/sci/ - Science & Math

Just doing some exercises.

Currently reading "a course in homological algebra" by Hilton and Stammbach

Does someone know where i can find some BSc theses to read? I still have to ask my advisor for a topic and wanted to ask about something homological algebra-related, is it a good idea? What's a nice subject in homological algebra that's interesting and suitable for a bachelor's thesis?

Just finished reading pic related. It was honestly difficult to comprehend, but the concepts got a bit easier to understand after the first few chapters. I've read a lot of high-level math texts in my years, but this one is by far the most rewarding.
Has anyone else picked this up? What did you think?

>>9702543
Seriously why do you want to do something “homological algebra” related. Homological algebra is just language and set of techniques which are usually trivial when considered in generality, they acquire life and interesting structure only when they arise from actual mathematical objects (I mean, unless your a category diarrheaist)

>>9702562
Well, i'm interested in its applications (hopf algebr[as/oids] in noncommutative geometry for example) but i'm still an undergrad, i need an undergrad thesis topic... (i don't really know much about undergrad theses desu so i was guessing)

Pappus - Collection (ancient geometry)
Vilfredo Pareto - Manual of Political Economy (the index is calculus)
Aristotle - Metaphysics (the foundation for most scientific understanding)

And I'm also reading The Federalist Papers, this is the only non-/sci/ thing I'm reading.

Soare's Recursively Enumerable sets and degrees. Just starting to get proirity arguments so it is real fun. Also been learning forcing in computable structure theory.

>>9702559
anon... that book is randomly generated...

>>9702562
>they acquire life and interesting structure only when they arise from actual mathematical objects
Not really. Something becoming significantly more interesting when applied to certain contexts doesn't make that thing dead and not interesting.

>>9702619

You tried starting a good /lit/ thread that went nowhere the other day, didn't you? Check the expired thread one more time if you haven't already.

>>9702619
Keep off-topic material contained to some other board.

>>9702637
I don't see it in the archive. You could link it

>>9702638
>doesn't understand minors or apotomes
>thinks what I'm posting isn't /sci/ related
L0L

>>9702658
>/sci/ related
Keep non-mathematical trash contained to other threads if you claim it's /sci/ related.

>>9702658

This: https://boards.4chan.org/lit/thread/11057457/

Started reading this. It's cute. Talks mostly about sangaku, geometrical theorems painted onto tablets and left at shrines as offering, but also goes into the history of mathematics in Japan/China more generally.

>>9702711
I have acquaintances, how about that? I also work a lot, which is why I didn't reply to that thread after I made it.

>>9702709
Pappus is very much mathematical. Do you even know what the Collection is? It's a commentary on all mathematics including Euclid, Archimedes, Apollonius, and Nicomedes. I mean honestly, this man is trying to claim Calculus and ancient geometry isn't mathematical. You should be castrated.

>>9701959
I completed 8th grade on Khan academy. Now I can do multiplication in my head and long division.

>>9702794
PS can anyone explain what a logarithm is?

>>9702636
Literally go read a book, dumbass. The source of every single homological construction is a geometric or topological functor and the study of their failure to be exact. The only general theorems are trivial ones that are consequences of couching things in the language of morphisms and functors. Any non-trivial construction comes from actual mathematics, as evidenced by literally all of mathematics since Leray invented sheaves

>>9703255
>Any non-trivial construction comes from actual mathematics
define "mathematics"

>>9703257
No thanks. If you don’t know what math is you can go chase tautologies via diagrams for the rest of your life for all I care

>>9702634
I want to believe.

>>9701959
>>9703392
H-how do I solve for [math]p_3[\math]?

[math]
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{11} \\ b_{12} \\ b_{13}
\end{bmatrix}

=

\begin{bmatrix}
\begin{pmatrix}
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{21} \\ b_{22} \\ b_{23}
\end{bmatrix}

\end{pmatrix}
\begin{matrix}
x_2 & x_3\\ y_2 & y_3
\end{matrix}
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{31} \\ b_{32} \\ b_{33}
\end{bmatrix}

[/math]

>>9701959
>>9703392

H-how do I solve for [math]b_3[/math]?

[math]
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{11} \\ b_{12} \\ b_{13}
\end{bmatrix}

=

\begin{bmatrix}
\begin{pmatrix}
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{21} \\ b_{22} \\ b_{23}
\end{bmatrix}

\end{pmatrix}
\begin{matrix}
x_2 & x_3\\ y_2 & y_3
\end{matrix}
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{31} \\ b_{32} \\ b_{33}
\end{bmatrix}

[/math]

>>9703414
There's no b_3 in your equation.

>>9703416
The vector at the far right.

>>9702797
integral from 1 to x of 1/x is log x

Just learned about dedekind cuts, now what? metric sapces? limits?

>>9703501
Just read rudin

>>9703525
fuck that, I'm better off learning from ProofWiki

Can some topology anon check my proof that the 0th singular homology group of any space is the direct sum over the path connected components? I've seen others that usually involve the first isomorphism theorem, but i think this one is conceptually easier:

Let [math]X[/math] be any topological space, and let [math]x\sim y[/math] iff they are on the same path component (it is an equivalence relation obv). Then consider the quotient [math]X / \sim[/math]. It is easy to see that it is the disjoint union of a representative point over all path components, that is [math]X/ \sim\; \cong \coprod_\alpha [x_\alpha] [/math]. The homology of a disjoint union is the direct sum of the homologies, so on one side we have [math]H_0(X / \sim)=\bigoplus_\alpha \mathbb Z[/math]. Now I claim that [math]X / \sim[/math] and [math]X[/math] are homotopy equivalent, and hence their homologies coincide.

Consider the inclusion [math]i[/math] of the equivalence class representative into the space [math]X[/math] (it is well defined up to homotopy) and the projection to the quotient space [math]p[/math] as the other map. We have that [math]pi[/math] is the identity clearly. Now let [math]\gamma^\alpha_x\colon [0,1]\to X[/math] be the path that takes [math]x[/math] to [math]x_\alpha[/math] for [math]x[/math] in the path component of [math]x_\alpha[/math], that is, [math]\gamma^\alpha_x(0)=x, \gamma^\alpha_x(1)=x_\alpha[/math]. In any path component, we can require that for any two distinct points [math]x,y[/math] that the two paths [math]\gamma_x^\alpha[/math] and [math]\gamma_y^\alpha[/math] are homotopy equivalent (just take the path [math]y[/math] to [math]x[/math] to [math]x_\alpha[/math]). Then the homotopy [math]H(x,t)=\gamma_x(t)[/math], defined to be [math]\gamma_x^\alpha[/math] on each component, takes the identity to [math]ip[/math]. [math]\blacksquare[/math]

>>9703501
Forget them because you will never think about them again.

