/mg/ math general -- gömböc edition

[Anonymous No.9475076]

talk mathematics (analysis also allowed)

[Anonymous No.9475135]

>>9475013

[Anonymous No.9475165]

>>9475076
Can anyone please recommend good Math I / II books in German? Thanks!

[Anonymous No.9475219]

>>9475076
Going to start a math major this year, I want to specialize in Analysis and maybe someday get a Ph. D, what advises can you guys give me?

Not to do well in evaluations, I think I'll be able to do that fine, but rather to become one of the bests (profs recognize you, uni gives scholarships to the best students etc)

[Anonymous No.9475426]

What's the official mathematical philosophy of /mg/?

[Anonymous No.9475506]

>>9475219
Uni scholarships (in undergrad) will depend on your grades or your extracurriculars, not how much dick you suck.
Getting your profs to like you and offer you opportunities is entirely reliant on your dick-sucking skills.

[Anonymous No.9475542]

I think I'm retarded because I don't understand what is meant by this :

>Carry out division with positive remainder and division with minimal remainder for the following :
>express am your answers as equalities of the form a=q (b)+r

Here's one of the problems
> 27 / 7
I don't know what it means at all.

How do I do it?

From a Person who came from a bad education who wants to do math

[Anonymous No.9475548]

>>9475542
It's asking you to perform long division with remainder, like you would have done in elementary school. Just in a more formal way.

[Anonymous No.9475558]

>>9475548
I think what's tripping me up is I'm interpreting it as two remainders ? A positive AND a minimal implies it's expecting different answers to me but I didn't see how there could be more than one remainder .

Also I've never phrased long division answers in terms of an equality. I don't have the book yet so I can't find what those variables are meant to represent

[Anonymous No.9475576]

>>9475558
>Also I've never phrased long division answers in terms of an equality. I don't have the book yet so I can't find what those variables are meant to represent
when dividing dividing a by b, find integers q,r such that
[math]a = qb + r[/math] such that [math]0 \leq r < |b|[/math]

[Anonymous No.9475580]

>>9475558
If you require positive and minimal _together_ then yes, there is only one answer.
Probably you at least know that you get 3 with remainder 6, meaning 7 goes into 27 3 times with 6 left over.
Symbolically this is 27 = 3(7)+6.
But you can make the remainder smaller if you allow negative numbers by writing 27 = 4(7)-1.
Usually this isn't how we divide things (the definition >>9475576 is accurate) but I don't know what your book is trying to say.

[Anonymous No.9475600]

>>9475576
>>9475580
Ah I see thank you all. This isn't that bad then but the wording and different way of doing division threw me off.

[Anonymous No.9475603]

>>9475076
link the previous thread

[Anonymous No.9475625]

>>9475603
i dont fuckin care about the previous thread

[Anonymous No.9475636]

>>9475135
In differential geometry, a choice of local coordinates is a choice of chart in the manifold's atlas.

A chart is some open subset together with a diffeomorphism to R^n.

[Anonymous No.9475726]

>>9475426
exist as abstract objects

[Anonymous No.9475729]

>>9475426
Also, that's not what formalism is. Formalism should go under the "meaningless question" part since it doesn't even acknowledge meaning.

[Anonymous No.9476551]

why are so many math professors uncharismatic

[Anonymous No.9476576]

Anyone knows where I can find Joseph Kitchen - Calculus of One Variable? Google search, libgen and scribd doesn't work, and I can't buy it. Looks like the book don't even exist on the internet.

[Anonymous No.9476600]

>>9476576
In the kitchen.

[Anonymous No.9476885]

>>9476576
https://math.stackexchange.com/questions/731087/joseph-kitchens-calculus-reference

It doesn't exist in PDF form you have to find a rare physical copy or just alternatives.

[Anonymous No.9476900]

>>9475219
>>9475506
Getting profs to like you is about adding to the class(think before speak, speak only when necessary), for maths just enjoy the class and murk the curve while everyone else whines and drops, the maths prof will like you

[Anonymous No.9476905]

How do I get set up doing research with a professor or getting research experience as a math undergrad ?
I'm not really an exemplary student really. The only thing that sets me apart is that I never complain or give up. Interns of actual achievements and intellect I'm pretty middle of the road.
There's only one math professor I'm on speaking terms with . I'm now in a class that's basically his research PHD topic with me and maybe 5 other people(it's not popular at all among math undergrads here) and of those people I'd argue I have some of the higher level of enthusiasm for the subject .

The thing is I'm so much of a dumbass that I don't see what I offer. At least in other sciences you can do lab monkey work and some other stuff for the professor and just get to observe and listen but I don't see what I can do as a dumb math undergrad.

[Anonymous No.9476920]

I was reading a book and pic related appeared, asserting [math]f(\emptyset) = \emptyset[/math], which I could not prove; actually, it's pretty easy find a counterexample: any constant function works. If, say, [math]f(A) = Y[/math] for any [math]A \in \mathcal{P}(X)[/math], then [math]f[/math] is a completely additive constant function, but [math]f( \emptyset ) = Y \neq \emptyset[/math]. Is there a missing hypothesis?

[Anonymous No.9476922]

>>9476905
How much stuff do you know? Assuming this is a pure math field, you won't be able to crunch any numbers or do any monkey stuff on a computer.

Honestly, because math is solitary, it's hard to do anything as an undergrad. Honestly, maybe just ask him? He might give you a "toy" problem to work on and you can try proving it and then writing a paper about it to cut your teeth on doing research

[Anonymous No.9476932]

>>9475219
To be recognized, you need to do a couple things

Destroy your classes
Ask well informed questions during lecture (because you read ahead and already know what is to come, but maybe there is something you didn't understand that the professor can clarify)

And maybe this is vague, but respect mathematics, in front of your professors. Don't ask stupid questions, do the reading, don't complain, don't give up, don't cheat, and don't trivialize anything.

Understand that a research professor in math has devoted his life to that craft. Don't diminish math and thereby him by complaining or acting bored, or any stupid bullshit your peers do. Be interested and be an adult

Do all that and you'll stand out

[Anonymous No.9476943]

>>9476920
Your counterexample doesn't work, it is mapping the empty set to something. Not mapping elements of the emptyset to something. Think about it.

[Anonymous No.9476951]

>>9476943
>it is mapping the empty set to something.
The empty set is an element of the domain

[Anonymous No.9476957]

>>9476922
I don't know a whole lot to be honest. Just finished with the absolute basics of a math degree and getting into more specialized subjects . I have done foundations of math courses and have experience in proof writing and formal logic.
Presumably by the end of the course I'll have dipped my toe into the basics of the field the professors PHD topic is in.

