who is scared by the concept of irrational numbers?
Also what's your recommendation for getting into abstract agebra?

>>11680663
Only preliminary knowledge of abstract algebra is linear algebra, so just start working through a book/course and learn. Its simple.

>>11680663
>Also what's your recommendation for getting into abstract agebra?

>>11680663
the ancients killed people for investigating the irrationals. we don't even know why they're called irrational

find me a way to represent them as exact constants. I'm obsessed with this task

do I really need to know trigonometry to understand calculus?

>>11680746
>we don't even know why they're called irrational
they're called irrational because they're not rational.

>>11680755
define rational, mathman

>>11680754
Technically no, but in practice yes.
Most of the concepts of calculus don't depend on trig, you really only need it to change coordinate systems.
But a big chunk of the examples/problems of calculus involve trig functions so you can't really DO much calculus without trigonometry.

>>11680761
ratio of two integers

>>11680767
funny the rational number as a concept didn't exist until after the irrationals were called irrational

>>11680775
Wrong.

>complex analysis
>partial differential equations

>combinatorics

Show S4 has exactly 3 subgroups of order 8 that contain N (the unique normal subgroup of S4)

I've established that S4/N is isomorphic to S3, and that the subgroups of S4 containing N are isomorphic to the subgroups of S3 (by correspondence theorem)
Thus since S3 has 6 subgroups, it must be that there exist 6 subgroups of S4 containing N.

I've totally hit a dead end at this point. Where did I go wrong / how do I proceed?
Thanks in advance fellas

>>11680763
>But a big chunk of the examples/problems of calculus involve trig functions so you can't really DO much calculus without trigonometry.
That's what I'm noticing. It's not actually relevant to the subject other than that being what many of the examples are (and knowing derivatives/integrals of the trig functions)

>>11680802
>Thus since S3 has 6 subgroups
and exactly 3 of them have order 2. 2*4 = 8.

Also, S4 does not have "a unique" normal subgroup. It happens to be clear from context that the question's talking about the order 4 one, but it's still a bad description.

>>11680837
Thank you!!!
I apologize about the poor way I posed the question, working for 10 hours straight has taken a toll on my brain :)

>>11680837
Followup question: why do we multiply 2 by (I assume) the order of the normal group N as you did here?

>>11680846
It's just Lagrange's theorem. The correspondence theorem sends H -> H/N. If you want to find |H| = 8, and you know |N| = 4, then you look for subgroups of order 2 in the image.

>>11680853
Makes sense, thank you so much.
I really should have started participating in this general sooner

>>11680856
you should take your algebra homework to the stupid questions thread instead >>11671932
that's kind of the entire point of /sqt/

>>11680794
>>11680801
>differential geometry

>>11680390
You are correct (assuming that a "generator" of a convex set is the same as an extreme point). Consider a diamond with corners at (1,0), (-1,0), (0,1) and (0,-1). when projecting to the x axis you get the interval [-1,1], and the points (0,1) and (0,-1) get sent to 0, which is not an extreme point of the interval.

>>11680493
>injective just means 1-to-1
No, that would already be a bijection.
You don't know, if the function is also surjective ("hits" all elements of Y), so you can't call it "1-to-1".

>>11680445
[math]x_1=x_2[/math] means [math]x_1[/math] and [math]x_2[/math] are referring to the same element.
If for an injective function the image of two elements is the same, that means those two elements are the same element.

>>11681091
bijective is 1-to-1 correspondence
injective is 1-to-1
surjective iscorrespondence
surjective & injective is bijective

>>11680586
>https://vixra.org/pdf/2004.0065v1.pdf
>https://vixra.org/pdf/1502.0038v1.pdf
This is why vixra is based

>>11671193
>textbook for students of mathematics
This is racism.

Is there any point to starting a math blog and just posting solutions to exercises mostly? Would anyone read it?

>>11681995
I would see such a blog be appreciated by people. Self studying is nice otherwise, but then you get stuck with a problem and there is not necessarily anyone to help you with it (in flesh and blood). Such a blog would be a good alternative to asking on StackExchange or /sqt/. Please do at least give it a try if you happen to have the time to do so.

>>11681995
Are we talking normal proofs or really slick proofs?

I can never remember the basic inequalities... how can I estimate |x^k - y^k| by |x-y|.