>>9703618
but I think about them all the time

Sobolev spaces... A threat or an opportunity? My instructor suggested me to write my Bachelor's thesis about Sobolev spaces based on what courses I have taken so far, but I don't know shit about Sobolev spaces apart from quickly skimming through the wikipedia article. Should I do it? Is it even math, or is it just engineering in disguise? Please help!

>>9702543
>Does someone know where i can find some BSc theses to read?
Google. ALGANT and other 'special' schools (as well as normal ones) publish all the theses.
>I still have to ask my advisor for a topic and wanted to ask about something homological algebra-related, is it a good idea?
Just ask him. I went with a topic and it was sadly rejected, and I was asked to write about something in a different field. Don't dwell on it too much.
>What's a nice subject in homological algebra that's interesting and suitable for a bachelor's thesis?
Maybe studying/comparing some (co)homological theories, such as Spanier and so forth? Or go for a topic which is in the expertise of your advisor.
Anyway, how much homo algebra do you know? Already seen derived functors? Applications in topology/algebra?

>>9703587
>Let [math]X[/math] be any topological space, and let [math]x\sim y[/math] iff they are on the same path component
Connected by a path.
> Now I claim that [math]X / \sim[/math] and [math]X[/math] are homotopy equivalent, and hence their homologies coincide.
Wrong. Not every path connected space is homotopic to a point. Take a circle for instance.
> Conceptually easier.
Wrong. Use
> The homology of a disjoint union is the direct sum of the homologies
and the usual derivation of 0th-homology for path connected space as in pic(Shastri).

>>9701959
Currently learning Mathematica and Linear Algebra
I have a long way to go, /mg/

>>9703699
To answer your question, and I say this without memeing as I speak from experience, you'll not even find an engineering student at Masters level who knows what a Gamma functions is.

>>9703815
There's a business school in spain that teaches you the gamma function in like first or second year lol, not that they do it very rigorously but yeah

>>9703699
>Is it even math
Yeah, of course, it isn't even that strange of a concept.
Not that I have heavily studied it, but the core concept seems not that strange.

>is it just engineering in disguise?
I doubt that.

>>9703815
I know engineering students below a master level who have heard of what Sobolev spaces are.

>>9703587
LMAO dude this is so wrong. your reasoning implies that every path connected space is contractible

Cohen's textbook on stable homotopy theory. The part where he starts talking about the Freyd category is quite interesting, as this happens to not only be an abelian category but Frobeni(u)s as well. Also a book on the localisations and extensions of Grothendieck categories. After these, a book on categorical constructions in stable homotopy theory. These should keep me busy for a while.

>>9703988
It's true in ZF

>>9701959
Baby Rudin

>>9703501
>dedekind cuts
are a spook and non rigourous

ok im having another brainlet moment. In need to show there exists two chain complexes and a chain map such that the degree 0 map is not the zero map, yet the induced map on homology is. My idea is to make it such that the homology group of the target chain complex is 0, so the map must be the zero map.

So we have two chains (assume where the first [math]\mathbb Z[/math] is in the first chain is degree 0):

[eqn]0\to 0 \to \mathbb Z\to 0\\
0\to \mathbb Z \to \mathbb Z\to 0[/eqn]
Where the horizontal and vertical maps [math]\mathbb Z\to \mathbb Z[/math] are just the identity (isomorphism ). The homology of the first chain is 0 everywhere and just
[math]\mathbb Z[/math] at degree 0, and the homology in degree 0 of the second chain is 0 since the image of the boundary map in degree one is an isomorphism. In particular, the induced map must be the zero map.


But homology is functorial?? How can the identity map be the zero map? Did i do something wrong?

>>9704096
The homology functor gives you the homomorphism you get by restricting the given homomorphism to the kernel of the boundary morphism, and then taking the quotient with respect to the image of the preceding morphism. You can make your idea work if you swap the complexes you have. Start from the 0 -> Z -> Z -> 0 and go map it to 0 -> 0 -> Z -> 0. If the morphism Z -> Z is the identity, then its image is the whole Z which is also the kernel of Z -> 0, and so ker/im = Z/Z = 0, and the induced morphism is 0.

>>9704078
>non rigourous
What do you mean?

>>9701959

>>9704078
njw pls go

>>9703414
>>9703416
>>9703417
I am but a humble brainlet. Someone pls halp.

>>9704357
I doubt that a unique solution has to exist.

Kolmogorov complexity is such a brainfuck to me

>>9704369
It should. It maps a point in one barycentric coordinate system onto another barycentric coordinate system using only the relative coordinates, so it has a pretty good geometric justification. I'm just not sure how to invert the 2x3, especially when one of its columns is defined as a matrix-vector product.

Does anyone else have some ideas?

>>9704381
>I'm just not sure how to invert the 2x3
You can not a non square matrix can't have an inverse.
Your system has 3 variables and 2 equations, I do not know how you have any expectations for a unique solution.

>Does anyone else have some ideas?
You won't find anyone to invert a 2x3 matrix.

>>9704388
There are side-specific inverses for non-square matrices, but I don't have enough experience with them. Also these are barycentric coordinates, so one of the rows is redundant.

Now does anyone have any advice?

>>9704229
>pop math

>>9704393
>so one of the rows is redundant
Just ignore one of these rows then.

If the third row is just a linear combination of the ones before there is no point in even considering it.
Ignore the last row of both matrices and the last entries of both vectors.

>>9701959

Found a website in my mother tongue explaining math from elementary school to entry level uni math and have finally gotten an understanding of math within just a few weeks. It's amazing how much faster my understanding of math is now that I know the theory behind what I'm actually doing, instead of just trying to apply formulas that some teacher threw at us with barely any context.

Stochastic processes are still a pain, but I hope to catch up to some of you math masters someday.

>>9704393
For example, you could assume that b31 is zero, then solve your (now uniquely solvable) problem, from there on you could also calculate the entire solution set.

Alternatively you could use the pseudo inverse, to find a unique solution.

>>9701959

What do you call a bijection between a set and itself? I would say automorphism but I'm not dealing with a bijective homomorphism because the set I have has no operation

Would this be a derangement?

>>9704446

But derangement doesnt seem right either because it isnt an ordered set, just a finite one

>>9704446
a permutation?