>math is solitary
Yeah. I'll clarify that he is a pure math professor and this is math so there isn't any monkey work for me to do. He's open to giving puzzles and thinking exercises to students but I don't think either he or grad schools would consider that "math research"

Finding research experience for a math undergrad who wants grad school is hard. Would I be better off contacting another field and being their monkey for a bit? I'm not horrible at bio, chem, and physics... maybe try to squeeze some pure math out of something in one of those disciplines that is less solitary ?

[Anonymous No.9477009]

>>9476957
I'm an undergrad doing research with a professor right now (pure math). A lot of what I'm doing is implementing stuff in sagemath and computing lots of examples of the algorithms and methods my professor and his coauthors have developed. There is quite a bit of monkey work involved, but it still requires some understanding of the theory. I feel in over my head a lot of the time but I am also surprised at the things that I can understand and do. I am probably going to get my name on a publication as an undergrad.

I'm good but not the greatest student. I also have a reputation as a hard worker. I like to hang out with my professors. My advice would be to just express your interest in these topics you like. Look up your professor's research/papers and ask them questions about it. Also you can let them know what your career goals are and ask for advice on that (getting into grad school, etc). Also treat your profs like people. Show respect but be personable too.

[Anonymous No.9477154]

I have an Oxford interview next week concerning a PhD position. Any fellow anons have any tips?

[Anonymous No.9477156]

>>9477154
Mmmh try MAHNZ ZAVIZI ZAVIZI MAHNZ

[Anonymous No.9477264]

>>9476920
>>9476951

You're correct, the book seems to have a mistake.

[Anonymous No.9477286]

>>9476951
Which is why it can map ot to the empty set. Reread the definition of the function

>>9477264
No

[Anonymous No.9477289]

>>9477286
>Which is why it can map ot to the empty set.
Who said it couldn't?

[Anonymous No.9477301]

>>9477289
The union of the power set of the empty set is equal to the union of the power sets of all sets of the empty set, or the empty set. Your counterexample doesnt make sense

[Anonymous No.9477316]

>>9477301
>The union of the power set of the empty set
This is a meaningless notion.

The counterexample was any constant function from P(X) to P(Y) whose image is not the empty set. For example if X=Y={1}, then you can take the constant function f: P(X) -> P(Y) defined by f(empty)=f(X)=Y, which is completely additive (see below) but does not satisfy f(empty)=empty.

f(empty union X) =
f(X) =
Y =
Y union Y =
f(empty) union f(X)

[Anonymous No.9477323]

>>9477316
Defined in that way for any power set that contained the empty set, or all of them, it would be false. That would be meaningless. I think we are talking past each other. Also meaninglessness comes about a lot with the empty set

[Anonymous No.9477325]

>>9477323
>Defined in that way for any power set that contained the empty set, or all of them, it would be false.
What are you trying to say here? Every power set contains the empty set.

[Anonymous No.9477326]

I'm a bit stupid but what is the union of (abb)* and (bab)* ?

[Anonymous No.9477327]

>>9477325
Yeah, I am saying that 0f = 0 would be false. I wasnt trying to imply that power sets didnt include empty sets

[Geometric Unity No.9477329]

[Anonymous No.9477441]

>>9476920
he probably considers unions indexed over the emptyset

[Anonymous No.9477465]

>>9477327
>Yeah, I am saying that 0f = 0 would be false
Why would it be false?

[Anonymous No.9477473]

I'm trying to prove linear algebra proposition this but I've reached a deadlock.
Let [math]T : V \mapsto V[/math] be an endomorphism such that [math]T \circ T = 0[/math], prove that [math]T + \itemrm{Id}_V [/math] has an inverse

So this is my attempt. Assume there exist an inverse of such function [math]S : T \mapsto T[/math], using the definition of inverse,
[math] S \circ (T + \itemrm{Id}_V) = \itemrm{Id}_V = (T + \itemrm{Id}_V) \circ S [/math] because T is an endomorphism, but
[math] (S \circ T) + (S \circ \itemrm{Id}_V) = (T \circ S) + (\itemrm{Id}_V \circ S) \\ (S \circ T) + S = (T \circ S) + S \\ (S \circ T) = (T \circ S) [/math]
Here I found myself stuck, I don't know if this reasoning is good, how do I finish?

[Anonymous No.9477474]

>>9477316
That's correct, this guy doesn't know what he's talking about>>9477286 >>9477301

[Anonymous No.9477476]

>>9477473
It was textrm, wasn't it? Damn

[Anonymous No.9477479]

>>9477441
hmm yeah, that should work.

[Anonymous No.9477483]

>>9477473
>Assume there exist an inverse of such function S:T↦T, using the definition of inverse,
>S∘(T+\itemrmIdV)=\itemrmIdV=(T+\itemrmIdV)∘S because T is an endomorphism, but
>(S∘T)+(S∘\itemrmIdV)=(T∘S)+(\itemrmIdV∘S)(S∘T)+S=(T∘S)+S(S∘T)=(T∘S)
This is all garbage, why do you write that S is from T to T? Why are you assuming S exists when that's what you're trying to prove? Why didn't you use the fact that T^2=0?

[Anonymous No.9477485]

>>9477473
Don't assume the existence of what you are trying to prove exists. Instead try to apply [math]a^2 - b^2[/math] in a suitable way to get you want.

[Anonymous No.9477520]

>>9477483
S is V to V, made a typo. I though by doing so I could reach some kind of identity statement of something like that.

>>9477485
What do you mean?

[Anonymous No.9477548]

>>9477520
Vector spaces over a field give you an abelian category, so the morphisms (which are linear maps) satisfy [math]g\circ (f_1 + f_2) = g\circ f_1 + g\circ f_2[/math], and vice versa, whenever the composites are defined. Now, let [math]T\circ T = T^2[/math] and see where this fact takes you.

[Anonymous No.9477580]

>>9477548
I think I got it, thanks.
Still new pls no hate

[Anonymous No.9477582]

>>9477580
You are my arch enemy now BITCH.

[Anonymous No.9477584]

>>9477473
Wow, I'm glad I don't have you as a student.

[Anonymous No.9477759]

Probability Theory is the queen of math

[Anonymous No.9477800]

>>9477759
I am the queen of math.

[Anonymous No.9477811]

>>9477759
>women in stem

[Anonymous No.9478354]

I want to kill myself, i forgot the [math]'[/math] in the equation (it was only a point on the paper) so i was trying so hard to verify if it was a group, I even tested it with a python program.