>>11682200
[math] |x^k - y^k| = |x-y||x^{k-1} + x^{k+2}y + \cdots + xy^{k+2} + y^{k-1}| \le |x-y||x^{k-1}| + |x-y||x^{k-2}y| + \cdots + |x-y||xy^{k-2}| + |x-y||y^{k-1}| = |x-y|(|x|^{k-1} + |x|^{k-2}|y| + \cdots + |x||y|^{k-2} + |y|^{k-1})[/math], I would say.

>>11673133
if phi is injective, the induced morphism isn't necessarily injective. consider m=n=1, R=Z, phi(x) =2x and m=2Z

>>11681268
but there's some guy autistically spamming the logic section with literally a thousand "papers" that are all just a page claiming the refutation of various theorems.

I've seen a lot in my life, but this is the most autistic effort
https://vixra.org/setlog/
All papers look like so:
https://vixra.org/pdf/2004.0469v1.pdf

Whenever I upload a clip on a topic, (say https://youtu.be/E7V36JvHbsA)) I take a cursory look at what's already available online and in particular on youtube.
So I discovered this deranged person uploading videos of her going through some math texts that she found somewhere, and she kind of reads through it and presents it as some kind of Alien message

https://youtu.be/SdgYuqA9oDM

>>11681995
imho blogs should either be theory building or, better yet, honestly approach non-trivial things you know about or like.

>>11680485
kek are you in warwick uni?

imo do Galois theory, rings and modules (if it is what i think it is), puts extra focus on non-commutative results, and if you do happen to be interested in rings and modules, can take commutative algebra, which is much more useful

>>11682258
So maybe a more tangible example here

https://vixra.org/pdf/2004.0117v1.pdf
>Remark 3.3: Eqs. 3.3.1.2, ..2.2, ..3.2, and ..4.2 as rendered are not tautologous and
not equivalent in pairs of states for light on or light off, refuting the definition of
Bayesian network. This further denies the conjecture for structural equations and
causal models.

This person goes through various texts and make screenshots or types it off, enumerates the propositions (even if the original text doesn't have an enumeration) and then writes a line or two about how in their logic or approach, the claims made in those texts (say something about Bayesian networks here, but no math subject is save) is "refuted."
It's really scary.

>>11681995
If the blog is a waste product of what you do on a daily basis anyway, then why would it trouble you, if people didn't read it?
Do it for yourself.

>>11682206
Your first equation is already wrong, my guy. Unless I am misunderstaning what the ... is supposed to mean.

>>11683043
Oopsie, should be minus instead of plus there.in the exponents, good point. What do you think the ... means?

What does it mean formally for a sequence of sequences to converge? The concept makes sense but the formal definition I've thought of doesn't.

>>11671193
Hello, I'm back, did you guys miss me?

>>11677314
GeoGebra?

What sort of topics are you covering?

>>11683258
I admit nothing.

>>11683126
The meaning has to be that a sequence
1+1/n, 2+2/n, 3+3/n, ... for n > 0
converges to the sequence
1, 2, 3, ... as n -> oo.
In which case the formal definition has to express that each element of the former sequence converges to the corresponding element of the latter sequence.

>>11683302
What are you studying?

Hey, I want to learn a lot more about math independently rather than watching popsci videos but I don't know where to start.

I studied Economics at university and when I was there I studied Linear Algebra and Calc I and II, but that's as far as my knowledge goes. Stuff like Number Theory or Real Analysis seems really interesting to me but I don't know if I'm going to order a textbook and I'm just not going to get it at all because there's prerequisite shit I've missed out.

Does anyone have any advice?

>>11683363
read Serre

>>11683363
>Does anyone have any advice?

>>11683363
>I studied Economics
>order a textbook
What the fuck, you literally didn't learn anything.
Walras would be rolling in his grave.

>>11683363
RA only has basic set theory as prereq. Algebra is self contained (at your level), and while some number theory rests on algebra, the entry level stuff can be done on ts own.

>>11682258
>>11682267
Logic is over my head, but BASED. I hope vixra always just lets whatever garbage stay there. The best comedy.

>>11683363
Calculus and Linear Algebra are the fundamentals of all modern math, although I believe you probably have a strong foundation in calculus, your Linear Algebra knowledge probably boils down to computational problems, so I'd suggest starting from proof-based LA and for that there are two good books:
>Linear Algebra Done Right (highly controversial buy pretty good nonetheless)
>Linear Algebra by Hoffman and Kunze
You could also strenghten your calculus basis using Apostol but it might be a waste of time if you still retain some knowledge of the most important theorems and definitions. After this you should tackle Real Analysis and Abstract Algebra.