>>9704416
I'm interested anon, sauce?

>>9703255
>Literally go read a book, dumbass.
I'll actually be reading some book on pure homological algebra right now just to spite you and other subhumans like you.
>couching things in the language of morphisms and functors
You mean expressing things in the usual and natural language of mathematics? Yeah, I see how doing that would make things much simpler.
>Any non-trivial construction
There are no non-trivial constructions in homological algebra if you are a non-brainlet (including those coming from "actual mathematics"). That doesn't somehow make it not interesting.
>construction comes from actual mathematics
Yes, some homological constructions do come from algebraic geometry and homological algebra itself. Both of which are "actual mathematics".

>>9704446
I think it would be a permutation. A derangement is a specific kind of permutation, where no cycle element is fixed, e.g.:

[math]\sigma_1 = (1 2 3) \in S_3[/math] is a derangement, while
[math]\sigma_2 = (1 2) \in S_3[/math] is not, because 3 is fixed.

nonetheless, both are bijections on the set.

>>9704456
behold the wild category diarrheaist in his natural habitat

>>9704475
What is your definition of "category diarrheaist"?

>>9704456
>I'll actually be reading some book on pure homological algebra right now just to spite you and other subhumans like you.
Why did you decide to mention this in a mathematics thread?

>>9701959
Studying a (pure) topos theory book just to make some people in these threads angry.

>>9704489
Why not? Seems like the perfect place to mention those kinds of things.

>>9704529
>Why not? Seems like the perfect place to mention those kinds of things.
Why would a mathematics thread be the perfect place to mention things unrelated to mathematics?

>>9704534
Why would things used in mathematics be unrelated to mathematics?

>>9704543
>Why would things used in mathematics be unrelated to mathematics?
They wouldn't be.

>>9704121
Both 0 -> Z -> Z -> 0, sorry.

>>9704121
i still dont get why my solution is wrong

>>9704766
It's almost correct.

>>9704797
yeah but i dont get what i did wrong

>>9704446
a permutation is the classical name. Alternatively, an automorphism of the category [math]\mathsf{Set}[/math]

Procrastinating on work for a couple of days. I learned the bare basics of coq, enough of the language to be able to know how to write proofs , axioms and theorems but likely not in the most ergonomic way.
I actually managed to formalize a proof of the exercise in halmos "naive set theory":
If (a,b)=(c,d) then a=c and b=d, where you define (a,b) in the usual way {{a}, {a,b}}. It was pretty neat to see how the "hint" in the book (try first the case a=b) actually simplified the work.

I think one could automate some of that stuff in a way similar to how people managed to teach a computer how to do indefinite integrals better than undergrads with surprisingly little rules and some ad hoc notion of complexity, (I think this was done in the early history of ai). I wouldn't be surprised if you could tell the program a theorem and a textbook style "hint" and it could automatically generate a proof. A less trivial example would be the theorem in this blog plost https://gowers.wordpress.com/2008/07/25/what-is-deep-mathematics/ that becomes "easy" once the hint (try the baire category theorem) has been provided.
The website for coq says that it's an automatic theorem prover so it's very likely that this has already been done, it just didn't appear in the very short tutorial I followed. It's a bit of a shame that I can't read longer manuals on it but I've procrastinated too much already.

can someone give me a pointer for this task:
I have given a parameterized field lines in 3d space. now I have to construct the vector field to it. I know field lines are tangents but Im not sure how to go on about it.

>>9704096
>But homology is functorial?? How can the identity map be the zero map?
Your solution is correct. Homology is a functor from chain complexes of abelian groups to the category of abelian groups. And your chain map is clearly not the identity chain map.

>>9704801
You get the zero morphism which is the identity of your 0th homology group there if you have 0 -> Z -> 0 for both complexes and use the identity morphisms as the boundaries and chain maps.

>>9704872
Discussing techniques of formalization of math is not math. Try >>>/g/ or >>>/lit/.

>>9704920
>Discussing techniques of formalization of math is not math.
define "math"

>>9704921
Discussions about defining non-mathematical terminology belong in >>>/lit/.

>>9704922
>Discussions about defining non-mathematical terminology belong in >>>/lit/.
define "non-mathematical"

>>9704926
Refer to >>9704922.

>>9704927
>Refer to >>9704922.
define "non-mathematical"

>>9704920
it's a science board, not a math board, idiot

>>9704934
Refer to the post you replied to.
>>9704938
If you can show that formalization of math is a science feel free to make a separate thread with your discovery of this fact.

>>9704979
>Refer to the post you replied to.
define "non-mathematical"

>>9704920
I take it you've never taken a first course in mathematical logic. Considering the cult of personality in mathematics you'll be glad to know that hilbert disagrees with you.
>https://en.wikipedia.org/wiki/Hilbert_system

Ok, I'm slightly interested in non-standard analysis.
Hyper-reals, with transfer theorems, are interesting, but it gets ugly as soon as you try to do some higher order theory (talking about topolpgy and so on).
How good is synthetic geometry, i.e, the analysis with topos theory? I mean, proving things in that context is totally equivalent to prove it in standard analysis? Can we solve all exercises using it?
What is the less shitty language to do it? Halp

>>9705012
>I take it you've never taken a first course in mathematical logic.
How is me taking philosophy courses relevant to the discussion? Feel free to make a thread on >>>/lit/ and link it here if you want to discuss this topic with me.
>you'll be glad to know that hilbert disagrees with you
Hilbert was known to be interested in many non-mathematical things. I don't see how this is relevant. If you wish to discuss what he did on the side further, try using >>>/lit/ or some other suitable board.

>>9705027
>philosophy courses
That's a standard undergraduate math course. You'll get there.
>Hilbert was known to be interested in many non-mathematical things.
Ok, how about the Hilbert's problem, the problems hilbert says he chose for what he believed would be their importance for mathematics in the 20th century.
>https://en.wikipedia.org/wiki/Hilbert%27s_problems#Nature_and_influence_of_the_problems
Now, look at the second one.

When 'cut the bullshit' edition?

>>9705019
I don't know about synthetic geometry, and not very much about hyper-reals, but have you tried non-standard analysis as in Robinson's book?. You can deal with higher order notions by considering "inner sets" instead of all sets, and that gives you things like non-standard topological spaces.
As for whether that's useful, I tend to think no. There are some theorems that are hard to prove otherwise (and even some new ones) that you can prove using that stuff but the proofs are long and complicated. There is no magic wand unfortunately.

this may sound like a dumb question but how do you go from the second to the third step? it's not obvious to me.

