/sci/ - Science & Math

>>11614104
You should probably choose the problems where you show that bijectivity is equivalent to having an inverse. If I remember correctly, the first example of a group Herstein gives is the set of bijections from a set to itself and their composition. The group structure depends on invertibility, so those problems are quite essential, unless you have already showed that those properties are equivalent on a course or something. Anyway, good luck to you and good night to everyone else (and to you later but not yet)!

I don't understand this proof clearly.
Why is n the smallest positive integer m such that a^m=e?

Consider the space of trace class (nuclear) operators [math]T(\mathcal{H})[/math] on a separable Hilbert space as the dual of the space of compact operators [math]K(\mathcal{H})[/math] on the Hilbert space.

Is the weak-* topology on [math]T(\mathcal{H})[/math] weaker or stronger than the weak operator topology (i.e. the topology induced by seminorms of the type [math]|\langle Tx,y\rangle|[/math] for [math]x,y \in \mathcal{H}[/math])?

0.999... = 0.999...

>>11614167
The first sentence

>>11614167
That is the definition of order.

What's a good book on the history of mathematics? I recently read A Concise History of Mathematics for Philosophers and it was quite a good read.

>>11614235
>>11614513
Isn't the order the number of elements that G contains?

>>11614556
Dieudonné wrote a bunch of books on the history of specific subjects. There's also Geometry by it's History from a Russian those name I couldn't spell even to save my own life. Schlager's Science and it's times also covers the history of mathematics, but it's not focused on it (it's also 8 volumes long)
Afaik my University's history of math course uses Boyer's book, but I never took ir, so dunno how good or bad the book i.

>>11614580
yes, but in this case, G is cyclic.

Official /mg/ Curriculum; Big Brain Only

High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

>>11614580
The order of an element g of a group is the number of elements on the subgroup generated by it (i.e. order of g is card(<g>))
Equivalently it's the smallest positive integer n such that g^n = e

>>11614596
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

>>11614596
>>11614603
fuck off

>>11614596
>>11614603
I adore these posts. They are really just a filter for Fields Medalists. If one is found to have completed such a curriculum within the stated chronological benchmarks, then they must be awarded the next Fields Medal.

>>11614601
>>11614595
I understand that G={e, g^1,...,g^n-1} are all distinct so g^n is equal to some value of G, but why e?

>>11614756

e is for element in this context not the limit of compound interest

how inherent is the connection between synthetic geometry and euclidean geometry? do trig functions hold up well in spaces other than euclidean?

>>11614603
>>11614596
Are you Russian hackers? Please stop messing with our elections.

>>11614767
I know that e is the identity element.

>>11614756
>>11613906

>>11614150
Havent gotten to those yet, also havent gotten to groups yet, but that seems a property I would never think of and will have to prove to believe it;
Did the mediums that interested me except last one that I couldnt figure out, also video was longer as I had a harder time explaining / remembering what I had done. Gn

>>11614130
How old is /mg/?
Who are the anime posting fags? Are you involved in something using math in some professional way?
Do you have a normalfag life (stable job + family)?

>>11614814
>anime posting fags

Son, do you know what website you're on?

>>11614603
What comes after Freshman?

>>11614768
>do trig functions hold up well in spaces other than euclidean?
There are hyperbolic versions of the trig functions (sinh/cosh/tanh). I think there is such a thing as spherical trigonometry as well, but it's basically the poster child of "dusty old shit nobody learns anymore". People learn the hyperbolic sine functions not for their geometry, but because they have really useful properties as functions in their own right.

>>11614130
Ahh finally a thread about graph theory, a branch of my speciality; computer science. Do any of you math brainlets need help with your computer science homework? I'm the top of my graph theory class.

I'm studying analysis from Rudin on my own. How do I ever learn how to start on problems?

>>11615255
>I'm studying analysis from Rudin on my own. How do I ever learn how to start on problems?
By closing that meme and reading a non-meme instead

>>11615255
1. Start
2. Stare for a while
3. Get an idea
4. Doesn't work
5. Stare for another 3 hours
6. Figure it out
7. Start the next one

For real, a lot of the exercises in Rudin are tough. Don't feel bad if one question takes hours.

>>11615269
I got the 2e too, which has different problems than 3e. No solution sets online.

>>11615271
rip

>>11615271
I mean, I think that 2e and 3e follow the same structure, so you could just do the problems from 3e instead

Good morning /mg/!

>>11614799
Some comments:
(12) You claim you changed the set, but that is actually not true. The set is the same. What you are trying to say is that you use the reduced form p/q of n/m, where the greatest common divisor for p & q is 1.
(13) What you are trying to say is that 2 and 3 are prime numbers. The reasoning is correct.
(14) Sufficient and necessary condition can be easily remembered if you look at the truth table of [math]p \rightarrow q[/math]. If we assume that the implication itself is true, then p being true is a sufficient condition for q to be true. Conversely, if the implication is true, then q being true is a necessary condition for p to be true.
(17) & (18) There is always the identity. There are actually m! bijections for a set of m elements. If m=2, you also have f(1)=2, f(2)=1. Note that if there are m! > 1 bijections from S to itself, then there must be more than 1 function from S to itself!
For the last one, I can't think of anything quickly. Keep up the good work!

>>11614814
I'm doing a PhD. I have a bf, but we are not in the same country.

>>11614756
Suppose [math]g^n = g^m[/math], for [math]m<n[/math], so that [math]g^m = g^m\cdot g^{n-m}[/math]. Inverting gives us [math]e \neq g^{n-m}=e[/math], a result most problematic.

>finish a math PhD, now 2nd year working as a postdoc
>have to look up basic facts in analysis because I don't remember shit from my undergrad
how the fuck do you retain all these things in your memory?

>>11615820
You don't, you review and forget until the subject soaks into your bones and you become the old russian professor who can pull proofs out of a tophat.

>>11614179

bros.... im not going to make it....

>>11615629
Are you a woman? If so, are you trans?
>>11615992
We're all going to make it!

>>11615978
this compares WOT with the weak* topology on [math]\sigma(T(H), B(H))[/math], but >>11614179
is asking about the weak* topology on [math]\sigma(K(H), T(H))[/math]

>>11615992
You will, and so do we all.

>>11615997
Let's say "yes".

If a test has 4 options, correct give +3 points, false gives -1 points, skipping the question is 0. How many options would you have to eliminate before it becomes more advantageous to guess than skipping the question? If you have no idea, would it be better to randomly guess than skip?

>>11616150

you eliminate 3 so there's only the right one left

>>11615956
I went to a high school with an olympiad training program taught by this ancient old dude. He was basically a wizard from the perspective of the kids; he had seen so many competition problems that when you showed him one he just reflexively knew 3 ways of doing it without even thinking about them anymore.

How many platonists are in /mg/? Put your hand up if you're a platonist.

>>11616401
nophilosophicalpositionchad reporting in

I recall reading that if the axiom of choice fails, then there exists a partition of [math]\mathbb R[/math] into strictly more than continuum equivalence classes. This sounds absurd, but things do get crazy when AC fails...

I can't find any reference on this however. Any ideas on how to prove this (assuming my memory serves me right)?

>>11615629
Understood everything, thanks
(13) I dont think i was thinking about prime numbers, but that makes more sense
(14) Makes sense, I forgot how I thought about it on those questions in HS but I always had them right
(17) yep, it was just that I couldnt think of a function that would do it, now I remember that we could do a 'function by parts', and what I mean by that is something like: f(s) = s if s < 5 and f(s) = -s if s >5, and this way we could get m! mappings that would be injective, and the same for subjective, and for each of those cases it would also be the other one, so ye would also be bijective (not an exercise)
Last one looks a bit like one of those open problems orso, going to look at it more later

what should i do to make it in mathematics?

>>11616554
close /mg/ and go study

>>11614768
>how inherent is the connection between synthetic geometry and euclidean geometry? do trig functions hold up well in spaces other than euclidean?
The first and second part of the question somewhat don't go together / ask about very different things.
To answer the first question as it stands, there was of course lots of work on synthetically axiomatizing geometry, see e.g.
https://en.wikipedia.org/wiki/Hilbert%27s_axioms
or
https://en.wikipedia.org/wiki/Tarski%27s_axioms

Tarski generally has a lot of funky stuff if you look into it, e.g.
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals

>>11616474
To cite the Wikipedia page of AoC
>In some model, there is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.

>Any ideas on how to prove this
I'm not aware of the proofs of this (although probably you find one in some of the AoC books, there's some popular ones).