[Anonymous No.9478365]

>>9478354
Wait, that tiny little dot next to "ye" is supposed to be a prime? Can't say that I blame you.
>>9477759
At least name a branch of mathematics that isn't just measure theory but with the names changed.

[Anonymous No.9478388]

>>9478365
Yeah, it's supposed to be a prime...

[Anonymous No.9478398]

>>9478365
>probability theory is just measure theory
Ok and field X in mathematics is just set theory in different clothes

[Anonymous No.9478400]

>>9478354
>I even tested it with a python program
>>>/g/

[Anonymous No.9478405]

>>9478400
>tfw those cute azunyan tea cups are ruined with some stupid logo inside them
I just wanted a cute cat cup.

[Anonymous No.9478425]

Is category theory part of Abstract algebra?

[Anonymous No.9478427]

>>9478400
I was mad at testing different value to test if it worked. But i admit it, i'm from /g/ (I'm still in pure maths though, not in CS).

[Anonymous No.9478434]

>>9478425
>Abstract algebra
What is "abstract algebra"? Are you implying there is "non-abstract" algebra?

[Anonymous No.9478437]

>>9478427
>I'm still in pure maths
That's even worse. We don't need CS monkeys such as yourself shitting the field up.

[Anonymous No.9478438]

>>9478434
2deep4me im just an undergrad

I just want to do if it's part of my update course since I can't find any details

[Anonymous No.9478442]

>>9478425
Possibly. If you want to include higher category theory, then it is closer to topology.

[Anonymous No.9478459]

>>9478438
You'll have to do it either way if you are interested in algebra or anything which makes substantial use of it, which is basically the entirety of mathematics.

[Anonymous No.9478487]

>>9478459
>or anything which makes substantial use of it, which is basically the entirety of mathematics.
Not really.

[Anonymous No.9478571]

>>9478354
>be /g/tard
>can't read
Yeah that's about right.

[Anonymous No.9478618]

>>9478354
Fuck, there is another fucking thing in this paper, is written that this group is not an abelian group. Fuck, i'm done.

[Anonymous No.9478628]

i find that math students are a lot more chill than the rest of STEM despite having equal or even higher autism levels, why is this?

[Anonymous No.9478650]

I got my complex number exam back (highschool final) and I got zero marks for my final proof. Should I send it back for resubmition or did I do it wrong (1/2)

[Anonymous No.9478655]

>>9478650
2/2

[Anonymous No.9478658]

>>9478655
Nevermind just saw the mistake lmfao

[Anonymous No.9478674]

Any help with proving (without truth tablee) that if p=>q holds, then (p^r)=>(q^r) is a tautology?

I know about the (p^q)=>p property but I don't know if it helps

[Anonymous No.9478701]

>>9478674
I'm not gonna start fighting with Latex, so my natural deduction will look a bit wrong, but pretend it looks like what it should.
[math][p\land r]_1\\ \ \ \ \ p\ \ \ \ p\rightarrow q \ \ \ [p\land r]_1\\ \ \ \ \ \ \ \ \ \ q\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ q\land r\\ \ \ \ \ \ \ (p\land r)\rightarrow(q\land r)[/math]
And you eliminate the assumption 1 in the end when you introduce the implication.

[Anonymous No.9478703]

>>9477473
If TT=0, then:

(T + id)(id - T)= TT + T - T + id = 0 + 0 + id

Was is that hard lol

[Anonymous No.9478765]

>>9475076
Why do geometers hate algebra so much?

>muh visualization

[Anonymous No.9478775]

>>9475426
Platonist master race

[Anonymous No.9478779]

>>9478765
You answered your own question. If you do geometry, you can easily spot some kind of algebraic structure in play, like for example hyperbolic stuff and the Möbius groups, but the other direction is when you need to develop a geometric interpretation. Without developing the interpretation, you can't visualize your algebra that easily.

[Anonymous No.9478782]

>>9478765
Reminder that:
>algebra = hell
>geometry = purgatory
>topology = heaven

[Anonymous No.9478803]

>>9478779
>>9478782
Am I the only person in the world who loves algebra, but thinks geometry is just meh?

[Anonymous No.9478807]

>>9478765
>>9478779
>>9478803
What is "geometry" though?

[Anonymous No.9478811]

>>9478803
No. I don't like geometry that much. Algebra and topology make a holy duo, adding geometry there would make it an unholy trinity.

>>9478807
I wish I knew.

[Anonymous No.9478820]

>>9478807

geometry is a type of topology

similar to how alzheimers is a type of dementia

[Anonymous No.9478827]

>>9478803
Geometry has never caught my attention really. At best I can see in beauty in applications like Dirichlet's hyperbola method.

>>9478807
>What is "geometry" though?
I agree with this. I think I have learned quite well that "algebra" is and same goes for other fields like number theory that still have dedicated courses at universities. But what *really* is geometry? Isn't geometry just visual math?

I mean, I guess you could say geometry is the study of shapes (and I mean shapes in the most abstract sense possible) but then that would imply geometry = analysis + algebra - fun which means geometry isn't even a core field itself, but an application of analysis and algebra. But many people will call geometry its own thing. And the ancients thought geometry preceded analysis and algebra. So what's the deal with geometry?

[Anonymous No.9478866]

>>9478827
>geometry = analysis + algebra - fun
Analysis and fun can be seen to be special types of algebra.

[Anonymous No.9478868]

Do you agree with the statment " pure (measure theory, group theory, number theory) is completely bland an unispiring except you apply it to other fields"?

[Anonymous No.9478869]

>>9478866
No way.
geometry = analysis + algebra - fun
geometry = algebra + algebra - algebra
geometry = algebra

But jokes aside
>analysis
>special type of algebra
There is no fucking way you are serious. Please explain.

[Anonymous No.9478874]

>>9478868
Not at all. I am pretty sure that once you are hooked on any of those fields you will probably get harder from any new developments about the inherent structures of the subject than about any applications outside the core field.

[Anonymous No.9478876]

>>9478628
they're happier
>don't have to sell their future to industry like engineering and CS students
>aren't annoyingly ambitious and constantly stressed premed students, who account for large portions of biology and chemistry majors
>aren't physi-shits who struggle with inferiority complexes

[Anonymous No.9478883]

>>9478869
>geometry = algebra
This is assuming "geometry" exists.
>There is no fucking way you are serious.
It's pretty obvious. Analysis is a special type of topology, which is a special type of algebra.

[Anonymous No.9478889]

>>9478874
What else is there to pure group theory? Lie groups aren't "pure".

[Anonymous No.9478894]

>>9478883
>Analysis is a special type of topology, which is a special type of algebra.