>>9702797
A logrithm is simple the reverse of an exponent, much like division is the reverse of multiplication.

y=x^n => n=log_x(y)

>>9705055
Yes, I have tried hyperreals = non-standard analysis from Robinson / Goldblatt / other notes. Those inner sets make me cringe, essentially you have to put the whole structure and its universe in the language, but this is ugly. I must admit I was amazed when I saw the non-standard correspective of continuity / uniform continuity / differentiability, but apart from that I don't see any reason to prefer it to usual analysis.
Why is there a good logic even for metric spaces (continuous logic) but not for topological spaces? It gets on my nerves.

This thread is fucking weird

>>9705067
You divide n! by (n-k)!
Now leave, combinatorics isn't mathematics.

>>9705041
>That's a standard undergraduate math course.
I pity your shitty "country" if it considers philosophy courses to be standard math courses.
>the problems hilbert says he chose for what he believed
Hilbert's personal views on philosophy aren't mathematics so they should be discussed at boards where it is suitable to do so. I personally recommend >>>/lit/11070539/.

>>9705041
>That's a standard undergraduate math course.
I pity your shitty "country" if it considers philosophy courses to be standard math courses.
>the problems hilbert says he chose for what he believed
Hilbert's personal views on philosophy aren't mathematics so they should be discussed at boards where it is suitable to do so. I personally recommend >>>/lit/11070539

>>9705082

Wait until you realize that group theory is really combinatorics in disguise

How is it possible that we know so little about PDEs

>>9705087
I'd rather trust Hilbert's view and what is mathematics rather than someone who does not even know mathematics subject classification. If you want to discuss why they MSC is also wrong I suggest you take it to wherever crack pots discuss their pet views.

>>9705067
>this may sound like a dumb question
More than "sound like".

Use /sqt/ next time.

I just saw a generating function proof that uses an infinite amount of variables and it made me happy cuz I tried to do that but didnt know how it's done

>>9703722
Thanks for the advice
Already seen derived functors and some applications, I've read J.P. May's a concise course in algebraic topology, some stuff on De Rham's cohomology, topological K-theory, cyclic homology and Hopf algebras

>>9705082
I'm sorry for being especially retarded but could you elaborate a little more. Are you multiplying by (n-k)!/(n-k)!?

>>9705102
>I'd rather trust Hilbert's view
Your and Hilbert's views on philosophy are best discussed in >>>/lit/11070539
This thread is strictly for mathematics discussion.
>MSC
You genuinely sound like a CS-tard, not that I'm surprised in the slightest. Maybe you should try discussing these topics in >>>/g/ as well? Surely there are other fellow programmers who would love to have deep discussions on the philosophical nature of mathematics.

>>9705077
Uhm. It's a bit of a weird coincidence, but I heard relatively recently that apparently there is something called "topological model theory" that sort of sounds like a "logic for topological spaces" similar to continous logic. I suppose it's more obscure/harder and with less applications, and it might not be the "correct" definition (whatever that means) but you might find it interesting.
https://www.springer.com/us/book/9783540097327

shit thread

>>9705124
Thank you, I will give it a try tomorrow.

For a group, I observed that

[math] \Delta G = G \times G [/math]
[math] (\Delta (f) ) (x,d) = (f(x), f(x+d) - f(x)) [/math]

has
[math] (\Delta (id) ) (x,d) = (x, x+d - x) = (x,d) [/math]
and
[math] ( \Delta (g) \circ \Delta (f) ) (x,d) = (\Delta (g)) (f(x), f(x+d) - f(x)) = (g(f(x)), g( f(x) + f(x+d) - f(x) ) -g( f(x) )) = (g(f(x)), g( f(x+d)) -g( f(x) )) = \Delta ( g \circ f ) ) [/math]
so that should make it into a nice functor.

Now since G has a special element (is pointed), there are some nice embeddings from G to the image of the functior (e.g. g to (g,0) or (g,g) or (0,g)) and there are several natural arros in the other direction (e.g. from (f(x), f(x+d) - f(x)) projecting out f(x) or projecting out both an adding them, resulting in f(x+d)).
And I have computed the concatenation of this Delta as an endofunctor (which should give a 4 tupple of various differences, in 3 directions), and I'm pretty sure we can make this into a nice monad, but I have no good idea how to do the co-unit.

Any takers?

>>9705116
But by saying what you believe is mathematics, you yourself are pressumming your philosophical views are important. As I do not consider such matters of philosophy important for mathematics I shall just ignore you and simply continue to discuss things that in my opinion are mathematics.

>>9705128
thanks for the input sweetie

>>9705159
>As I do not consider matters of philosophy important for mathematics
Good. So you should move to >>>/lit/11070539 to discuss philosophy of mathematics and you should move to >>>/g/ to discuss certain techniques of formalizing mathematics. I'm glad we came to an understanding.

>>9705163
Of course there is no understanding. My point is that you may well be wrong about what is and isn't mathematics. Are you wrong though? I don't care, discuss it if someone is willing in that lit thread. I will continue to post things that in my opinion are of relevance to mathematics just here. Or I would, if I frequently posted in this site. Hopefully now you get my meaning.

>>9705115
Just open the factorials and remove the terms that cancel each other out, I'm sure you're aware of the absolute basic arithmetics required ( (xy)/(xz) = y/z ). If the abstract version is difficult to follow, try inserting some actual numbers and see if that helps. Example: n = 10, k = 7
[math]\frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = \frac{10*9*8*7*6*5*4*3*2*1}{(7!)*3*2*1} = \frac{10*9*8*7*6*5*4}{7!} [/math]
If this didn't clear this up, you should drop what you're trying to do.

>>9705168
>what is and isn't mathematics.
We can discuss this topic in a thread where it is appropriate to do so. I suggest using >>>/lit/11070539
>I will continue to post things that in my opinion are of relevance to mathematics just here.
Your opinion is simply wrong if you consider things like "programming" or "philosophy" to be mathematics. Feel free to discuss how this isn't an abhorrent abuse of language and why they should be considered mathematics in a proper thread such as >>>/lit/11070539

>>9705160
It really is, I can elaborate: no math discussion, no problems, thread evenly split between categorical bullshit, ad hominems and random shit

>>9705183
>thread evenly split between categorical bullshit, ad hominems and random shit
>evenly split
Clearly you are reading some other thread.