>This sounds absurd
I doubt it's problematic, it's probably an artifact of Cantors cardinality definition.
For two generic sets X and R, you have that |R|<|X| if you can inject R into X but there's no bijection. Now if the objects are weird enough so that you provably can't find a bijection without non-constructive means, then you're already there. I could imagine the proof of the relation goes like that.

Note that there's other size-comparison relations, e.g.
https://en.wikipedia.org/wiki/Subcountability

Also, note that ZF (with or without ZFC) can't even prove that |Y|>|X| implies |P(Y)|<|P(X)| or give any hint on how |R| compares to the cardinalities of the ordinals (apart from being bigger than |\omega_1|)

>>11616574
apart from being bigger (or equal of course)

>>11616574
apart from being bigger (or equal of course), in the last sentence.

And maybe to re-emphasize again, the issue I put forward is that Cantors formalization of "bigger" is in terms of properties of functions between them.
So it's not like "hey those two sets can't be in bijection because the one is bigger" but instead "hey, this set is bigger because (this just means) there's no bijection but only an injection" and, in this case this degenerates to "hey, I guess we have to say that this set is bigger because we know we can't find an existence proof that they are of equal size"

>>11616574
apart from being bigger (or equal of course), in the last sentence.

And maybe to re-emphasize again, the issue I put forward is that Cantors formalization of "bigger" is in terms of properties of functions between them.
So it's not like "hey those two sets can't be in bijection because the one is bigger" but instead "hey, this set is bigger because (this just means) there's no bijection but only an injection" and, in this case this degenerates to "hey, I guess we have to say that this set is bigger because we know we can't find an existence proof of a bijection (i.e. a proof that they are of equal size)"

That proving < comes down to proving there's no bijection also makes it plausible how the equal-size proof would relate to axiomizing choice functions into the picture

Hey does anyone know what happened to xamuel.com? I remember he used to have some pretty good stuff.

Will mathematics fill the gaping hole in my soul?

>>11616652
no

>>11614556
Klein’s Development of Mathematics in the 19th Century

>>11616542
>(13) I dont think i was thinking about prime numbers, but that makes more sense
Primes are used for arguments like these because of the uniqueness of the factors. For example, if you want to show that [math]\mathbb{N}^n[/math] is countable for any [math]n\in \mathbb{N}[/math], it suffices to find an injection. In order to construct one, grab any n distinct primes [math]p_i[/math] and define a function [math](k_1, \dots, k_n) \mapsto p_1^{k_1}\cdots p_n^{k_n}[/math].

I think you could probably move forward and read the next chapter. If some claim seems odd, then you can return to the mapping chapter, as (if I remember correctly) the first example of a group he gives you will be automorphisms, and the claims he makes will be about bijections (or can be verified more concretely using those), like for example associativity. Keep up the good work, anon!

>>11616652
It filled mine! So perhaps!

Help please
For each statement (a) through (g), if it is true, write the word "True." Otherwise, write the word "False." No explanation is necessary.

(a) The graph of a rational function can never cross its oblique asymptote.

(b) If the real number LaTeX: cc is a zero of the denominator of a rational function, then the line LaTeX: x=cx = c must be a vertical asymptote of the function's graph.

(c) A real number LaTeX: cc is a zero of a polynomial function LaTeX: f\left(x\right)f ( x ) if and only if LaTeX: x-cx − c is a factor of LaTeX: f\left(x\right)f ( x ).

(d) Every polynomial function of degree LaTeX: nn (LaTeX: n\ge1n ≥ 1) has LaTeX: nn real zeros.

(e) Every graph of a polynomial function of degree LaTeX: nn (LaTeX: n\ge1n ≥ 1) has LaTeX: n-1n − 1 turning points.

(f) Every real number is a complex number.

(g) LaTeX: \sqrt{-25}=\pm5i

>>11616753
I got that primes made more sense because by multiplying primes you will get the same divisors (1,p,q,p*q), had to google this to be sure.
So do you suggest to skip to the next section instead of doing the hard problems?
T-thanks y-you too..

>>11616796
>skip to the next section instead of doing the hard problems?
NGMI

>>11616796
>So do you suggest to skip to the next section instead of doing the hard problems?
Not necessarily skip, but more like reading the next chapter for the first time and then returning to those if you feel like it. The main theory is what you should focus on, not the problems themselves!

>>11616850
Don't bully!

>>11616854
>The main theory is what you should focus on, not the problems themselves!
spoken like a true anime tranny
go whack off to some more nlab articles instead of trying to groom freshmen into being as incompetent as yourself

>>11616863
>voluntarily using nlab
Do people actually do that?

>>11616854
Ye makes sense, Ill read next section and come back to exercises after doing my daily morning routine that I couldnt do today, brb.
I just dont want to get fucked by thinking I understand but not be able to solve any exercise in future sections, thats why I wanted to do all (interesting) exercises.

>>11616850
Im going to read next section for curiosity but come back for these exercises, then reread before starting new section problems, I have done this in the past

>>11616767
Describe how you felt before and how you feel now

>>11616924
wat

>>11616978
>I just dont want to get fucked by thinking I understand but not be able to solve any exercise in future sections, thats why I wanted to do all (interesting) exercises.
I see what you mean, and I try to do the same as often as possible, but I also think it is a good idea to check the next chapter quickly before returning to the undone problems.

>>11617006
Such a horrendously ugly site that is.

I just realized the importance of trying to do the proofs of a book before reading them.

Why didn't you tell me this before, /mg/?

>>11617060
what?

for each statement (a) through (g), if it is true, write the word "True." Otherwise, write the word "False." No explanation is necessary.

(a) The graph of a rational function can never cross its oblique asymptote.

(b) If the real number c is a zero of the denominator of a rational function, then the line x = c must be a vertical asymptote of the function's graph.

(c) A real number c is a zero of a polynomial function f(x) if and only if x-c is a factor of f ( x ).

(d) Every polynomial function of degree n(n ≥ 1) has n real zeros.

(e) Every graph of a polynomial function of degree n(n ≥ 1) has n − 1 turning points.

(f) Every real number is a complex number.

(g) \sqrt{-25}=\pm5i

>>11617098
we saw you the first time faggot
nobody wants to do your homework for you

>>11617105
;(

>>11617098
>>11613906

>>11617105
This

>>11617098
if you are a cute girl post your feet (with a stamp) and i might solve your problems with nice explanations

>>11616474
>>11616474
There are lecture slides about this called "How to have more things by forgetting where you put them" by Oliver. You might also be interested in definable cardinalities. Basically you want to see how you can compare the cardinalites of various sets of reals by Borel functions.

>>11614756
g^n = g^k for some k such that 0 =< k =< n-1, right. If k is not 0, then 1 =< n-k =< n-1. But g^(n-k) = e, so the order of G being n is contradicted. Then k=0, so g^n = e.

>>11617059
>his issue is with how nlab looks
>>11617060
Pretty sure we not only told you, but your professors also told you, and it's written in the preface of a few books you read.

>>11617059
Does it look bad, though?
I mean 4chan looks very much the same as it did in 2004 and the color choice and waste of space are both feature and bug I suppose.

>>11617128
http://math.yorku.ca/~moliver/how.pdf
Is there a recording?

>>11614130
>professor sends you back to your seat midway through your proof
>says you have terrible calligraphy and penmanship and that you're not allowed back up until it improves
What the actual crap. I'm good at math. How do I get a semester-long permaban from the chalkboard just for bad handwriting?

>>11617234
based professor

>>11617192
>>his issue is with how nlab looks
I don't have an issue with its contents, as I hardly ever use it for reference. I know a priori how the grains of knowledge I could/would find in the article are buried under a mountain of HCT jargon in many cases, so nlab does not surprise me with how little I get from it. It's probably good for (higher) category theorists.

>>11617215
The diagrams are absolutely horrible in my opinion. It just doesn't please my eye the same way even Wikipedia does. A matter of taste.

>>11617240
It actually came up in my evaluations. Like, I'm pretty it's on a written transcript somewhere now.

Rate this typo.

>>11617707
so minor my brain autocorrected it while reading
2/10 would not send angry email to author

>>11617707
Where's the error?

>>11617715
>sending angry emails over typos
Do people really?
>>11617727
I'm personally used to seeing [math]\overline{X} = X/G[/math].
Wikipedia says that this notation is normal, now that I've checked it out. Don't think I ever saw it before.