I disagree fundamentally with this. Topology is typically applied to solve problems of interest in analysis but as a whole topology is unequipped for answering the questions of analysis. The analysis is not a subset of topology, it is quite distinct from it.

[Anonymous No.9478918]

>>9478889
representation theory is a subcategory of group theory

[Anonymous No.9478929]

>>9478889
>What else is there to pure group theory?

Well, there is quite a lot. Not everything about the classifications of groups is settled. And I also disagree with your following point.

What is pure is the study of the structures for their own sake. But let's say that you gain interest in lie groups specifically because of their wide applications elsewhere. This makes lie groups a hot topic of research and I would still consider it pure to study lie groups. Especially if you study lie groups for their own sake, not seeking to really establish anything outside of "this is a property that this class of object has".

[Anonymous No.9479035]

Hey /mg/, I'm gonna be starting as a math major this year. What graphing calculator should I get?

[Anonymous No.9479039]

>>9479035
>math major
You wouldn't need a "calculator".

[Anonymous No.9479043]

>>9479035
>math major
>calculator

[Anonymous No.9479143]

Is there any actual use for polar equations?

[Anonymous No.9479176]

>>9479035
that's for engineers
or for some god forsaken applied math (interpolation)

[Anonymous No.9479333]

>>9475076

Why has analysis been disparaged thus?

[Anonymous No.9479372]

>>9479333
How is allowing its discussion even when it's not related to the subject matter of this thread "disparaging"?

[Anonymous No.9479580]

>>9479143
>Is there any actual use for polar equations?
No.

[Anonymous No.9479692]

>>9479035
None. Just buy a normal pocket calculator for exams and shit and use your computer for the rest.

[Anonymous No.9479704]

>>9478354
Identity: (x,y)(0,0) -> (x+0, y*e^0 + 0*e^x)
Inverse: (x,y)(-x, -ye^-2x) -> (x-x, ye^-x -ye^-2xe^x ) = (0, 0)
Associativity: (x,y)[(a,b)(f,g)] = (x,y)(a+f, ge^a+be^f) = (a+f+x, ye^[a+f] +[ge^a+fe^b]e^x ) = (a+f+x, ye^[a+f] + ge^[a+x]+be^[f+x] )
and you can clearly see that this is symmetric from permuting any pairs so fuck doing it the other way.

QED

[Anonymous No.9479760]

I just spent an hour trying to prove a map was invertible when I was actually just supposed to prove the image of a certain element was invertible the ring. If I had taken a minute to think of a concrete example I would have realized my mistake, as the statement I was originally trying to prove is false. Kill me.

[Anonymous No.9479806]

>>9479333
alg*brainlets can't handle real manly math so they delude themselves into thinking their abstract nonsense actually means something

[Anonymous No.9479810]

>>9479806
true patricians recognize the value of both despite whatever their personal preferences may be

[Anonymous No.9479938]

I'm retarded
How does consensus theorem work?

[Anonymous No.9479950]

is there a general way to determine how many homomorphisms exist between two arbitrary groups?

[Anonymous No.9480030]

just dropped my set theory course

[Anonymous No.9480107]

>>9479950
Not in general, no. There's not even a general way to determine whether two groups are isomorphic.

[Anonymous No.9480185]

>>9479810
>true patricians recognize the value of both despite whatever their personal preferences may be
There's no "value" in alg*bra.

[Anonymous No.9480186]

>>9479938
>How does consensus theorem work?
Have you tried reading a proof?>>9479760

[Anonymous No.9480190]

>>9479950
>is there a general way to determine how many homomorphisms exist between two arbitrary groups?
You probably won't get a good answer for "arbitrary" groups, but if you restrict to something like finitely generated abelian groups you can get a very concrete answer.

[Anonymous No.9480621]

>>9478883
More like the other way around:

- Abelian Groups are 0-truncated Spectra
- Rings are 0-truncated A_∞-Ring Spectra
- Commutative Rings are 0-truncated E_∞-Ring Spectra
etc.

[Anonymous No.9480628]

>>9480621
>- Abelian Groups are 0-truncated Spectra
Can you detail how Z/2Z is a 0-truncated spectrum for someone who doesn't know anything about spectra?

[Anonymous No.9480634]

>>9480628
Brown Representability Theorem says spectra are essentially the same thing as generalized (co)homology theories.

To any abelian group, we can associated a 0-truncated spectrum called the Eilenberg-Maclane spectrum.

This is the spectrum correspond to singular (co)homology with coefficients in that abelian group.

[Anonymous No.9480652]

>>9480628
>>9480634
Particularly, a spectrum is a sequence of pointed spaces such that nth space is weakly equivalent to the looping of the (n+1)th space.

The eilenberg-maclane spectrum associated to an abelian group A is given by the sequence of eilenberg-maclne spaces K(A , n).

Eilenberg-Maclane spaces K(A,n) are characterized by having homotopy concentrated in degree n equal to A.

The nth homotopy group of a spectrum is defined as a colimit over k of the (n+k)th homotopy groups of K(A,k).

So the only non-zero homotopy group of the Eilenberg-Maclane spectrum is π_0 , and it is equal to A.

Thus it is 0-truncated.


It defines singular cohomology by H^n(X;A) = [ X , K(A,n) ]

[Anonymous No.9480670]

>>9480628
>>9480634
Particularly, a spectrum is a sequence of pointed spaces such that nth space is weakly equivalent to the looping of the (n+1)th space.

The eilenberg-maclane spectrum associated to an abelian group A is given by the sequence of eilenberg-maclne spaces K(A , n).

Eilenberg-Maclane spaces K(A,n) are characterized by having homotopy concentrated in degree n equal to A.

The nth homotopy group of a spectrum is defined as a colimit over k of the (n+k)th homotopy groups of the kth space.

So the only non-zero homotopy group of the Eilenberg-Maclane spectrum is π_0 , and it is equal to A.

Thus it is 0-truncated.


It defines singular cohomology by H^n(X;A) = [ X , K(A,n) ]

[Anonymous No.9480675]

>>9480652
>>9480634
I'm a brainlet who don't follow

Could you make this construction explicit for the easiest possible group (not the identity)? It sounds like you're constructing the spectrum based on the abelian group ("0-truncated Spectra are Abelian Groups"?) and not the other way around which is what I thought you meant by "Abelian Groups are 0-truncated Spectra"

[Anonymous No.9480691]

>>9480675
A spectrum [math]E[/math] is a sequence [math]\left\{ {{E_n}} \right\}[/math] of pointed topological spaces such that the maps [math]{E_n} \to \Omega {E_{n + 1}}[/math] induce isomorphisms on homotopy groups.