>>9705172
lol thanks. I was confusing myself expanding the upper part. I was retarded and didn't realize that (n-k)! comes after the (n-k+1) term. I thought it was reverse and that's what confused me. Ty kind soul!

>>9703634
Stop that, it's not fashionable.

>>9704456
>I'll actually be reading some book on pure homological algebra right now just to spite you and other subhumans like you.

What kind of dumbass needs a book to learn homological algebra? At least read Grothendiecks Tohoku paper, although unless you've never been exposed to any actual non diarrhea math, such as Algebraic Topology or Algebraic Geometry, most of it should be review.

>You mean expressing things in the usual and natural language of mathematics? Yeah, I see how doing that would make things much simpler.

of course it makes things simpler. That's why there is a generation of category diarrheaists attracted to the practice of reinterpreting the work of actual mathematicians categorically, without having to do any real work.

>There are no non-trivial constructions in homological algebra if you are a non-brainlet (including those coming from "actual mathematics"). That doesn't somehow make it not interesting.

If they are trivial they are by definition uninteresting, unless you are retarded.

>Yes, some homological constructions do come from algebraic geometry
Some? You just sound like some high schooler who chased down the snake lemma diagram, and thinks it's cool because there are a bunch of arrows and cool symbols. ALL of them come from topology, algebraic geometry, or group transformations.

Shitposting mostly aside, how would you define math?

>>9705694
Seriously, try asking in >>>/lit/11070539

>>9705694

Don't give in to this person's reductionist frame. Instead, talk past him in this thread about whatever you care to discuss, and ignore his unintelligent repetition to which he has now fully committed.

Moreover, continue this strategy in future threads with total disregard for his and like opinions, as it cannot fail to raise the content of substantive mathematical discussion above that shown in this particular thread thus far. If he actually knew what math was, and if he actually cared as much as he claims to about purity of discussion in the given thread, then he would recognize the diminishing returns and mathematical obviation of his repeitions (imagine the problem: insistence on content in a forum on a certain topic, with the diminishing return of "rule-enforcement" posts to the point that the topic never actually gets discussed) t. guy posting in this thread for the first and last time, feeding the tenacious troll.

>going to college
>haven't read Wealth of Nations somehow
>fagfriends Hassel me about it

so I've been binge-reading some Adam Smith. peer pressure isn't all bad.

>>9704917
>>9704914
yeah im retarded, forgot that all maps in the chain map needed to be the identity, not just the one lmao

>>9705787
>Seriously, try asking in >>>/lit/11070539
But this is the maths thread.

>>9705694
>Shitposting mostly aside, how would you define math?
The study of sets.

>>9705101
cos they boring as fuck to study

>>9705101
cause they're fuckin super hard

>>9705869
Now you know better. It's mistakes like these that teach us things.

How the fuck do you compute this without creating a monster chain complex with 18 1-simplices and 7 0-simplices

>>9706027 cells*
my only other idea would be to just state that the complex is path connected, so H_0 = Z, and the inner paths do not touch the 2cells, so H_2 must be also Z (identical to the two-sphere), and then say that H_1 is the abelianization of the fundamental group, and since this is a wedge of 8 circles, then it must be Z^8

>>9706027
Are you allowed to use the Mayer-Vietoris sequence? The "cross" consisting of the three axes is homotopy equivalent to a point, and the intersection with the sphere just consists of 8 points (you have to adjust the subspaces a little bit in order to make them fulfill the conditions for applying the Mayer-Vietoris sequence but essentially this is the correct decomposition).

>>9706056
I was thinking about it but it's an exam paper and the next one required me to use MV sequence, so i assumed this one would be testing something else. Regardless, does my description work?

>>9706059
I don't know - my MV sequence ends in
[math] 0 \to H_1(X) \to \mathbb{Z}^6 \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to 0 [/math], so the rank of [math] H_1(X) [/math] should be 5 (I haven't done this for long, so there is no guarantee that I'm not merely talking bullshit unintentionally).

>>9706059
By the way - the intersection consists of 6 points, I recently saw.
tfw doing homology but being to dumb to count.

Brainlet here, how would you prove this?
I also can't really imagine such a function other than the obvious [math]f(x)=k,\quad k\in \mathbb{R}[/math].

>>9706155
I'm still thinking about a proof but you could define
f to take the value 0 for x=0 and 1 everywhere else.

>>9706155
>Brainlet here, how would you prove this?
By contradiction.

>>9706155
>I also can't really imagine such a function other than the obvious f(x)=k,k∈R.
you can have things like pic related (round edges mean endpoint not included, square edges mean included)

>>9706155
Just an idea: for each [math] y \in \mathbb{R} [/math] the set of all [math] x [/math] such that [math] f(x) \geq y [/math] is open (same for [math]f(x) > y [/math]. I don't know if this topological consideration helps.

>>9706155
>the range is the reals
>the image is countable
If I read such shit, I simply refuse to care about the exercise. This is math, not humanities 101. Consider it solved.

>>9706165
>>9706169
>>9706185
>>9706187
Thank you very much

>>9706197
>If I read such shit
What do you mean?

>>9706155
Not true. Consider [math]f(x)=0[/math] for rational x and [math]f(x)=x[/math] for x irrational

>>9706224
>Not true. Consider f(x)=0 for rational x and f(x)=x for x irrational
Wrong, not every point is a local minimum.

>>9706230
how so

>>9706239
>how so
f(-1/2) = 0 and f(x) = x < 0 for irrationals close to -1/2.

>>9706239
any open interval around an irrational point x contains a rational point q, and hence f(x)>f(q), contradiction

>>9706242
for x positive of course

>>9706242
>any open interval around an irrational point x contains a rational point q, and hence f(x)>f(q), contradiction
That's not a contradiction to local minimality.

>>9706246
yes it is. for an irrational x>0, f(x)=x>0=f(q) which is in the interval (since the rationals are dense in R). In particular, f(x) is not a local minimum

>>9706155
Let [math]f(x)=0[/math] for [math]x\leq0[/math] and [math]f(x)=1[/math] for [math]x\geq1[/math].
let [math]C[/math] denote Cantor's set. Choose [math]c,d\in C[/math] st there is no [math]y\in C[/math] strictly between c and d and let [math]f(x)=c[/math] for [math]c\leq x\leq d[/math]
[math]f[/math] is piecewise constant, so it has the property that for every x it has a local minimum there, but it's image contains entire Cantor's set, which is uncountable.