>>11617747
Holy how did I not see that

>>11617756
No idea.
Wanna try to redeem yourself by figuring out the intuition behind this and explaining it to me?

>>11617761
The intuition is that you took the correct definition and dropped all the higher coherences because you're working 1-categorically.

>>11617761
No idea, but you can try >>>/out/

>>11617747
[math]N_G(H)\setminus H[/math] orbits or difference? I think I saw something like this in a paper, and it had me confused for a moment.

>>11617792
Absolutely no idea tbqh.

>>11614179
tl;dr I think they are equivalent.
I'll write [math]f[/math] for an element of [math]K(H)^*[/math], and [math]T_f[/math] for the same element when treated as a trace class operator. These two representations are connected by
[math]\langle T_f x, y \rangle = f(g \mapsto \langle g, y \rangle x)[/math], the argument of f here is a rank-one operator.
Note that both topologies can be induced by a countable number of seminorms so it is enough to consider sequences.

Sequence [math]f_n \rightarrow f[/math] converges in weak*-topology iff [math]f_n(T) \rightarrow f(T)[/math] for any compact operator T.
Sequence [math]T_{f_n} \rightarrow T_f[/math] converges in WOT iff [math]\langle T_{f_n}x, y \rangle \rightarrow \langle T_f x, y \rangle [/math] for any [math]x, y \in H[/math].
So weak*-convergence implies WOT-convergence - just plug the operator [math]g \mapsto \langle g, y \rangle x [/math] as T in the first definition.

The implication in the other direction also holds. If [math]T_{f_n} \rightarrow T_f[/math] in WOT, then you get [math]f_n(T) \rightarrow f(T)[/math] for any finite-rank [math]T[/math]. Recall that
1) finite-rank operators are dense in K(H);
2) if [math]T_{f_n} \rightarrow T_f[/math] in WOT, then the sequence [math](||T_{f_n}||)[/math] is bounded.
Then you can make a simple argument with limits.

>>11617761
I CAN'T EVEN MAKE SENSE OF THE FUCKING COCYCLE CONDITION DAMN IT
HOW THE FUCK DO I COMPOSE THOSE FUCKING MORPHISMS

>>11617825
I'll be honest with you.
I don't wanna take my time to solve anon's problem, but the sequence [math]T_n[/math] of trace-class operators which are just [math]1/n[/math] on the first [math]n[/math] diagonal entries is really firing my paranoia since it doesn't converge to zero in trace and I don't wanna double check stuff.
Do it for me kudasai.

>>11617820
Only from context later was I able to deduce it meant the orbits. Nasty notation, isn't it?

>>11617849
Yeah, horrendous stuff.

Where can I find the proof of this for the infinite-dimensional case?

Lord Dupuy has clapped back at Scholze

>>11617917
Assuming the Axiom of Choice, every linearly independent set of vectors X of a vector space V can be extended to a basis of V.
If V is a vector space and U is a subspace of V, let X be a basis of U which exists by the above result taking the set to be {}. Then X can be extended to a basis of V, call it Y. Then Y\X spans some vector subspace U' of V, and you obviously see that U (+) U'= V.

>>11617917
Fix a basis for U and extend it to V, the vectors not generating U generate W.

>>11617917
Zorn Lemma, probably

>>11617960
You mean the ultrafilter theorem?

>>11617059
Ok Im going to start now, was koding and still want to rewatch a 1h30 lecture of a class

>>11618016
Show those problems who's the boss! If you make another video, I'll check that out in the morning. Have fun!

Good night /mg/

>>11617937
based

>>11618047
good night based niceposter

>>11614788
>not using 1 for the identity element
ngmi

>>11616150
if you're only caring for expected value, it doesn't matter whether you skip or guess if you have no idea

What are some of the best textbooks in applied math? Doesn't matter what the topic is (so long as it falls within applied math), I'm just interested in seeing what you guys think are the best books for a given topic and why you think it's so good.

>>11617849
>>11617895
Isn't this the same as Γ\H, as in
https://en.wikipedia.org/wiki/Modular_curve
?

>>11618050
who's dat

>>11618102
I like this
https://www.amazon.com/Probabilistic-Robotics-INTELLIGENT-ROBOTICS-AUTONOMOUS/dp/0262201623

>why you think it's so good
A good book is like porn, I know it when I see it.
More seriously, I think, just like with prose in novels, there's a thing to be said about pace and clarity that makes a good textbook - be it applied or not.

I have 3 weeks of stay-at-home vacation in front of me, what should I do bros

Does anyone know if there exists a direct proof that [math]n_1^{1/q_1},n_2^{1/q_2}...n_k^{1/q_k}[/math] are linearly independent over Q? (you can assume what you want about n_i. technically all you need is that n_i^a =/= n_j^b for any integers I think, but you can make them pairwise coprime or even prime if that helps).

I've found find two proofs of this online, but the simpler one does some polynomial wizardry with the reducibility of [math]x^{q_k} - n_k[/math] over [math] \mathbb{Q}[n_1^{1/q_1}...n_{k-1}^{1/{q_{k-1}}}][/math], and the other one is some Kummer theory bullshit. Neither actually ever refer to a specifically _linear_ relation at any point, it's just a special corollary of a theorem about fields.

>>11618591
Have you tried being a faggot and using the nullstellensatz?

>>11618591
if the degrees of a of the polynomials in a set are all coprime the degree of the extension is the product of the degrees of the polynomials, you can get linear independence of the roots from that, but it doesn't really answer your question I guess

>>11614596

>>11614130
I'm disappointed with my high school math education because it was not rigorous enough. I was never shown any proofs. Would reading over "Basic Mathematics" by Serge Lang be a good choice? I was thinking about Gelfand series, but I think it is far to basic.

I also find geometry quite interesting.

>>11618653
>Would reading over "Basic Mathematics" by Serge Lang be a good choice?
Lang is a meme.

>>11618656
>Lang is a meme.
Instead of being a faggot, tell me what I should so instead.

>>11618659
Read Lang.

>>11618659
>faggot
Why the homophobia?

>>11618676
Why the homophilia?

>>11618680
why the philophobia?

>>11618692
Why the degeneracy?

>>11618047
Posted the video, I feel im rambling too much and losing my pace of thought, hope you can undertand from what I say and have written, exercise 30 looks wrong to me or Im missing something, read next section and will start its exercises tomorrow, gn.

>>11618676
>>11618680
>>11618692
>>11618695
Why the samefagging?

>>11618701
Why the avatarfaggotry?

>>11618706
I'm not sure how you can even call posting assorted 2hus avatarfagging, but because they're cute.

>>11618701

>>11618717
>ruins the meme chain to get defensive
dumb anime tranny

>>11614130
What does /sci/ think of OpenStax textbooks?

>>11618676
Are you a bot?

>>11614130
Random graphs are neat.

>>11618829
>Are you a bot?
No.

>>11618653
>I was never shown any proofs.
take a look at Velleman's How to Prove it and Smith, Eggen & St. Andre A Transition to Advanced Mathematics. Those are perfect for an introduction to rigorous math.

What are some cute math textbooks?

>>11618887
Lang's Algebra

>>11618887

>>11618879
Huh. Whenever I see
>>faggot
>why the homophobia?
I assume that it's a bot. It would certainly be easy enough to code. Bot-like behavior in any case.

>>11615255
https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf
This is a nice supplement I used when going through Rudin if you don't want to cheat and look up answers. The added problems, especially at the start of a chapter, are nice to get you to the ideas you need for exercises.

A good morning /mg/!

>>11618700
Some comments again:
(25) It is the exponential function indeed. This will be important later when you have the additive group of reals and the multiplicative group of positive reals. This will then give a homomorphism, or a function compatible with the operations of those groups. This is not unique, though. Any a>1 can be used as the base. The inverse is merely the a-base log.
(27) In (a) f is indeed a constant function. In (b) g is constant. These would hold for any composable functions, not just from S to S. You mentioned the special case S={x}. Note that f is then necessarily f(x)=x, so that is not a counter example!
(28) Correct.
(30) That is not wrong. For a finite set S, a function from S to S is injective iff surjective (onto) iff bijective. Try doing (29) to confirm this.
Otherwise good job!

>>11618887
My diary, desu.