We can define homotopy groups of spectra by [math]{\pi _n}\left( E \right) = {\operatorname{colim} _k}{\pi _{n + k}}\left( {{E_k}} \right)[/math] .

A spectrum is said to be 0-truncated if all its higher homotopy groups are 0. Technically there can be spectra with homotopy in negative degree, but most of the time we only care about connective spectra (which are spectra with no negative degree homotopy).

Every spectrum defines a cohomology theory by [math]{E^n}\left( X \right) = \left[ {X,{E_n}} \right][/math].

The classical cohomology theory is singular cohomology [math]{H^n}\left( {X;A} \right)[/math] .

This is represented by a spectrum [math]HA = \left\{ {K\left( {A,n} \right)} \right\}[/math] . Where the spaces [math]{K\left( {A,n} \right)}[/math] are called Eilenberg-MacLane spaces, and are characterized by the fact [math]{\pi _k}\left( {K\left( {A,n} \right)} \right) = \left\{ \begin{gathered}
A\,\,\,\,\,k = n \hfill \\
0\;\;\;\;otherwise \hfill \\
\end{gathered} \right.[/math] .

So [math]{\pi _k}\left( {HA} \right) = \left\{ \begin{gathered}
A\,\,\,\,\,k = 0 \hfill \\
0\;\;\;\;otherwise \hfill \\
\end{gathered} \right.[/math] .

So abelian groups can be realized as 0-truncated spectra.

[Anonymous No.9480709]

>>9480691
For a more categorical perspective.

The ∞-categorical analog of Abelian Categories are Stable ∞-Categories.

i.e. Has all finite limits/colimits, has kernels/cokernels, zero objects, etc.


The category of abelian groups is universal amongst abelian categories.

The ∞-category of spectra is universal amongst stable ∞-categories.

[Anonymous No.9480724]

>>9480709
>>9480691
>>9480621
>>9480670
>>9480634
What are the applications of this?

[Anonymous No.9480741]

>>9480724
Traditionally spectra came out of trying to study cohomology theories.

An instance of where studying the representing spectrum of a cohomology theory is far easier than studying the actual cohomology groups is with Cobordism.

I don't really know where the idea of rewriting algebra as a special case of "higher algebra" defined in terms of spectra originated, but it is well-developed in Lurie's work.

[Anonymous No.9480747]

>>9478618

But it is abelian unless x and y are matrices or quaternions.

[Anonymous No.9480822]

empty set is subset of every set

but is it an element of every set?

[Anonymous No.9480844]

>>9480822
>but is it an element of every set?
No, consider the empty set which has no elements.

[Anonymous No.9480886]

>>9480724
Spectral sequences (e.g. Leray, Gysin, etc) can be used to evaluate hypercohomology groups of sheaves, which characterize diffeomorphism classes of Hermitian line bundles on paracompact (or ILH) spaces. These diffeomorphism classes present obstructions (the [math]H^1[/math] and [math]H^2[/math] terms) to the existence of quantomorphisms (lifts of symplectomorphisms onto Hermitian line bundles) via Kostant's construction, as well as obstructions (the [math]H^3[/math] term of the Hochschild cohomology) to deformation quantization a la Weyl-Moyal. These quantization conditions are necessary (and sufficient in the case of symplectic manifolds) conditions for your manifold to have "quantizable" physical observables.

[Anonymous No.9480903]

>>9480886
Spectral sequences aren't the same thing as Spectra.

Although they can be interpreted as sort of sequences of spectra.

[Anonymous No.9480915]

so im sure some of you have seen infinite series video on metallic means where the host gives the open problem of "does the exist any regular polygon which has a diagonal to side ratio of a metallic mean higher than sigma_2"
ive tinkered a bit with it and came up with an equation that will map an n-gon and a sigma to a diagonal index. see pic related. however its domain isnt restrained to the integers.
do you think im on the right track or not?

[Anonymous No.9480917]

>>9480903
Yeah I figured. Your definition of a spectrum is different from what I've seen for spectral sequences but since you used the same symbol [math]E_n[/math] I thought that I might as well flex myself a bit.

[Anonymous No.9480918]

>>9480886
Get some new material I'm tired of seeing this pic.

[Anonymous No.9480928]

>>9480918
It was only the third time I posted that though.

[Anonymous No.9480981]

9480917
This is embarrassing even for a physishit.

[Anonymous No.9481012]

>>9480886
>""physical"" ""observables""
>>>>/x/

[Anonymous No.9481137]

>>9480981
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

[Anonymous No.9481147]

>>9480928
>>/sci/?task=search2&search_filename=__yagokoro_eirin_and_yakumo_yukari_touhou_drawn_by_unya__6952b0ba0a950b326e9a4026e86d2142.jpg

Um?

[Anonymous No.9481168]

>>9481147
Have you ever been to a CS conference? Their shit's even more fucked up.

[Anonymous No.9481174]

>>9480652
>a spectral sequence is a sequence of pointed topological spaces
No. Sounds like what people would gather from skimming Griffith. Go read an actual QM book like Townsend, Sakurai or Landau-Lifshitz.

[Anonymous No.9481182]

>>9475426
What is the difference between an abstract object and a mental one?

[Anonymous No.9481186]

>>9481182
This is explained in Sakurai - Intuitive Topology for Physicists.

[Anonymous No.9481207]

>>9479950
Yes. It is intuitively clear using the physical interpretation of groups.

[Anonymous No.9481211]

>>9480915
>do you think im on the right track or not?
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

[Anonymous No.9481222]

>>9481211
any idea how i would go about prooving it?
ive tested all of the metallic ratios below 100000 and only the 0th, 1th, and 2nd have had any solutions within a 1E-6

[Anonymous No.9481224]

>>9481174
Spectrum =/= Spectral Sequence

Neither have anything to do (a priori) with physics.

[Anonymous No.9481225]

>>9478701
>natural deduction
The cobordism hypothesis can be proved, so where are the proofs of this natural "deduction"?

[Anonymous No.9481228]

>>9477473
>have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. I mean, this is essentially what's happening in QFT, right?
Not really. Read Sakurai - Visual Homological Algebra for Physicists

[Anonymous No.9481229]

>>9481224
>Spectrum =/= Spectral Sequence
Of course not, because spectral sequences are the cornerstone of something concrete (i.e. TQFT) while these "spectra" are the cornerstone of absolute algebraic wank.

[Anonymous No.9481233]

>>9481222
>any idea how i would go about prooving it?
I don't think this even requires proof. It's immediately obvious using the physical interpretation of metallic ratios given by Sakurai.