>>9706259
[math]f(x)=c[/math] for [math]c\leq x < d[/math]

>>9706259
>Choose c,d∈C st there is no y∈C strictly between c and d
is this possible?

>>9706266
Yes, for example 1/3 and 2/3. But the issue is that the function posted does not satisfy the local minimum everywhere condition.

>>9706270
why? it would look somehow like pic related, but with uncountably many lines between 0 and 1 and for every x in the neibourhood of x f is either constant or has a discontinuity

>>9706278
None of these left endpoints are local minima since points to their left have smaller images.

>>9706282
what about this?
for every point in Cantor's set f is equal to some a>1, and c for [math]c<x<d[/math], on every segment where f is linear every x from that segment is a local minimum, and for x from C they're isolated point so we don't care about values of f near them

>>9706292
Now none of those points in Cantor's set are local minima since their image is a>1 while all the points around them have image in (0,1).

Protip: If a question says the author and another mathematician have solutions, you're better off looking for proofs than counterexamples.

>>9706299
the point you marked with blue arrows aren't minima if we choose [math]\varepsilon=a-1[/math]

>tfw cant even solve a simple problem
Might be time to abandon math studies desu

>>9706307
>the point you marked with blue arrows aren't minima if we choose ε=a−1
The arrows were meant to signify that the point they're coming out of has larger image than its surroundings, so can't be a minimum.

>>9706310
but we don't care about the surroundings if it's sufficiently separated

>>9706312
>but we don't care about the surroundings if it's sufficiently separated
What do you mean? If you're looking at a point, then you have to care about the intervals immediately next to it.

>the Platonic realm

>>9706315
(not true by the way)
given epsilon small enough all the points you indicated with arrows are ruled out and all we care about are the individual points

>>9706155
I think this should work: we have that the real numbers are Lindelöf (i.e. each open cover has a countable subcover). This can be used to prove that each uncountable set has an uncountable intersection with some [math] \varepsilon_x [/math]-ball. Fix those [math] \varepsilon_x [/math].
Take an uncountable set [math] M [/math] on which [math] f [/math] is injective.
Take an [math] x_0 \in M [/math] such that the [math] \varepsilon_{x_0} [/math]-ball around [math] x_0 [/math] contains uncountably many points of [math] M [/math]. Also, find an [math] x_1 \in M \cap B(x_0,\varepsilon_{x_0}) [/math] with [math] x_0 \neq x_1 [/math] fulfilling the analogous condition.
Now chose [math] x_{i+1} \in M [/math] strictly between [math] x_{i-1},x_i [/math] such that there are uncountably many points in [math] M \cap B(x_i, \varepsilon_{x_i}) [/math].
Then there is an [math] x [/math] contained in all [math] B(x_i, \varepsilon_{x_i}) [/math] and [math] f(x) > f(x_i) [/math] for all [math] i [/math]. But any neighborhood of [math] x [/math] contains some [math] x_i [/math] which is a contradiction.

>>9706318
>given epsilon small enough all the points you indicated with arrows are ruled out and all we care about are the individual points
The points I indicated with arrows weren't the issue, it's the points in the Cantor set that the arrows are pointing out of. The image of the Cantor set points are surrounded by images in (0,1) which is smaller than a>1.

>>9706322
assuming axiom of choice

>>9706326
Assuming existence of fucks I give.

>>9706328
>Assuming existence of fucks I give.
Do you need to swear?

>>9706330
Sorry, I went too far

>>9706330
> Do you need to swear?
No. But do you need to avoid the axiom of choice?

>>9706202
It's pretty much the standard to consider the range to be the image, not the codomain.

>>9706346
>It's pretty much the standard to consider the range to be the image, not the codomain.
It's really only an issue if you're autistic or a brainlet.

Im fucking retarded, please help, im gonna fail this exam

>>9706353
No. You should do it.

>>9706353
>>9706358
i dont know where to even begin, what spaces should i use for the MV sequence?? The condition of the homeomorphism just makes it more confusing

>>9706366
If you want to us MV try to understand what is that homeomorphism of the torus. What does it means that the map S^1->E_2 induces the 0 map in homology? Then try taking a little more than the boundary to use MV.

>>9706379
nigga you think i know? ive spent over an hour trying shit out before i asked it here

>>9706381
Think.

Holly shit, why my uni doesn't explain the relation with the power series? It makes so fucking sense, that's beautiful.
We literally learned the formula before power series, that's fucking stupid. No wonder why modern education is for brainlets.

>>9706512
I mean if you are really curious you could also just google it or look it up yourself without being a brainlet

>>9706524
yeah and that's exactly what i'm doing, that's why i'm self learning on top of my uni work.

>>9701959
I've been reviewing for finals and considering my summer reading. Currently looking for a good Algebra book. I've got a copy of Artin from an upperclassmen, but I'm not super fond of it. My older brother used Jacobson, so I have vol I and II. The adviser recommended I get a much easier book, like Pinter, and use that. But I'm not really knowledgeable enough to know which would be best.

So I'm up in the air about what I should be reading right now.

>>9706259
>Cantor's set
Prove that it exists.

>>9706512
Euler's formula is almost always given before they actually derive it because it makes everything else easier.

>>9706537
>Prove that it exists.
https://en.wikipedia.org/wiki/Cantor_set

>>9706562
>>9706537
For the exercise you are not supposed to know the Cantor set. The answer should be one-line only.

>>9706583
>For the exercise you are not supposed to know the Cantor set. The answer should be one-line only.
It exists for He has a name.

>>9706537
The process of proving existences has no place in a mathematics thread. The proper place for such discussions is >>>/lit/.

>>9706512
Yeah, it's a shame they don't teach stuff like that in calculus more clearly.

>>9706155
Ok, so i checked the solution from Spivak, here it is if you are interested.

Why is differential equations taught when so early? Any time I ask why something works, or what the actual fucking statement of a theorem is, my professor waves it off with a "here be analysis." What's the fucking point?

>>9706689
Welcome to modern education.

>>9706695
>>9706689
And that's because algebra is not taught earlier.

>>9706689
Physics and engineering need formulas to solve their equations, so they colonized the math department. Same reason that Linear Algebra is taught as "math that CS majors need" at most universities.

I bet you go to a school with a large/well-funded engineering department.

>>9706719
I'm not him but we learned differential equations for the first time during physic classes at my university, nobody understood it at first. I'm now fully in math without physic but we still learn it at the same time of linear algebra...

>>9706735
Pure math is going to die within the next 100 years at all but the top institutions.