For real analysis you might want to use Pugh or Tao, they're both good, not very easy, but also not Rudin level approaches. If even then you're struggling with Analysis, then maybe use Abbott which is a fairly handholding approach to the subject.

For abstract algebra there are several good books such as Dummit and Foote, Artin, Aluffi, Herstein, Rotman, Jacobson etc whichever you choose is good enough.

After that maybe you could be interested in Differential Geometry, but the only book I used for this was do Carmo and it was pretty good.

There's also ODEs and PDEs at the undergraduate level, but I can't recommend anything on those since they're too boring for me.

>>11683387
Oh I also forgot to metion that you should study Calculus III before DG.

>>11683351
Trying to apply homotopy with coefficients to get a reduced version of a nice theorem by James (or actually its generalisation) to work in the setting I am interested in. The original one says that if we want to have the space [math]S^m \cup_\alpha e^n \cup_\beta e^{m+n}[/math], where the dimensions would satisfy [math]n > m+1 \ge 3[/math], then this space exists iff a certain multiple of the Whitehead product of [math]\alpha[/math] and either of the generators is in the image of the homomorphism [math]\alpha_* \colon \pi_{m+n-2}(S^{n-1}) \to \pi_{m+n-2}(S^m)[/math] induced by [math]\alpha[/math]. This can be generalised by replacing the sphere of smallest dimension with a CW-complex, and now I would like to see if the integers in the theorem may be replaced with [math]\mathbb{Z}/p\mathbb{Z}[/math]. The coefficient is given by the cup product of the cohomology classes carried by the sphere and the n-cell, so it could very well be the case that it can be done in the mod p version. How about you?

>>11683383
>I hope vixra always just lets whatever garbage stay there. The best comedy.
So much this. I think I saw a paper on how to calculate 1+1 with a Casio, and it contained an open problem whether that can be done with a Texas Instruments calculator.

>>11683381
>Not having a beautiful math shelf on your house
Your life must be pretty boring

>>11683396
>How about you?
Damn, I've been NEETing myself reading manga all day for the past two weeks

>>11683398
I have a beautiful shelf of books stolen from my university's library (they literally let me borrow 10 books, for some reason or another), Goosebumps volumes I got as gifts and old books that came to me some way or another.

>>11683402
Did the semester end or something? If not, prepare to be beaten up, and if it did, prepare to be beaten up if you don't reactivate during the next week. It's easy to get rusty, you know!

>>11683371
This seems above my level lole

>>11683381
I know kek but I find that I can motivate myself to read or learn something more easily with a physical book in front of me. I don't understand why that's the case psychologically or whatever, but I know that for me it is. Also, I don't have access to a university library any more.

>>11683374
>>11683383
>>11683387
Thanks for your help. I think I have a better idea about what I should learn first now.

>>11683414
Quarantine. Yeah, I'll make sure to study properly tomorrow, senpai!

>>11683411
You really can steal books with a straight face, tohouposter? I'm completely disappointed on you right now, shame on you, those books are for everyone

>>11683411
>beautiful shelf of books
>university's library
pick one
ex-library books are all beat to shit and covered in stickers and tape and stamps

>>11683418
>Takes minutes to make a good, thoguhtful and helpful post just for this anon's sake
>Give good recommendations and tips
>He groups my post with two others instead of acknowledging it separately
Go fuck yourself, I'm never helping you again

>>11683421
That is a vow not easily broken. You will make me proud, will you not?

>>11683418
you should probably know that >>11683374 is a meme and you probably shouldn't follow it. >>11683387 is good.

>>11683430
I'll train hard, senpai, until I reach the point where I can finally surpass you, then I'll become the King of Mathematicians

>>11683427
Yeah, the undergrad books absolutely are.
But quite a bit of the grad stuff is in excellent condition.

>>11683432
I was well aware that the topics listed in the image were more difficult than the level of education they were listed under, but the stuff under "high school" seemed worth looking into at first glance, but upon closer inspection maybe not that helpful lel

>>11683429
Yeah, your post was definitely the most useful. Not sure why I did that.

>>11683436
Then I will have to work even harder to make sure I will not be surpassed! Very well, let the battle begin!

And /[math]\mathfrak{gnmg}[/math]/

>>11683460
Good night, my dear.