>>11618529
>Isn't this the same as Γ\H, as in https://en.wikipedia.org/wiki/Modular_curve?
Yes but with the ambiguity added by the fact that the normaliser of a subgroup contains the subgroup itself, and so it is unclear from the notation whether it is the set difference or the orbits of the conjugation action.

yo nerds, I need some help understanding something:
Are these two equivalent:
∃x[Fx ∧ ∀y(Fy x = y)]
∃x∀y(Fy x = y)
I know that the former means 'there is exactly one' and that the latter signifies uniqueness. But are they interchangeable for all purposes? My solution book uses the former but I have a habit of using the second. If they are different, then what am I misunderstanding about them? Thanks.

>>11619047
fuck, the conditionals are missing
∃x[Fx ∧ ∀y(Fy only if x = y)]
∃x∀y(Fy iff x = y)

>>11619048
They're logically equivalent, though the first one is clearer

[math] 420^{69} [/math]

>>11619063
thanks friend

What plan B should I have in case I get no Ph.d offers?

I am strong in Numerics. What sort of career path should I aim for? Industries to check out?

I am about to read Infinitesimal Approach by Keisler to teach myself calculus. (I did take some calculus classes in school a decade ago but I don't remember fuckall, skipped precalc and had a teacher who didn't know what she was teaching, so I didn't really get a grasp of the content, so I am basically starting from nothing)
I have a few questions. They won't really affect what I do right now, but I just want to know for future reference.
>After reading Infinitesimal Approach, would it be worth going through a more standard approach introductory calculus book (i.e. Calculus Volume I by Apostol), or would that be redundant and a waste of my time?
>Infinitesimal Approach touches on vector calculus and differential equations, and is self-described as covering "the usual three or four semester sequence". While I am studying on my own, in this specific regard it is helpful to compare to school classes. It is my understanding that a 4 semester sequence would be calculus 1 and 2 (which would comprise volume 1 of a more traditional calculus text i.e. volume 1 by Apostol), followed by multivariable and vector calculus, and then differential equations and linear algebra. Yet Infinitesimal Approach looks to only touch briefly on vector calculus and differential equations, and it doesn't look like it even mentions multivariable calculus. So why does it describe itself as a 3 or 4 semester book? After reading Infinitesimal Approach, should I then move on to something like Volume 2 by Apostol?
If it makes any difference, I want to get a solid understanding of calculus, and then move into physics and chemistry, so if wanting to later apply calculus to physics/chemistry affects the answers to these questions, please answer accordingly.
Thanks.

>>11618887
Fuck textbooks, watch YouTube videos senpai-kun! =^-^= uvu
https://www.youtube.com/watch?v=5wFDWP5JwSM

>>11614130
Many, many years ago I had a mind-numbing lecture on Fermat's Last Theorem. Lecturer claimed that there is a proof that Fermat coudn't make a correct proof of his theorem. (17th century mathematics wasn't advanced enough.) Any info on that?

>>11619825
his name could just be pierre de fat lmao

>>11619746
sounds like you're overplanning this, just read for a month and the reevaluate

>>11618989
stop waking up

>>11615997
probably looks like kyriakos grizzly

>>11619746
>infinitesimal approach
Meme book. Read Spivak or whatever.

>>11616410

This. Euler got so fed up of philosophy pseuds at germany's court he fucked off back to russia.

>>11616410
quite unironically based my man

It's been a while since I've done this type of stuff. Please help me.

How did they get that the roots of x^4+x+1 are w,w^2,w^4,w^8 where w is a 15th root of unity?


I know that it's obvious that any root it has will be a 15th root of unity. I know that you could check this by finding a primitive root of unity and testing that its 1st, 2nd, 4th, and 8th powers solve this polynomial. I guess what I'm asking is "why do these powers in particular have to be the root of the same polynomial?"

>>11620429
[math]
x^{15} - 1 = 0
[/math]

>>11620454

But why wouldn't they be w^5,w^14,w^12,w^3 or anything else?

>>11620472
[math]
x^{15} = 1
[/math]

>>11620486

Why does your mother suck so many black cocks?

>>11620429
You know that this equality is over [math]\mathbb{F}_2[x][/math], not polynomials over R, right?
It's simple computation, if w^4 + w + 1 = 0, then
w^8 + w^2 + 1 = (-w-1)^2 + w^2 + 1 = 2w^2 + 2w + 2 = 0.

Why don't you guys do CS theory where you might do work that matters? It's still math

https://s.wayne.edu/mts2020/
Peter May included!

>>11620041
>stop waking up
Ok, but I'll first learn how to sustain myself without sleeping.

>>11620604
>another seminar uses fucking zoom
fucking americans why do they use this garbage software

>>11620619
I don't know. I wonder if there is some reason why they won't use MS Teams instead. Have people register their email on a list that will then add those emails in a team or whatever you are supposed to call those groups. It would seem pretty easy, but I'm pretty dumb, so yeah. Anyway, Peter May!

>>11620658
>MS Teams
I mean sure that's better but I for one would love a cozy discord server being made to host a seminar

>>11620683
Hah, that would be pretty funny. H3NT41M4ST3R420 giving a talk on equivariant cohomology etc.

My wife, Lang's Algebra, is so cute.

>>11620712
ngl I'm going to do just that one day
but with my regular anime online persona

>>11620735
Please do keep me informed on your plans. I don't want to miss this.

>>11620746
not going to happen anytime soon, but will inform my lovely cu/mg/uzzlers of the event when it happens

>>11620755
I'll wait. Even if it takes a few millenia.

>>11620596
>It's still math

To be fair: That's no argument, since literally everything can be interpreted as maths.

>>11620963
>literally everything can be interpreted as maths.
[citation needed]

>>11617747
You've never seen left division by a group before? Obviously the correct notation for a left action quotient is left division.
How else do you plan to take quotients by groups acting on either side, e.g. G \ X / H.

>>11614130
Why do you need to convert differentiated fractional powers to root/radical form?

>>11621151
reference?

>>11614167
Is this Judson? I'm reading this book in my class and reading Artin for depth, but judson is so much more clear.

Anyone have the tier list of the different math fields/subjects (like the image I posted but instead of majors its ranks Complex Analysis, Topology, Mat. Logic etc.) saved? I never saved it, but I remembered seeing it around here some time ago and I wanted to see where Differential Geometry fell on it so I can feel like my irresponsible life decisions have been validated (or shit on) by the autists of /sci/

>>11621493
You got it all wrong, m8. There are no tiers of fields of maths, only tiers of mathematicians and tiers of their work. In any field of maths there is shit-tier research and god-tier research.
You should be guided by what interests you and what natural questions you want to find answers to.

>>11621493
Do you mean the deep sea trench image?

>>11621502
I appreciate your post and I certainly agree with what you said, I simply only want to revel in my insecurities right now, that's all
>>11621516
Nope, that one is easy to find, I am talking about the exact same format as the one in my previous post

>>11621493
Hahahahaha here you go friend.

>>11621547
Half a minute template btw.

Someone help me improve this btw.

>>11621587
Unbelievable suicide tier, nice.

>>11621547
>SVC
woudn't even have taking me till the last row to know this comes from /mg/

>>11621608
>svc
That's a typo, it should've been scv.
I'm now trying to recall what the fuck is svc.

>>11621626
SVC could have been several variable calculus.

>>11621626
singular value decomposition

>>11621638
singular valuede composition*

>>11621626
I usually think of support vector machines first, when reading those letters

>>11618989
Ok so:
(25) Ye makes sense, I didnt think of it because I started with log and then tried with the inverse of log, and in my clg all logs would be considered of e, we wouldnt have log of 4 on a base of 3 for example, but I know what it is from HS;
(27) Got it
(30) I think I forgot to mention my main question: [redacted for rambling], so after seeing question 29 I understand, but does f: S -> S mean that all elements of S are used in both size? What does it mean for S to map into itself? It means that f(S) is contained in S, but doesnt mean that f(S) = S, otherwise it would mean its onto, right?

>>11621810
Forgot to mention, since its relatively early and I have nothing that I need to do if you want we can talk about anything

>>11618887
My favourite.

>>11621810
Hi, I was just thinking about you(r progress).
(30) If you have any finite set S and a function f from S to S, then f(S) is contained in S, yes. Now, if f is injective, then the number of elements in f(S) is the same as the number of elements in S, and so f is onto. If it is not injective, then it is not onto either. Since the problem assumes the function f is 1-1, it is bijective by 29. This result is related to the automorphism groups, where (for a finite set) the group itself is finite, and so you can compose an automorphism with itself a few times to get the identity function. Do you want a hint for this?