>>9706784
It's already happened. My university (UT) has more than 50,000 students. There are, to the best of my knowledge ~20 pure math majors every year.

>>9706533
Use Dummit and Foote

im learning about gaussian integers in the context of number theory and rings and polynomials, its pretty neat working with complex numbers after ignoring them for so long

>>9706801
ut math professors are legit, i've really enjoyed taking classes with dudes like blumberg and kidwell

>>9706825
>>9706533
How's Nicholson's Introduction to Abstract Algebra?

>>9706801

That actually makes me really happy

>>9701959
Could someone help me ? We had a test today and there was a question I couldnt solve.

It asked whether the eigen values of AB equal the eigen values of BA assuming B and A are symmetric matrices, any chance someone could help ?

>>9706957
Why's that?

>>9707032

Less normies

>>9701959
Critique of Pure Reason.

How many UT people are there here?

>>9707059
>How many UT people are there here?
femanon phd student here

>>9707062
There are female students in the math department? I've legitimately not seen one since my first year here.

>>9707059
Junior here. Don't take Algebra with Keel.

When do we make a math contest? Show your honor anon.

>>9705113
>topological K-theory,
This doesn't really have much to do with homological algebra.

It is a "homology theory" in the axiomatic sense of algebraic topology, the study of such things is much more homotopical algebra than it is homological algebra.

>>9707084
Not him, what is homotopical algebra? If I recall correctly there was something called like that in Algebra 0, homotopy between complexes and so on, isn't that still explained in homological algebra books?

>>9706924
quadratic number fields are pretty neat

>>9707069
*Professor* Keel here, come to my office I will fuck you up

>>9707091
It has nothing to do with homotopies of chain complexes.

Homology theories in the axiomatic sense are equivalent to things called "spectra".

These spectra play the role of abelian groups in a form of algebra where all the usual properties only hold up to coherent homotopy.

The equivalent of commutative rings are "E_∞ -rings". The spectra representing stable homotopy, singular homology w/ coefficients in a commutative ring, and topological k-theory are all examples of such.

Stable ∞-categories play the role of abelian categories, with the stable ∞-category of spectra playing the role of abelian groups.

There homotopy theory built into stable ∞-categories makes developing a higher notion of a derived category unnecessary.

This is what (in the modern sense) is meant by homotopical algebra. More classical concepts objects of focus in homotopical algebra would be model categories.

>>9707109
is there some book on model categories ?

>>9707167
There is the original "Homotopical Algebra" by Quillen.

There is "More Concise Algebraic Topology" by May, for more topological focus.

There is "Simplicial Homotopy Theory" by Goerss, for a more simplicial focus.

Couple weeks into calculus 1 now, doing well, already past the chain rule and beyond. Quotient rule was a joke. Product rule remains my specialty.

I ask my professor his thoughts on quantum mechanics and partial derivatives. He's impressed i know about the subject. We converse after class for some time, sharing mathematical insights; i can keep up. He tells me of great things ahead like series and laplacians. I tell him i already read about series on wikipedia. He is yet again impressed at my enthusiasm. What a joy it is to have your professor visibly brighten when he learns of your talents.

And now I sit here wondering what it must be like to be a brainlet, unable to engage your professor as an intellectual peer. All of the deep conversations you people must miss out on because you aren't able to overcome the intellectual IQ barrier that stands in the way of your academic success... it's so sad. My professor and I know each other on first name basis now, but i call him Dr. out of respect.

And yet here you brainlets sit, probably havent even made eye contact with yours out of fear that they will gauge your brainlet IQ levels.

A true shame, but just know it is because i was born special that i am special. I can't help being a genius, nor can my professor. Two of a kind is two flocks in a bush.

>>9707230
6/10 - Lacking originality

W = Span{v1, v2 ,v3}
v1 = { 1; 1; 1; 0; 0; 0; 1 }
v2= {0; 1; 1; 0; 0; 1; 1}
v3 = {0; 1; 0; 1; 1; 1; 1}


Can you guys help me with this basis/dimension problem?

So dimension of W is 3=d. How can I use that information to tell how many vectors are in W?

Also, wouldn't there be a whole lot of elements in W?

>>9707265
However many elements are in [math]\mathbb{F}^3[/math], clearly.

>>9707032 >>9706801
>~20 pure math majors among 50,000 students
That means less competition

>>9707265
>How can I use that information to tell how many vectors are in W?
[math] |\mathbb{F}|^3 [/math] where W is a [math] \mathbb{F} [/math]-vector space

I just found a rational representation of sqrt(2) (didn't need to assume axiom of choice), what do I do now?

>>9707032 >>9706801
>~20 pure math majors out of 50,000 students
Less competition

>>9707296
>>9707300
Okay, I think that makes sense if I look over vector space and surrounding definitions of span and basis.

I'm asked to also write out every element of W, would the elements of W just be v1, v2, and v3?

>>9707301
send it to wildberger and tell him his theories are unnecessary

>>9707331
no. every element will take the form [math] \lambda_1v_1+\lambda_2v_2+\lambda_3v_3 [/math] where [math] \lambda_{1,2,3}\in\mathbb{F} [/math]

>>9707297
>>9707303
But there isn't any less competition--the dumb people are just BA or teaching track.

Post anime girls with cool advanced math books

>>9701959
Hey, I just accepted a university offer I received for Mathematics over one I had for Chemical Engineering. Help me justify why I did this in my own mind so I can sleep again.

>>9706353
why are you taking theoretical math courses if you can't do it lmao

>>9707514
Math is beautiful, while chem eng is dull.

>>9701959
Calculus - Michael Spivak
The Idiot - Dostoyevsky

>>9707581
Seems about right

>>9707301
THERE ARE NO RATION REPRESWNTATIONS OF ROOT 2 I PROOFED TIS MY FORST DAY OF ANALYSIS OWO

>>9707539
Why are you taking math classes that you can already do?

>>9707605
>Why are you taking math classes that you can already do?
Easy grades.

>>9707605
I'm an engineer.

>>9704475
>diarrheaist
redpill me on diarrheaism

>>9705090
could you fuck off

>>9708285
>could you fuck off
Do you need to swear?

>>9707514
You fucked up completly, math major its for those intelcktual yet idiots

>>9707766
>redpill me on diarrheaism
diarrheaism = anything that leads one astray from the one true path (analysis)

>>9708422
Some people simply aren't interested in engineering.