[Anonymous No.9481246]

>>9480628
>can't understand basic quantum mechanics
>wants to study string theory

[Anonymous No.9481268]

>>9481222
>ive tested all of the metallic ratios below 100000 and only the 0th, 1th, and 2nd have had any solutions within a 1E-6
Are you sure this is correct? My physical intuitions don't match up with this.

[Anonymous No.9481288]

>>9481229
Well Spectral Sequences are just a tool for computing cohomology, and Spectra are equivalent to cohomology theories.

So that makes no sense.

[Anonymous No.9481345]

>>9481233
how so?

>>9481268
it doesnt sound right to me either.
this is my code for checking it, bounds are based on observations from my gif

https://pastebin.com/BeV78gYr

[Anonymous No.9481437]

>>9481288
You're talking to an imposter btw. Since this >>9481137 post.
Some non-functioning autist apparently took a liking to impersonating me.

[Anonymous No.9481446]

>>9481437
How could someone be impersonating you? Are they using your nickname?

[Anonymous No.9481453]

>>9481446
It's someone I know IRL. He's apparently trying to mess with me.

[Anonymous No.9481464]

>>9480634
>Brown Representability Theorem says spectra are essentially the same thing as generalized (co)homology theories.
This is incorrect, spectral sequences in general don't correspond to cohomology theories.

[Anonymous No.9481467]

>>9481453
That's a new Yukari pic, mind if I save it?

[Anonymous No.9481473]

>>9481467
Can't tell you since I've never used any.

[Anonymous No.9481477]

>>9481473
I liked the one where Yukari and Yuyuko raped this one village boy.

[Anonymous No.9481732]

What would be a good research question for studying improper integrals? I want to know what kind of thing i should focus on, thx

[Anonymous No.9481754]

>>9481732
All research problems in this area were solved in Sakurai & Ballentine - An Intuitive Approach to Improper Integrals for Physicists.

[Anonymous No.9481827]

>>9481453
>He

[Anonymous No.9482043]

>>9476920
What the fuck
Never before have I seen this notation for the power set of X

Why can't he use P(X) or 2^X like a normal person

[Anonymous No.9482286]

Is there anything like axiomatic math logic? I'm fucking tired of all the mathematical logic books defining stuff by intuition. Do you guys know any book that explains logic in an axiomatic way, not an intuitive one?

[Anonymous No.9482318]

>>9482286
Read Wightman - Spin, Statistics and All That.

[Anonymous No.9482716]

>>9477441
Did you mean this?:
Since [math]f( \bigcup_{i \in I} A_i) = \bigcup_{i \in I} f(A_i)[/math] then, for any [math]K \subseteq I[/math], [math]f( \bigcup_{i \in K} A_i) = \bigcup_{i \in K} f(A_i)[/math] is true, so is for [math]K = \emptyset[/math], then:
[math]f( \emptyset ) = f \left( \bigcup_{i \in \emptyset} A_i \right) = \bigcup_{i \in \emptyset}f\left( A_i \right) = \emptyset[/math]

This suggests not to allow the index of the union to be the empty set becuase, again, given a constant function, [math]f(x) = c \neq \emptyset[/math] this could lead to a contradicion:
[math]f( \emptyset ) = c \neq \emptyset = f( \emptyset )[/math]

But what about if we consider the empty set as a family of subsets of [math]X[/math]? if I'm not mistaken, [math]\emptyset[/math] can be seen as an indexed family, namely [math]\{ E_i \colon i \in \emptyset \}[/math] (where all [math]E_i[/math] turn out to be the empty set), and again [math]f( \emptyset ) = \emptyset[/math] even if [math]f[/math] is defined to be a constant [math]c \neq \emptyset[/math].

[Anonymous No.9482723]

>>9482716
f(x) = c does not imply that [math]\text{Im}[f](\emptyset) = c[/math]

[Anonymous No.9482739]

>>9482723
What does [math][f][/math] stand for?

[Anonymous No.9483277]

Is there any difference between vector calculus, multivariable calculus and differential geometrY?

[Anonymous No.9483285]

>>9483277
>Is there any difference between vector calculus, multivariable calculus and differential geometrY?
Yes.

[Anonymous No.9483329]

>>9481477
Anime was a mistake.

[Anonymous No.9483336]

>>9482739
It's what some use in place of parenthesis for functors like [math]\text{Im}[\cdot][/math], though of course Im itself is nonstandard for the image since most people just write f(U) and the like and leave it at that.

[Anonymous No.9483352]

>>9481186
Neat-o, thanks.

[Anonymous No.9483613]

>>9483336
>functors
Ok, but the question was about sets, not categories.

[Anonymous No.9483631]

>>9483613
Any set is a functor since the category of sheaves on the one-point space is equivalent to the category of sets.

[Anonymous No.9483637]

>yfw IUT papers have been out for 5 and a half years and still no major error found

[Geometric Unity No.9483651]

>>9483637
>mfw

[Anonymous No.9483724]

Let I [math]=*\leftarrow*\rightarrow*[/math].

(i) Give an example of a diagram [math]A:\text{I}\rightarrow \text{Top}[/math] such that there exists a diagram [math]X:\text{I}\rightarrow \text{Top}[/math] so that [math]p: \text{Hom}_{\text{Ho}(\text{Top}^{\text{I}})} (A,X) \rightarrow \text{Hom}_{\text{Ho}(\text{Top)}^{\text{I}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^\text{I}) \rightarrow \text{Ho}(\text{Top})^\text{I}[/math] is not an equivalence of categories.

(ii) Suppose [math]A \in \text{Top}^{\text{I}}[/math] is given by [math]A_{10} \leftarrow A_{00} \rightarrow A_{01} [/math] with both maps Huerwicz cofibrations. Show that [math]p: \text{Hom}_{\text{Ho}(\text{Top}^{\text{I}})} (A,X) \rightarrow \text{Hom}_{\text{Ho}(\text{Top)}^{\text{I}}} (A,X)[/math] is a surjection for all [math]X:\text{I}\rightarrow \text{Top}[/math]. Is it always injective?

[Anonymous No.9483740]

Let I [math]=*\leftarrow*\rightarrow*[/math].

(i) Give an example of a diagram [math]A:\text{I}\rightarrow \text{Top}[/math] such that there exists a diagram [math]X:\text{I}\rightarrow \text{Top}[/math] so that [math]p: \text{Hom}_{\text{Ho}(\text{Top}^{\text{I}})} (A,X) \rightarrow \text{Hom}_{\text{Ho}(\text{Top)}^{\text{I}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^\text{I}) \rightarrow \text{Ho}(\text{Top})^\text{I}[/math] is not an equivalence of categories.