>>11621868
No sorry I think I got this but wanted clarification on the definitions, while I was complaining about the exercise right now about why it doesnt work I think i got it

>>11621820
Also if you want to talk about progress or something check this

>>11621892
Okaydokay! I have roughly an hour before I should be going to bed, so that I'm not a total vegetable in the morning, so I can very well talk. Anything in mind?

>>11621908
I was thinking of voice chat and basically talk about progress / my mathematical background and if I would be missing anything and anything else you like

>trying to understand the applications of set theory
>years go by, wondering what it all means
>pea-brain finally manages to crank out some electrical activity
>in a fever dream a distant voice calls out "computational set theory"
>next morning more excited than i've been in over a decade
>google computational set theory
>it's just SQL
>mfw

>>11621942
Haha relations are sets!

>>11621942
That's not a very constructive interpretation (pun intended).
I've been shilling CZF every now and then here, come on board.

>>11621928
>voice chat
Sorry but I am too shy for that, or maybe I just am too self conscious about my voice and pronunciation and so on, I don't know, plus it's midnight and my neighbour is either sleeping or trying to do that and the wall separating the two of us isn't soundproof (I can hear her voice through it when she's talking on the phone etc.), so I don't want to disturb her sleep. Sorry.

What is your background, though? I notice in the videos that the tab says "mathematics for computer scientists" and you mentioned coding. Are you a CS guy?

>>11621942
extremal set theory is cool
(it's just fucking combinatorics though and that's what makes it good fuck set theory lmao)

>>11621975
Ye alright no problem;
I dont know about that tab I cant find it, but Im taking telecom+computer engineering, the closest to 'maths for cs' would be downloading concrete mathematics by knuth and reading like 5 pages; so uh im in my 2nd of 3 years for bachelors, my degree does more engineering maths than what usually is shitposted by /sci/, not flexing, is just so youre not thinking >CS maths, I have taken calc1, calc2(multivar), complex analysis and dif eqs (same class), topics of discrete maths, linear algebra, physics on mechanics and waves, physics on eletromagnetism and optics, topics of numerical analysis, and this semester taking prob+stats and signals+systems (laplace transform, fourier series and transform), so thats my background, I think it 'matters' because Im always considering switching to mathematics on and off, and I will probably change (do masters in maths) after the end of next year and get absolutely destroyed by classes on masters level maths, and they are on every major subject, like 2 required classes in algebra+topology, 2 geometry, 2 analysis, and 2 DEs, so Im a bit worried on that, also will have to change campus, rent will be double, and all that kind of bad stuff;
For progress I feel im being slow for doing like most of the exercises, but I always do the mistake of only reading the theory and never practicing and I end up getting some exercises and confused on how I would apply that theory and end up understanding I havent actually learned anything; a good thing is today govt said 4th of May daycare should open so my baby brother wont bother me all day, hopefully, as I think thats my biggest time goblin as of now.
Dw about voice chat, I just thought it would be easier and Im not that shy so yea
Also sorry to all maths guys for this block of unrelated text

>>11622044
>I dont know about that tab I cant find it, but Im taking telecom+computer engineering, the closest to 'maths for cs' would be downloading concrete mathematics by knuth and reading like 5 pages; so uh im in my 2nd of 3 years for bachelors, my degree does more engineering maths than what usually is shitposted by /sci/, not flexing, is just so youre not thinking >CS maths, I have taken calc1, calc2(multivar), complex analysis and dif eqs (same class), topics of discrete maths, linear algebra, physics on mechanics and waves, physics on eletromagnetism and optics, topics of numerical analysis, and this semester taking prob+stats and signals+systems (laplace transform, fourier series and transform), so thats my background, I think it 'matters' because Im always considering switching to mathematics on and off, and I will probably change (do masters in maths) after the end of next year and get absolutely destroyed by classes on masters level maths, and they are on every major subject, like 2 required classes in algebra+topology, 2 geometry, 2 analysis, and 2 DEs, so Im a bit worried on that, also will have to change campus, rent will be double, and all that kind of bad stuff;
Topology has its own tricks you will get used to doing, but the prerequisites are mostly just set theory and maybe a bit of analysis for intuition, but I am fairly sure you will get the hang of it reasonably quickly. You should also have a good starting position on diff eqs with all that signal and wave stuff! I don't know what analysis means in your country, but for me it meant Lebesgue measures, Egorov, Lusin and Fubini's theorems etc. Those may be a bit more challenging, at least they were causing me a lot of headache, so don't get shocked! The algebra stuff is something you will develop a nice intuition by practicing, and Herstein really likes making people work. Do you think you could get funded by your uni?

>>11622120
>For progress I feel im being slow for doing like most of the exercises, but I always do the mistake of only reading the theory and never practicing and I end up getting some exercises and confused on how I would apply that theory and end up understanding I havent actually learned anything; a good thing is today govt said 4th of May daycare should open so my baby brother wont bother me all day, hopefully, as I think thats my biggest time goblin as of now.
It's good to do a lot of exercises, but I think it's also good to have a quick look on the next chapter when you feel like you are about ready with the problem set. Memories and stuff, people hardly remember anything from the last night binge study, but repetition helps. I can imagine a sibling who wants to play be quite bothersome if you are trying to get stuff done. How would you explain that your studies must be prioritised over playtime?

Anyhow, it's time for me to rest. Good luck to you and good night in general!

>>11622124
Gn, well I dont have a good starting position on DEs but I think ill get to it when I complete my degree;
> I don't know what analysis means in your country,
I did calc1, calc2, and Complex Analysis and Differential Equations, I guess they didnt want to name it calc 3 and I never heard complex calculus, always complex analysis, so thats why there is analysis in the title but yea I haven't done anything; also we used fubini in calc2, and I think teacher gave a proof of it but that teacher was kind of hardcore, but exams were typical engineering calc exams;
If you meant on a how hard it will be way, Ill have to take real analysis and topology fundamentals or complex analysis, and next sem functional analysis; real analysis has all those theorems in the curriculum, with lusin and egoroff being optional in the measure part;
> Do you think you could get funded by your uni
Uhh really really doubt it, RN Im getting supported by my parents, and I dont even have a maths degree so it would be next to impossible to find funding for a masters in a situation like that even in a lower tier uni, and Im in arguably best uni of my country, but not necessarily for math but I still want to take math here instead of somewhere else;

>>11622124
Alright I just feel bad for being slow in advancing, and you prob didnt expect it would take this long;

I cant really explain that, too young, but if he goes to daycare from the 4th of May Ill have way more time, I could do my daily study session and watch online classes and chill during afternoon then come back for algebra at night;
Gn

Does this actually work?

https://www.youtube.com/watch?v=gZU9nWAuE8I

>>11622287
yes

>>11621547
This is a good list.
>>11621587
Funny enough, so is this.

>>11622124
Uploaded video, I messed around too much doing nothing today, finished ex29 30 31, did like 3 of S(A) section and wrote some questions, not sure if missing a basic property on the last exercise I tried. Gn

>>11621975
>tranny insecure about his voice
what a surprise

I'm trying to show that no monic quadratic over Z9 is invertible. I don't really know how to proceed with this. If you suppose it is, then there should exist [math] x^2 + a_1 x + a_0 x = 1[/math]. If you suppose it could be factored, then you get (ax + 1)(bx+1) = 1. I don't really know where to go from here. Pls help

>>11623364
Hint: 1 is not a zero divisor.

Good morning, /mg/!

>>11622237
Now that you explained the contents of the courses, I am quite sure you will make it through them. Hopefully you will find the rent money somewhere. Maybe the uni can offer some jobs like tutoring or watching over exams. Those are not too bad!
>Alright I just feel bad for being slow in advancing, and you prob didnt expect it would take this long;
To be honest at first I thought you had no coursework left and thought you'd be faster, but that turned out not to be the case. After realising that, I lowered the expected speed, as you have other stuff to do, too. I think you are doing fine!
>>11622757
For 31, try finding how many permutations of m elements there are. I claim that number is quite important in this situation, For 5, what do we have in the middle of [math](fg)^2[/math] if we expand that? The other two were correct. Keep up the good work!

>>11623351
I'll unleash this ferocious beast upon you.

How do I get good at mathematics?
Does the sticky a good guide?

>>11623615
You will never get good at mathematics. No one can, because it's too vast of a field. You can get good at a specific subfield of mathematics, and for that you need to develop obsession with it. Textbooks, the way you study are irrelevant and won't make you better or worse at maths. The only thing that matters is interest- an obsession.