>>9707514

not gonna lie man that was pretty stupid

>>9708422
Analysis really is the one true path, as it is a direct path to ruin and ruin is the ultimate end for all of us.

>>9708515
Doing analysis is like spending your time in purgatory in advance. Willingly spending your time on that shit will redeem even the most horrendous of sins.

Functional analysis, after some terrible texts I'm finally understanding some stuff using Kreyszig's book. I was curious to see what kind of person was Riesz, and looking up the wikipedia article I found this, what a lad:

>He had an uncommon method of giving lectures: he entered the lecture hall with an assistant and a docent. The docent then began reading the proper passages from Riesz's handbook and the assistant wrote the appropriate equations on the blackboard—while Riesz himself stood aside, nodding occasionally.

>>9708586
Truly a man of men

How difficult is it to get an academic job in mathematics?

>>9708832
>How difficult is it to get an academic job in mathematics?
Not that hard.

>>9708832
Don't get you hopes up.

>>9708409
> Do you need to swear?
Is this a meme or is one single troll writing this everytime?

>>9708949
It's a meme

>>9708949
You could avoid to reply if you just have to swear.

>>9708949
Is swearing in maths threads a meme or is one single troll swearing everytime?

>>9708840
>>9708848
But who should I believe?

>>9708995
Me

>>9709018
But which one are you?

>>9709018
I'm not sure, I think the other person does have a point.

>>9708967
>>9708958
Thanks for explaining! (no sarcasm)

>>9701959
Operator theory. Physicsfag here, so i got into an interesting conversation with my prof about the algebra of operators and such.
Also studying the module of smooth vector fields on a smooth manifold. Pretty cool

>>9709139
What's interesting about that module?

>>9709150
Τhat is hasn't got a basis! (Not every module has a basis, unlike vector spaces. Thats why they even have a special name than just "Vector spaces over rings"). After that if you consider the sphere as a smooth manifold [math]M=\mathbb{S}^2[/math] comes the famous hairy ball theorem.

>>9709161
You mean it has no finite basis or no basis at all?

>>9709173
Most modules don't have any basis at all.

>>9709173Since the set of all smooth vector fields [math]\Gamma (TM)[/math] on a manifold [math]M[/math] has the structure of a [math]C^{\infty} (M)[/math]-module and [math]\left( C^{\infty}(M),+,\bullet \right)[/math] is not a division ring, it has no basis at all. Modules have a basis iff the underlying ring is a division ring.

>>9709188
>Modules have a basis iff the underlying ring is a division ring.
This is false.

>>9708967
dont think its gonna happen any time soon bud

>>9709198
no its not.

>>9709205
The true thing is that if the ring is not a division ring then there do exists non free modules.

>>9709207
well my prof told me that, if its incorrect could you provide a proof/counterexample?

>>9709213
[math] \mathbb{Z} [/math]

>>9709213
Why doesn't your professor provide proofs of his/her claims?

>>9709237
we aimed to focus on [math]\Gamma(TM)[/math], its not an algebra class.

>>9709225
can you elaborate?

>>9709248
Not him, this is false, take the rationals for instance.
If R is a ring (with the i) then every R-module is free iff R is a division ring. Proof by algebra book.
This still does not answer the original question.

>>9709248
>can you elaborate?
[math] \mathbb{Z} [/math] is a free [math] \mathbb{Z} [/math]-module

>>9709188
>Modules have a basis iff the underlying ring is a division ring.

>generate free module on a set
>doesn't have a basis
dumb animeposter once again outs himself as a retard

>>9709161
>Not every module has a basis, unlike vector spaces.
But not all vector spaces have a basis either (unless you assume they do).

>>9709270
i didn't say anything about free modules

>>9709352
>i didn't say anything about free modules
Please stop posting about things you don't have the mathematical maturity to understand.

>>9709358
okay i'll stop. i know its a crime to post here while LEARNING the subject

You are all a bunch of meanie, stop bullying >>9709366
If you are such big shots why haven't still given an answer?

>okay i'll stop. i know its a crime to post here while LEARNING the subject
Don't mind them, it is fine. Anime posters are always welcome. Nobody here is any better than you.

>>9709636
>If you are such big shots why haven't still given an answer?
Answer to what?

>>9709650
>Answer to what?
All started from
>You mean it has no finite basis or no basis at all?
Is the module of vector fields free? No answer so far.

i will always assume a basis exists and you f*ggots can't stop me

>>9709358
Free modules are free, but the question was are all (left) modules over a ring free if a condition is imposed on the ring. If it is a field, then all modules are free. Is commutativity requires here? No, so division rings satisfy the condition too. I don't quite see your reason to be grumpy here.

>>9709636
>Anime posters are always welcome
Anime is redundant. Only the quality of the posts matters. t. ex anime poster

>>9709708
>the question was are all (left) modules over a ring free if a condition is imposed on the ring
This came up later. The answer to the previous question has little to do with this. And it does not seem so trivial.

>I don't quite see your reason to be grumpy here.
The rudeness in >>9709270 >>9709358. He might have made a mistake, but I don't see why they have to repress him so much, without teaching anything above all. It is the worst thing to do to a fellow student.

>Only the quality of the posts matters
Right, by bad.

>>9709775 again,
>>9709708:
sorry, I mistook first quote for a (you).

>>9709789
that's ok animeshitter
we know you have special needs

>>9709775
>>9709789
I don't know who's who anymore.

>>9709817
Does not matter. I confused 9709358 for a 9709695 (who was me). I was also 636, so the quote popped up. Sorry for the hell.
Anyway at this point if anyone is interested
https://math.stackexchange.com/a/1612824
Free iff parallelizable, at least for finite dimensional manifolds. Gn.

Is there an equation or proof for this?

>>9709862
https://en.wikipedia.org/wiki/Mathematical_induction

>>9709708
>Free modules are free, but the question was are all (left) modules over a ring free if a condition is imposed on the ring. If it is a field, then all modules are free. Is commutativity requires here? No, so division rings satisfy the condition too. I don't quite see your reason to be grumpy here.
What grumpyness? The poster said "i didn't say anything about free modules" after saying "Modules have a basis iff...".

>>9709849
>>9709874
I don't care. Forget my posts and carry on.

>>9709862
Factor x^n - 1

>>9709161
>Τhat is hasn't got a basis!
That's pretty common. How does that make it interesting?

>>9709636
>>9709695
>>9709708
>>9709775
("Animeposter" here)
Thank you anons, your replies are appreciated.

>>9710272
This is the first time I heard something like this. I didn't know it could be possible.