(ii) Suppose [math]A \in \text{Top}^{\text{I}}[/math] is given by
[math]A_{10} \leftarrow A_{00} \rightarrow A_{01} [/math]
with both maps Huerwicz cofibrations. Show that [math]p: \text{Hom}_{\text{Ho}(\text{Top}^{\text{I}})} (A,X) \rightarrow \text{Hom}_{\text{Ho}(\text{Top)}^{\text{I}}} (A,X)[/math] is a surjection for all [math]X:\text{I}\rightarrow \text{Top}[/math]. Is it always injective?

[Anonymous No.9483861]

>>9483631
Yes, but, (aside from the fact I don't know nothing about categories) we are specting to solve the problem using the given theory and no other.

[Anonymous No.9483894]

Guys how do you find the sum of this series?
I've tried using the power series of ln(1+x) and the closest ive come is to represent it as:

[math]\int_{-1}^{1} x^2ln(1+x)dx= \sum_{2}^{\infty} \frac{(-1)^n}{n^2+n-2} [/math]

Any help or indicaiton as to where I'm going wrong?

[Anonymous No.9483916]

Test

[Anonymous No.9483937]

>>9483894
Guys I did it! I integrated between 0 to 1 instead. Is this justified?

[Anonymous No.9483948]

>grading for a discrete math class
>hw problem asks if 2 is in the set containing 2
>many students answer no

[Anonymous No.9483967]

Let I = ∗ ← ∗ → ∗.

(i) Give an example of a diagram A: I → Top such that there exists a diagram X: I → Top so that [math]p: \text{Hom}_{{\text{Ho}(\text{Top}^{\text{I}})}} (A,X) \rightarrow \text{Hom}_{{\text{Ho}(\text{Top)}^{\text{I}}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^\text{I}) \rightarrow \text{Ho}(\text{Top})^\text{I}[/math] is not an equivalence of categories.

(ii) Suppose A ∈ [math]\text{Top}^{\text{I}}[/math] is given by [math]A_{10} \leftarrow A_{00} \rightarrow A_{01} [/math] with both maps Hurewicz cofibrations. Show that [math]p: \text{Hom}_{\text{Ho}(\text{Top}^{\text{I}})} (A,X) \rightarrow \text{Hom}_{\text{Ho}(\text{Top)}^{\text{I}}} (A,X)[/math] is a surjection for all X: I → Top. Is it always injective?

[Anonymous No.9483968]

Let I = ∗ ← ∗ → ∗.

(i) Give an example of a diagram A: I → Top such that there exists a diagram X: I → Top so that [math]p: \text{Hom}_{{\text{Ho}(\text{Top}^{\text{I}})}} (A,X) \rightarrow \text{Hom}_{{\text{Ho}(\text{Top)}^{\text{I}}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^\text{I}) \rightarrow \text{Ho}(\text{Top})^\text{I} \text{ is not an equivalence of categories.}[/math]

[Anonymous No.9483969]

Let I = ∗ ← ∗ → ∗.

(i) Give an example of a diagram A: I → Top such that there exists a diagram X: I → Top so that [math]p: \text{Hom}_{{\text{Ho}(\text{Top}^{\text{I}})}} (A,X) \rightarrow \text{Hom}_{{\text{Ho}(\text{Top)}^{\text{I}}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^\text{I}) \rightarrow \text{Ho}(\text{Top})^\text{I}[/math] is not an equivalence of categories.

[Anonymous No.9483970]

Let I = ∗ ← ∗ → ∗.

(i) Give an example of a diagram A: I → Top such that there exists a diagram X: I → Top so that [math]p: \text{Hom}_{{\text{Ho}(\text{Top}^{\text{I}})}} (A,X) \rightarrow \text{Hom}_{{\text{Ho}(\text{Top)}^{\text{I}}}} (A,X)[/math] is not a bijection. Deduce that [math]p:\text{Ho}(\text{Top}^{\text{I}}) \rightarrow \text{Ho}(\text{Top})^{\text{I}} [/math] is not an equivalence of categories.

[Anonymous No.9483975]

>>9483970
(ii) Suppose A ∈ [math]\text{Top}^{\text{i}}[/math] is given by [math]A_{10} \leftarrow A_{00} \rightarrow A_{01}[/math] with both maps Hurewicz cofibrations. Show that [math]p:\text{Hom}_{\text{Ho(Top}^{\text{I}}}(A,X) \rightarrow\text{Hom}_{\text{Ho(Top)}^{\text{I}}}(A,X)[/math] is a surjection for all X: I → Top. Is it always injective?

[Anonymous No.9483981]

>>9483970
(ii) Suppose A ∈ [math]\text{Top}^{\text{I}}[/math] is given by [math]A_{10} \leftarrow A_{00} \rightarrow A_{01}[/math] both maps Hurewicz cofibrations. Show that [math]p:\text{Hom}_{\text{Ho(Top}^{\text{I})}}(A,X) \rightarrow\text{Hom}_{\text{Ho(Top)}^{\text{I}}}(A,X)[/math] is a surjection for all X: I → Top. Is it always injective?

[Anonymous No.9483983]

>>9483981
ok i'm going to kill myself

[Anonymous No.9483987]

>>9483970
(ii) Suppose A ∈ [math]\text{Top}^{\text{I}}[/math] is given by [math]A_{10} \leftarrow A_{00} \rightarrow A_{01}[/math] both maps Hurewicz cofibrations. Show that [math]p:\text{Hom}_{\text{Ho(Top}^{\text{I}})}(A,X) \rightarrow\text{Hom}_{\text{Ho(Top)}^{\text{I}}}(A,X)[/math] is a surjection for all X: I → Top. Is it always injective?

[Anonymous No.9484299]

>>9483983
Please do.

[Anonymous No.9484857]

>>9478365
it's not, measure theory doesn't concern independent sets, which are central in probability theory

[Anonymous No.9485320]

>>9483894

[Anonymous No.9485658]

How can I tell if something is really maths or not?

This past week I've become more and more skeptical that category theory isn't really mathematical

[Anonymous No.9485720]

>>9485658
lubos motl said category theory is powerpoint mathematics
anyway, it's math if it ultimately reduces to a counting problem of some kind

[Anonymous No.9485733]

>>9485720
>powerpoint mathematics
What did he/she mean by this?

[Anonymous No.9485852]

What's the most emotionally dishonest field of mathematics?