>>11683460
the manga girl posting in this thread really got out of hand in the last view hours

gnmg weeb

>>11683470
>Elon using /pol/ slangs

>>11683491
what's funnier is that the Trumpy gal probably didn't get the socialism rose - but that makes no fucking sense

fuck I love math
lectures are so comfy
there's nothing to argue about, it just is what it is

>>11683470
>mfw people said the anime roleplay circlejerk cancer was going to go away once lockdowns ended

>>11683502
Maybe you're the one who should go away, just saying man, maybe this is not your place, maybe try Reddit or Facebook? Just saying.

>>11683500
>there's nothing to argue about, it just is what it is

Is it the case that, given any pair of two 15x15 square matrices [math]X[/math] and [math]Y[/math], they can either be multiplied (possibly with repetition, e.g. [math]X \cdot{} X \cdot{} Y \cdot{} X [/math] or [math]Y \cdot{} X \cdot{} Y \cdot{} Y\cdot{} Y\cdot{} X [/math], etc.) in a way that yields the zero metric, or they cannot?

>>11683502
Who said that? That's a bold claim.

>>11683502
The lockdown hasn't ended, tho.

>>11683500
>there's nothing to argue about, it just is what it is

Is it the case that, given any pair of two 15x15 square matrices X and Y, they can either be multiplied (possibly with repetition, e.g. "X⋅X⋅Y⋅X" or "Y⋅X⋅Y⋅Y⋅Y⋅X", etc., etc.) in a way that yields the zero matrix, or they cannot?

>>11683502
Who said that? That's a bold claim.

Incidentally, what really is a proof? Is it just common sense stemmed from the game of words invented by the human mind as a failed attempt to produce an objective definition of reality known as logic? There's no real beginning to a proof since it stems from axioms, i.e., common sense, there are no rules other than to make sure it can't be proved when creating an axiom, therefore no proof has a definitive basis or standing to rely upon, as it is now it's a complete self-deception that mathematicians use in order to keep playing their puzzles. When will a foundation be found that completely addresses this problem? Right now set theory and category theory should be seen as nothing more than failures since both rely on 'magical tricks' to build mathematics, they're as useful as an out of tune piano, no matter how good you are, you'll never be able to produce the right sound.

>>11683542
You don't seem to ask a question, this just ends up being rhetoric. This last half of this makes it feel wrong to even respond to this.
As far as formalism goes (math as word games), I don't think the notion of proofs is illdefined or problematic. Yes, proof is conditional on the axioms.
I'm not sure why you say
>there are no rules other than to make sure it can't be proved when creating an axiom

>>11683579
I was just messing around

>>11683587
m-merely pretending

Mathematics comes from the heart, from feeling the problems, not from endless studies, that's bad for your health and will only destroy you slowly.

>>11684031
wow you must be very Smart But Lazy

I'm this guy >>11681995
I started the blog! I already have a few posts. I don't expect it get a lot of views but it's already giving me purpose to continue moving forward. Rudin is such a slog on your own. I hope it will blossom into a more in depth blog once I get to the stuff I want to do (topology, algebraic topology, computational topology) and I will use it explore theorems, problems, and applications very in depth.

>>11684066
Well, and where is it?

>>11684054
It's not laziness, it's studying from the heart, I don't force myself to study if I'm not feeling like it, I only do it when I know I'll absorb it in my spirit

>>11684080
I found this cool blog guys! https://themathlabyrinth.wordpress.com/

The author, totally not me, seems like a decent guy!

>>11684107
>themathlabyrinth
Cute, I'll put it in my favorites

>>11684107
Maybe you should knownthat there's already a solution's manual to Rudin.

>>11684118
>Maybe you should knownthat there's already a solution's manual to Rudin.
Rudin is a meme.

>>11684118
Wait, really?
Where?

>>11684173
https://www.google.com/search?q=rudin+solutions+manual

>>11684179
Whoa.

Thanks I guess.
Pic is you btw.

>>11684184
Maybe you could solve Pugh's book? Some problems are even harder than Rudin (and ignore the research-level ones)

>>11684197
I wasn't actually reading Rudin at all :D
Sorry for asking for the solutions manual, I was just curious.
But thanks for your advice, I am reading Understanding Analysis by Abbott and I am halfway through and I promised not to change books.

what branch of math is the most popular these days?
i wanna know what's on the cutting edge, what are researchers getting hyped about

>>11684607
Cutting edge? It may sound crazy, but there are some researchers at the bleeding edge who have started investigating quadruple integrals, but don't expect them to make much progress for decades, if not centuries. Most Ph.D.s find double integration too difficult right now.