>>11623615
Why do you want to get good at mathematics?

>>11623615
spend 10000 hours doing math
you can start with Jech Set Theory and some logic book like "How to prove it"

>>11623645
>>11623935
Entertaining answers.

He'll sure fall into a Von Neumann despair cope.
He'll sure like Jech introducing transfinite recursion on page 10.

>>11623946
Can't say anything about Jech or How to prove it, but the other post is pretty much on the nose.

>>11623956
>pretty much on the nose
You mean on point?

I mean yeah there's something to say about that, but it's a bit too much of an "atheist skeptic autist" sort of answer.
>How to get good at mathematics?
>Nobody can get good at it because..
Yeah, sure, but there's an intention behind the literal question that has been sidestepped through this answer.
That elaboration aside, I meant it when I said I find it entertaining - I liked those answers.
And in any case, pleb questions don't belong to /mg/ anyhow.

Also, yes is not a difficult book per se, but by no means an entry point to math.
Indeed, ZFC is to math what a basketball is to soccer. You can bring them together and have a lot of fun, but it's all a bit off.

>>11623983
How much of Jech have YOU read?

>>11623983
On the point, yes sorry. I guess I didn't wake up this morning. The obsession part is what I meant, it is a helpful property for one who wants to get decent in maths. The nobody-part is debatable, and so on, but this isn't relevant. The obsession part is good.
>soccer
I do believe you mean football.

>>11623983
er, mostly just the parts that interested me, the first two chapters and some on L

>The obsession part is what I meant, it is a helpful property for one who wants to get decent in maths.
I'm afraid that's true.

Fußball

>>11623995
er, mostly just the parts that interested me, the first two chapters and some on L

>>11624001
>The obsession part is what I meant, it is a helpful property for one who wants to get decent in maths.
I'm afraid that's true.

Fußball

Hey ZFCbros, how does this proof work? As far as I can tell, Jech only proved that it's [math]\Sigma_1[/math]. Why is it also [math]\Pi_1[/math]? This is page 187 of Jech.

After watching (language: German):
https://www.youtube.com/watch?v=TCXKWIRBhZY

Is [math]\{x \in \mathbb{R}: x \leq 5\}[/math] really technically wrong, and it's merely become an accepted shortcut for [math]\{x: x \in \mathbb{R} \wedge x \leq 5\}[/math]?

>>11624024
I'm unlikely to be able to help you, but I don't see where your pic relates to Pi_1

>>11624036
to me, the first one is correct and the latter is wrong/sloppy

>>11624036
I've only skimmed the video, but it seems he is making a valid point, while your example is unrelated to what I saw in the video.

Both
>{x ∈ R: x ≤ 5}
>{x : x ∈ R ∧ x ≤ 5}
denote the same set and will be understood.
Yes, the last one can be said to be more casual.

As far as FOL set theory is concerned, either notations are mere shorthand.
I think in Hilbert style calculus there's really only closed propositions and in natural deduction terms may pop up in existential instatiation, but otherwise there's no standalone terms/objects like "{n | n ∈ N ∧ n > 9000 }"

Read brackets
[math] X=\{a,b\}\equiv \forall c.\ ( c \in X \Leftrightarrow (c = a \lor c = b)) [/math]
as shorthand, as well as
[math] X=\{x|P(x)\} \equiv \forall x.\ (x\in X\Leftrightarrow P(x)) [/math]
and
[math] \{ x | P(x) \} = \{ x | Q(x) \} \equiv \forall x.\ \left(P(x) \Leftrightarrow Q(x) \right). [/math]
or
[math] \{f(x)|P(x)\} \equiv \{y|\ \exists x.\ (y=f(x)\land P(x))\} [/math]

Pic related is some notes on the topic I took a while ago (must clean up the formatting)

where [math] \equiv [/math] is just introduction of names and notation

>>11624050
A sentence is traingle_1 if and only if it's bothe pi_1 and sigma_1. You said you've read the chapter on L, which is the chapter I'm asking about. I thought that having read it, you would have understood it.

>>11624017
Fußball is much besser than soccer.

>>11624142
No sorry, but we can try if you dare us

>>11624182
I dare you!

>>11617098
What do you feel is the right answer?

>>11624196
I'm out in the sun and don't have the book.
What's Theorem 13.12 and is this about the defining definition about the graph of this map. And does that help us?

>>11624036
I haven't watched the video, but the second notation is basically nonsense tbqh.
The axiom schema of replacement guarantees that [math]\{ x \in A : \phi (x) \}[/math] is always a set, but it doesn't guarantee that [math]\{ x : \phi (x) \}[/math] is. Thus, the issue.

>>11624216
That's a fancy church you are having there. Germany? For some reason the photo sends Germanic vibes and the time zone would be correct. I'll go out, too. Inspired by you!

>>11624231
https://en.m.wikipedia.org/wiki/St._Stephen%27s_Cathedral,_Vienna

Good, but let's not give the problem up for adoption

if I started going through a pretty basic maths textbook (specifically Elements of Abstract and Linear Algebra by Connell), which doesn't have an answer key for the proofs, would /mg/ let me post my proofs here to be critiqued or would you bully me and tell me to stop spamming?

>>11624245
>>>/sci/sqt
>>>/wsr/

>>11624228
>replacement
Surely you mean separation/specification?
>>11624216
Hi Nikolaj, thanks for offering me help. Pic related.

>>11624245
>most of my proofs
no, I don't want to read 19 baby algebra proofs a day
if you want to post one or two you're really not sure about people will probably only call you stupid every once in a while

>>11624276
>specification
Yes.

>>11614603
>>11615098

Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

Urgh, I was being harassed by insects. At least I managed to do some solar powered homology.

>>11624242
Almost correct. Cute doggy.

>>11624245
If you did like this other anon does and posted videos where you tell what you did today and what was unclear, then I can take a look at those and beat your self confidence to pulp with my critique, as long as the videos are less than 15 minutes.

>>11624315
Misha privet!

>>11624315
now create lists for discrete and applied math

>>11624276
I'm home and looking at the chapter.

Lemma 13.10 (vii) says if [math] F [/math] is [math] \Sigma_n[/math] , then to prove [math] F[/math] is [math] \Delta [/math] , we only need to show it for the domain. Although the prove of that statement itself is awfully short.
(This kind of thing was what I meant with using the graph-like aspect of the function set.)

And I suppose the [math] \alpha[/math] in [math] \alpha\mapsto L_\alpha[/math] is an ordinal, and the first line of proof 13.12 tells us that this domain, [math] {\mathrm Ord} [/math], is even [math] \Sigma_0 [/math]

>>11624276
I'm home and looking at the chapter.

Lemma 13.10 (vii) says if [math] F [/math] is [math] \Sigma_n[/math] , then to prove [math] F[/math] is [math] \Delta_n [/math] , we only need to show it for the domain. Btw. this kind of thing was what I meant with using the graph-like aspect of the function set, I just looked out for a hint on that.
Although the proof of that statement itself is awfully short.

And I suppose the [math] \alpha [/math] in [math] \alpha\mapsto L_\alpha[/math] is always just the ordinal (although the book is not super clear on that either, but the +1 in Definition 13.1 gives not much other options). And the first line of proof 13.12 tells us that this domain, [math] {\mathrm Ord} [/math], is even [math] \Sigma_0 [/math] .

>>11624628
danke

>>11624520
>list
>for discrete math
here's your list
1. any textbook with the word combinatorics in the title
2. that's it, go do research

>>11624642
applying the same logic to the mg curriculum topics would take all the fun out, no?

>>11624667
That list is not from /sci/ originally. It is by Misha Verbitsky.

>>11624667
No, because in order to do the kind of complex geometry that list assumes you want to do you actually DO have to read a bunch of prerequisite books

>>11624633
Bitte.

Take this on your way

[math] z\,\sum_{n=0}^\infty\dfrac{1}{(z+n)^2}=\sum_{n=0}^\infty B_n^+\,\dfrac{1}{z^n} [/math]

[math] z\,\left(1+\dfrac{\cosh(z)}{\sinh(z)}\right)=\sum_{n=0}^\infty B_n^+\,\dfrac{(2z)^n}{n!} [/math]

>>11624882
this kills the category-theorist

>>11624895

>>11624882
Very few things in mathematics are as satisfying as a good generating function

>>11624895
lel

but then again, not really,
pic related

>>11624922
*rubs hands in <analytic combinatorics>*

>>11624937
shalom shalom

>>11624937
mmmmmmm

[eqn]\sum F_n^2 z^n = \frac{z(1-z)}{(z+1)(z^2-3z+1)}[/eqn]

>>11624990
dammit hiro you ruined it, it worked in the preview
[eqn]\sum F_n^2 z^n = \frac{z(1-z)}{(z+1)(z^2-3z+1)}[/eqn]

>>11624922
I think applying the Leray-Serre SS to a case where the fibre or the base space is a homology sphere is also quite satisfying.

>>11625007
[math]

>>11624520
>now create lists for discrete and applied math
why

>>11625047
I might be horribly confused here (as evidenced by repeated LaTeX botching) but aren't the math tags supposed to be for inline text [math]like this[/math] and eqn tags for
[eqn] stuff like this [/eqn]

>>11625171
of course it fucking works when I'm just posting garbage

>>11625171
It's $ vs. $$.

>>11625047
post

>>11625349
Modern?

>>11625370
if you will

>>11625410
What did you want me to post?

>>11625419
the thing that animates you

>>11625419
I think he wants you to explain what happens to the Serre spectral sequence when the base or fiber is a homology sphere.

>>11625438
>Serre
Leray-Serre.

>>11625438

>>11625438
In that case, let's start with an actual sphere of some positive dimension (at least 2 if it is our base space). There is the nice fact that the only non-trivial columns (base space) or the only non-trivial rows (fibre) are the 0th and the one corresponding to the sphere's dimension. If the space in question is a homology sphere, then its (co)homology is that of a sphere of some positive dimension, and the same things apply. The only non-trivial differentials can then be the ones that reach far enough horizontally (base) or vertically (fibre).

I stumbled across a situation like this today with a space I needed to consider consisting of a bunch of cells of dimension at most 8 and one of dimension 10. Pinching the 8-skeleton to a point gave me the 10-sphere and a map from the original space into this sphere. Taking the fibre of that map, I was in the situation where the base space ended up giving me a nice 9-periodic cohomology for the fibre.

>>11625437
I wish I knew. I'd keep that stuff close to myself every day. At the moment I am animated by caffeine.

Is Fractional Calculus/Fractional Differential Equations an active area of research? It seems interesting but I'm wondering if it's dead.

>>11623371
I still don't really understand how to do this. How does 1 not being a zero divisor help me here?

>>11625507
You say "is dead", but I'm not aware it was every much alive to a good degree.
I knew one, two Master students who wrote something in that direction. Also I am interested in it for it's connection to Fucker-Planck and Feynman–Cuck formulas

https://en.wikipedia.org/wiki/L%C3%A9vy_flight

Read up some on it, write it up and report back.

Are homeomorphisms of the 2-sphere preserving antipodal pairs rotations?

>>11625635
no

>>11625546
>How does 1 not being a zero divisor help me here?
Suppose you try to multiply your quadratic by a*x^n+(lower degree shit). What is the coefficient of the (n+2) degree term?

>>11625507
Papa flammy did his master's thesis on the fractional derivatives of [math]\zeta[/math], so not dead?

>>11625683
Ahh I see it now. If I suppose a polynomial [math]x^2 + a_1x + a_0 [/math] is invertible, then there exists a polynomial [math] p(x) = sum_i^n b_nx^n[/math] such that [math] (x^2 + a_1x + a_0)(p(x)) = 1[/math]. But then we require [math]1*b_n = 0[/math] which is never true over [math]Z_9[/math]

best Graph Theory texts ? i am currently reading Graph Theory and Combinatorics but i'd like to have a book or two more to reference if something doesn't click ( :

>>11625700
>Papa flammy did his master's thesis on the fractional derivatives of ζ
Ah, I remember heaving heard that.
Just looked it up.
https://www.researchgate.net/publication/339627752_The_Fractional_Derivatives_of_the_Riemann_Zeta_and_Dirichlet_Eta_Function

Impressions:
70 pages. Fractional derivative is only introduced on page 42. I see he doesn't really choose a formally explicit theorem-proof declaration style. I notice he switching between "I" and "we" and "paper" and "thesis".
p. shows it's about
https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative
(only 1 edit in 2019, sad)
Seems to work out ideas presented in a paper one can also find online.
He bashes the author, lel
Looks like the [math] \alpha [/math]'th derivative of the zeta function has a shifted critical strip and a functional equation/reflection formula (that is quite complex, in fact it's a tripple sum).
It's mostly looking at the series you get if you apply that fractional derivative operation to Hurwitz-type functions.

tl;dr It has not too much about the theory of fractional derivatives in general, but aims at parametrizing zeta-function results with a parameter [math] \alpha [/math] in the way implies by that operation.

>>11625825
i am a world-class expert at graph theory and combinatorics, just use /mg/ as your reference bro

good point but i wanna see the Current Day Takes /mg/ers have about graph theory texts

>>11623404
>jobs like tutoring or watching over exams
Hm I doubt it but I can always try, I could also try for some TA-like position, hope Ill figure it out..
> For 31, try finding how many permutations of m elements there are
Uh Im not sure why or how it is related to the problem but Ill recheck it, is my solution wrong?
> For 5
Makes sense, got it
Also I asked a question on for example f: S -> S, what if f(s) = 1/s; would f(0) return undefined? So later today I realized that we would have to say 0 doesn't belong to S, problem solved.

How do I change my brainlet into a giga mathbrain?

>>11626015
install gentoo

>>11624024
Set theoryfag here. Every [math]\Sigma_1[/math] function with a [math]\Pi_1[/math]domain is [math]\Delta_1[/math]. Consider the formula
[math]y=f(x)\longleftrightarrow x\in \mathtt{dom}(f)\wedge(\forall z)(x=f(x)\to x=y)[/math]. The domain of your function is an ordinal so it is [math]\Pi_1[/math]. So the right side of the forumla is [math]\Sigma_1[/math] and the left side is [math]\Pi_1[/math], hence [math]\Delta_1[/math]. Also you might want to use a better text to learn constructibility theory. Try Devlin's Constructibility book (there is an error in this book but it is not significant for later results) or Kunen's Set Theory book. Jech is better as a reference once you have learned stuff.

>>11626028
>(there is an error in this book but it is not significant for later results)
not him, just curious; do you mean it proves an actual false theorem? It's not clear to me what you'd be referring to since every math book ever has an error of some kind in it

>>11626028
Thanks. Ive started reading Kunen. Its more comprehensible.

>>11626035
His book is really in the nuts and bolts of definability. The very weak system of BS set theory he uses to do the encoding of formulas and the satisfaction relation in the beginning of the book is not sufficient to do what he wants to do. Mathias fixed it by adding as axiom that says for every natural number its subsets exist. The paper that fixes it is called "Weak Systems of Gandy, Devlin, and Jensen."

Hi smart people of /sci/,

does anyone know where I can find a logic formula generator? I'm having a lot of fun proving whether arguments are valid or invalid in propositional logic, but I'm running out of exercises and just want to run through a shitload of them with a generator. Anyone? I tried google and all I got was truth table generators which are cool, but not what I need. I tried looking for random theorems on proofwiki but most of them are numerical.

Or maybe someone can make one for me? :)) lol

>>11626097
Loop through strings of characters. It will generate all logic formulas, among other things

>>11626028
Rereading my post there are two typos, the x=f(x) on the right side should z=f(x), and when explaining the complexities I guess I don't know left from right. The right side is [math]Pi_1[/math] and the left is [math]Sigma_1[/math].

>>11626008
Hiya!
>Hm I doubt it but I can always try, I could also try for some TA-like position, hope Ill figure it out..
Not a bad option.
>Uh Im not sure why or how it is related to the problem but Ill recheck it, is my solution wrong?
It's not wrong, but this is more concrete. Given any bijection from S (m elements) to itself, it fill fix n elements (could be 0) and permute the rest m-n elements. This actually goes the other way around, too. Given any n elements in the set S, there is a bijection permuting precisely those elements and keeping the rest fixed. Using this idea, it is reasonable to think that m! would work for any such bijection. I'll leave the details to you.
>Also I asked a question on for example f: S -> S, what if f(s) = 1/s; would f(0) return undefined
And I forgot to answer that... sorry about that.

>>11626015
Get obsessed, get possessed! Try going outside with a book, something to write on (Or is it in if it is a notebook? You can close it, so the text would be inside...) and your preferred writing tools. A nice park, a forest, your local shore, whatever. Just minimise the distractions caused by your zoomer mind's internet addiction.