>>11684707
>by the regurality of X
This always gets me.

If you're not good enough for research, what the fuck do you do with a maths MSc?

https://sites.google.com/view/nialltaggartmath/oats
>Emanuele Dotto (University of Warwick)
>Title: Witt vectors with coefficients and characteristic polynomials over non-commutative rings
>Abstract: The characteristic polynomial of a matrix with entries in a commutative ring R naturally takes value in the ring of Witt vectors of R. In joint work with Krause, Nikolaus and Patchkoria, we extend the classical Witt vectors construction to allow as input pairs of a ring R and a bimodule M. I will explain how this construction relates to topological Hochschild homology, the Hill-Hopkins-Ravenel norm, and the characteristic polynomial.
3 hours and 20 minutes till showtime. Boohoo Zoom.

To the german anons or any anon that knows these books.
Are Fischer Lineare Algebra and Forster Analysis actually good books for self study? I see theyre often referenced as Standardwerke so they should be good or would I be better off if I would go with different books.

>>11684876
Just use the english standards like Rudin and Axler

>>11684835
learn to program in python and learn ML, become a ML engineer. high paying jobs once you get experience.

>>11685125
Are rudin and axler good for self study? Why would you pick them over These german books?

>>11685238
Local terminology may differ from English terminology quite drastically. I don't know how different German and English nomenclature are, but it may be the case that you would need to learn quite a lot of new names for things when moving from German books to English ones. Been there, done that, and it took some time to get used to that.

>>11685238
The anglophone world currently dominates math and other STEM fields, I think it's pretty normal to consider the recommended biography in the US/UK as being the definitive ones for those who really want to reach the top.

>>11671193
I just told my friend a maths joke I thought of I said addition isn't useful I've only used it a few times, I've used it one time a bit ago and then once after that so I've only used addition twice

do you get the joke?

>>11685278
You just added a lot to my suffering.

>>11685285
but do you get why it's funny

>>11685289
I am sorry to inform you that your joke falls into the unfunny valley.

>>11671193
Is Gorodentsev a good intro to Abstract Algebra?

>>11684707

why are weebs obsessed with little girls doing math? let alone something as autistic as group theory?

>>11685573
How is that pic related to group theory?

>>11684876
Fischer isn't great...I much prefer Liesen/Mehrmann (cleaner proofs, nicer structure, more modern presentation, better selected material)
Also Amann/Escher Analysis is a lot better than Fischer, Fischer is more approachable and more of a "standard" book (especially if you're a physics student) but as a meth guy Amann is just more material and a nicer approach.
Also Rudin and Axler unironically are much worse books than Amann/Escher and Liesen/Mehrmann (and I generally prefer english books- just not for those subjects)

>>11684852
how was it?

>>11685663
Thanks, why are Rudin and Axler worse?
Lineare Algebra 1 und Analysis 1 is pretty much the same everywhere in Germany right? So the books you mentioned should cover all topics I will have to know next Semester, yes?

>>11685691

what's drumpfie doing there

>>11685691
I don't know. I actually forgot the whole thing and went for a walk instead. They are being recorded, so maybe there is a video somewhere.

>>11685695
Just personal preference: mostly selection of the topics, writing style and prefer the exercises. Additionally in Germany you start undergraduate mathmatics by taking Analysis 1-3(1-4 in some unis) and Lineare Algebra 1&2.
In the US (as I've heard- no idea if this is correct) you take analysis later (not in the first semester of undergraduate) and the courses are shorter (1-2 semesters) so it's expected the german books will be a bit better suited for the german curriculum which is longer and starts earlier.
I'd say Ana/Lina courses are 80-90% the same but there's definitely some small differences between professors. It would probably be best to find out the name of the professor for next semester and check if you can find an old course page with the used literature.
Anyway if you study any of those books well b4 next semester you'll be running laps around your peers.
Just remember (if this is your first semester) those books will be quite a difference to high school material; don't worry if u struggle a bit. Just remember to read the exercises first, then the chapters and do at least some of the exercises for each chapter. Also try to prove statements yourself b4 reading the proofs in the books. Have fun! (also in the last post I obviously wanted to compare Amann/Escher to Forster, not Fischer)

New thread >>11685818