Suppose I have [math] K \leq G[/math], [math] N \trianglelefteq G, \pi : G \to G/N [/math]. I'm trying to show [math] \pi^{-1}(\pi(K)) = KN.[/math]. If I'm understanding this correctly, this is just writing out the definitions of these sets. So [math] \pi^{-1}(\pi(K)) = \{ x\in G: \pi(x) \in \pi(K) \} = \{ x \in G : xN \in \pi(K)\} = \{kN : k\in K\} = \{kn : k\in K, n \in N\} = KN [/math].

Here is where I am having trouble. I'm trying to show that [math]\pi^{-1}(\pi(K)) = K iff N \leq K[/math]. Can I get some help here?

>>11626112
Have you tried showing that [math]\pi(x) = \pi (xN)[/math]?

>>11626112
Suppose [math]N\not\le K[/math]. Then there is some [math]n\in N\setminus K[/math] and [math]kn\not\in K[/math] for [math]k\in K[/math] (why?). Conclude the result.

>>11626112
N < KN, since e is in K
so if KN = K, then N < K
and if N < K, then N < KN < KK = K

/mg/ has gotten a lot worse over the last 2 or 3 editions, and I think this one takes the cake.
I'd rather have racism and people flinging shit at each other than this garbage.
We traded one tranny avatarposter for another, but the new one is a lot more annoying. At least yukari was kind of a /mg/ meme, the new one is just a massive faggot. Speaking of which, the retard going through whatever utterly trivial set theory shit and engaging with the tranny is probably the biggest faggot I have ever seen on this website. /mg/ is not your personal playground or private chat, retards. You're shitting up the thread to the point that people are shying away from it and choosing to no longer engage, which is why the thread has been so dead. Even the "work with physishits" guy is gone.
All trannies should honestly just 41%

To make this more on-topic: I've taken a course (undergrad) on Riemann surfaces and really enjoyed it. I'm doing my masters next year, is there any course/ area other than complex manifolds that keeps building on the subject?

>>11626213
the new avatarposter is better than yukaritranny

>>11626232
*just as bad as

>>11626213
compact Riemann surfaces are projective varieties, you can do algebraic geometry with them.

>>11626213
seethe

>>11626213
And topology?

I don't know if you mean me with the set theory, but at least I'm sure the work with physicists guy will be back.

>>11626213
>Even the "work with physishits" guy is gone.
I'm not a "guy".

>>11626037
>>11626028
>Kunen
Hell yeah.

>>11626213
>is there any course/ area other than complex manifolds that keeps building on the subject?
>building on
I'm really not sure what you mean with building on, do you mind elaborating?
Have some links in the meantime.
https://en.wikipedia.org/wiki/Teichm%C3%BCller_space
https://en.wikipedia.org/wiki/Modular_curve

>>11626247
And the noncompact ones are Stein spaces.

>>11626213
Do Lie groups while waiting for your masters to start. They are pretty neat and somewhat related to your interests!

Anyway, good night /mg/!

>>11626247
I don't wanna go down that rabbit hole desu. I guess I should have specified that I didn't do the alg geo course offered this year so I might not have the background for that next year anyway.
>>11626262
I don't even know who the set theory faggot is. All I know is that he makes youtube videos and that the retarded tranny keeps making a million posts saying "good work! here's how to do this trivial garbage you missed". I stopped reading posts about this shitshow after one about proving that a function is bijective iff it has an inverse. Come on, that's middle school level and doesn't belong here - the fact that the tranny is encouraging this level of discussion on /mg/ is pathetic.
>>11626294
There's only one thing I could possibly mean, dumbass.
Me take course Riemann surfaces this year. Me ask what courses most similar next year?
Seriously, what broad area of modern maths does this fall under, and what related subjects should I study?
(11626384)
I hope you die in your sleep.

>>11626535

algebra of sets is cool as shit. who gives a fuck if it's an entry level questions. /mg/ is a thread of peace, take your bs somewhere else.

>>11626535
Yeah that's me.
I don't know who says "good work" though or about the bijective story, I never read that here.

>>11626535
A bit rude, no?
There's various discord server with more math content and faster banter if that's what you look for.

>>11626535
>what broad area of modern maths does this fall under
>implying maths is neatly divided into several subjects
>what related subjects should I study?
Stein spaces (as mentioned before, non-compact Riemann surfaces are Stein manifolds), Riemannian geometry, Kahler geometry, Teichmuller theory, commutative algebra, algebraic curves, complex geometry, more complex analysis in one variable (read the full Conway course), elliptic pdes (remember the result about how solutions to Cauchy-Schwarz are full on analytic functions? It generalizes) and J-holomorphic curves come to mind in particular.
But I'm more of a symplectictard than anything else, I'm mostly recommending you stuff because you're going out of your way to be unlikeable and I find it endearing.

>>11626598
Forgot to mention length spectra and systolic geometry.

>this is a thread of peace
>using discord
Is every single one of you a tranny now? I sincerely, genuinely, unironically and from the bottom of my heart wish that you die slow, painful deaths. You are worthless scum.
Since I'm getting banned anyway, I might as well extend my above wishes to niggers, kikes, faggots and jannies

>>11626535
Dude the set theory that was discussed was stuff most undergrads don't see. I'm not that Nikolaj guy but am a grad student in set theory. Why do you have such a bad attitude? Ignore posts that don't concern you and post stuff you want. If this thread Is so pathetic them leave loser.

>>11626598
>Cauchy-Schwarz
Cauchy-Riemann.
I rate my own typo a solid 9/10.

>>11626604
/mg/ has been and always will be opposed to discussions about foundations. I'm sorry your life is so sad that you're doing set theory in grad school (lmao), please make your eventual suicide entertaining at least

>>11626628
Keep seething loser

>>11626604
I gather by the bijection comment he meant the discussion about equivalence classes on R that have more elements than R.
Also I've never took a course in set theory, it's just a topic where people have often problems wrapping their heads around things and so you can explain things in a sentence.

Nobody on 4chan (probably also not discord, for the most part) puts in work of working things out for other people (since often, when you do, the OP wouldn't respond anyway and you might as well talk with a wall). So it's mostly chill banter. (And yurakari or what's it called overshoots in an annoying way where he just uses the questions to rant about vaguely related things he takes and interest in.)

>>11626628
I'd argue the opposite is the case. In 2015-2018, I felt there was a lot of light category theory mumbling.

At this point you have to be trolling. No one can possibly be this retarded.
Literally the FIRST post in this thread is the one about bijectivity and having an inverse. The reply chain is an example of the shit I'm against. Saying "your argument for this question is just that 2 and 3 are prime numbers" or shit like that is utterly retarded and moronic. Then the youtube faggot even said he hasn't gotten to groups yet.
What kind of absolute lunatic do you have to be to support a guy without working knowledge of high school math posting REPEATEDLY in this thread? Him and the tranny turned this thread into a public one-on-one chat.
I was never ranting about the serious set theory talk in this thread. It's customary to shitpost and flame people discussing foundations but I don't really care about it. My gripe is the kindergarten level faggot and his tranny caretaker shitting up this thread.

>>11626706
what do animes have to do with trannies?

>>11626628
/mg/ is opposed to foundationalism itself, but is (should be) always open to discuss it

>>11626716
Autokawaiiloliphilia

>>11626722
>foundationalism
define

>>11626731
Do you think that some people out there are genuinely that fucked in the head?

>>11626384
Updated, by the end things that I had done some minutes earlier stopped making sense so I just stopped. Gn

A good morning /mg/!

>>11627031
(10) Like I mentioned before, 6=3! makes every composite the identity. In this case it is also the smallest option.
(13) It's worth checking that it really does the trick. Just remember that all permutations are just products/composites of cycles of different length and then you want to kill all the cycles.
These things you are verifying now for these symmetry groups will be discussed more generally later. The cardinality of the group as a set is called its order, and the smallest positive integer (or infinity) turning an element to the identity element is called the order of the element. Later in the book, Herstein will show that the order of an element divides the order of the group in the finite case, but this does not work the other way around though. There are, like you wrote them down, only those 6 possible permutations of 3 elements, but none of them has order 6. Just something to keep in mind when you get there!

>>11626232
>>11626247
>>11626254
>>11626262
>>11626271
>>11626294
>>11626384
Fucking retards replying to a whiny faggot. You've now ensured he will post in every thread about how waaah there are some posts he does not like. Just ignore that retard and move on.