[Anonymous No.9485857]

>>9485852
psychology

[Anonymous No.9485873]

>>9485733

https://motls.blogspot.com/2004/11/category-theory-and-physics.html

[Anonymous No.9485970]

>>9485873
>physics
>>>/x/

[Anonymous No.9485978]

For two finite dimensional [math]\mathbb C[/math]-vector spaces [math]V,W[/math], what is the "usual" topology on [math]\hom(V,W)[/math]?

[Anonymous No.9485983]

>>9485978
https://math.stackexchange.com/questions/568529/topology-on-hom-mathbb-cv-w

[Anonymous No.9485990]

>>9485983
Yeah I got there too, coincidentally cos im reading the same book. However I don't know which of the topologies I'm supposed to use still: the compact-open? or the "normables" topology which I have no clue what it is

[Anonymous No.9486010]

>>9485990
although im guessing the one from [math]\mathbb C^{\dim V}\otimes\mathbb C^{\dim W}[/math]

[Anonymous No.9486107]

>>9475076
Retarded Social studies major here, how do I learn quadratic equation factoring. I haven't taken an algebra class in almost 3 years.

I literally can't solve this without having a breakdown.

-4x^2 + 23x +6 = 0
like I'm stuck here
(-4x + ?)(x + ?) = kms

[Anonymous No.9486126]

>>9477326
(abb)* | (bab)*

[Anonymous No.9486160]

>>9486107
>Retarded Social studies major
The "retarded" part is superfluous, it follows from the rest.

[Anonymous No.9486177]

>>9486160
Hahahaha lol you aren't fucking original, you autistic piece of shit. You probably don't know jack shit about the world you cock sucker, go solve some imaginary numbers you faggot, cause that's all you'll ever have, just like your imaginary girlfriend you jerk off. Any of you cunts that look down on any social learnings are ignorant subhumans, I bet you don't even realize how your social maladjustment has lead you to lead a life of eternal virginity you worthless STEMcel. Your a fucking dud, a fucking impotent is what you are, beating the slave drums, go get a STEM degree and compete with Pajeets you cock sucker, you are a fucking stooge for Global technocrats to exploit you barren wombed fuck! So don't reply to me, EVER, unless you want to help me with my math homework FAGGOT

[Anonymous No.9486183]

>>9486177
>FAGGOT
Why the homophobia?

[Anonymous No.9486186]

>>9486183
Just explain why this is?? Why, what does this shit even mean, I put in numbers, but I don't know how they work. I can't just guess forever.

[Anonymous No.9486192]

>>9486183
WHAT FUCKING PATERN IS THIS????

[Anonymous No.9486194]

>>9486192
>WHAT FUCKING PATERN IS THIS????
Do you need to swear?

[Anonymous No.9486197]

>>9486194
lol i'm just gonna drop out of this class, fuck university slavery and their bull shit, I'm getting a degree in German. Fuck math and this irrelevant trash, fuck business wasn't gonna do that shit anyways.

[Anonymous No.9486200]

>>9486197
cool down Hans

[Anonymous No.9486205]

>>9486200
Fuck off cunt, have fun slaving like a good goy faggot for your technomasters while I have a good time writing manifestos and ranting aginst (YOU)

[Anonymous No.9486209]

>>9486186
>>9486192
I'm sorry to hear about your disability, anon.

[Anonymous No.9486214]

>>9486209
oh i think I noticed it. I'ts just divided by 2 twice huh?

[Anonymous No.9486219]

>>9486214
2 is equivalent to 0 in the complex numbers, so that wouldn't make much sense.

[Anonymous No.9486225]

>>9486219
No no no, Like the number under the root gets divided by 2 and then the resulting number is divided by 2 and then you just pop a 2i in front of whatevers left under the root

[Anonymous No.9486234]

>>9486225
>divided by 2 and then you just pop a 2i i
2 and 2i are not equivalent in |C

[Anonymous No.9486236]

>>9486225
Sqrt(-4*7)=sqrt(-2^2 *7)=sqrt(-2^2)*sqrt(7)=?

[Anonymous No.9486247]

>>9486236
2i*sqrt(7) cause 2i is just -2^2 right? That's all way too complex man, but I appreciate it.

[Anonymous No.9486250]

>>9486247
Bingo

[Anonymous No.9486281]

>>9486107
>>9486177
>>9486186
>>9486192
Get your pacifier out babby, momma needs to teach you your 123s and how to not be a cunt because you posted on 4chan and expected anything other than "fuk u brainlet"-tier replies.

-4x^2 + 23x + 6 = 0
>The 'x' portion can only be broken down three ways:
(4x + ?)*(? - x)
(? - 4x)*(x + ?)
(2x+?)*(? - 2x)

>The '6' portion can be broken down a few more ways
(? + 6)*(? + 1)
(? + 3)*(? + 2)
(? - 6)*(? - 1)
(? - 3)*(? - 2)

At this point, a newb brainlet is going to have to fiddle around with these possibilities in free-form creative ways to try and eliminate some obviously incorrect combinations of this. A more practiced mathie is going to have some honed intuition when it comes to figuring out the pairing. For instance, looking at the presence of the '23x', I have a hunch the answer will involve a '+4x' and a '+6'. I proceed from those options.

Complex roots are incredibly easy honestly.
>Remove 'i' first
5 - √(-28) = 5 - i*√(28)

>Simplify any root like normal
5 - i*√(28) = 5 - i*√(4)*√(7) = 5 - 2i*√(7)

[Anonymous No.9486299]

>>9478765
>>9478782
>In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

[Anonymous No.9486432]

>>9486250
Ding, ding, ding.

[Anonymous No.9487180]

What's an example of a series [math](a_{n})[/math] such that [math]0\leq a_{n} \leq \frac{1}{n}[/math] but [math]\displaystyle \sum (-1)^{n+1} a_{n}[/math] diverges? /sqt/ was pretty indecisive on this and I'm drawing a blank.

[Anonymous No.9487185]

>>9487180
Take a_n = 1/n for even n and 0 for odd n

[Anonymous No.9487189]

>>9487180
>>9487185
Also an is a sequence not a series

[Anonymous No.9487240]

For any two propositions [math]P[/math] and [math]Q[/math], if [math]P\to Q[/math] is true then [math]Q[/math] is true.
(The proof relies on the law of excluded middle, hence this result is not valid in intuitionistic logic.)

Proof: By excluded middle, the proposition [math]P[/math] is either true or false.
If [math]P[/math] is true, then by the hypothesis [math]P\to Q[/math] and modus ponens, [math]Q[/math] is true.
If [math]P[/math] is false, then by the principle of explosion ("ex falso quodlibet") anything follows: in particular, [math]Q[/math] is true.
In either case, [math]Q[/math] is true, regardless of the truth-value of [math]P[/math].
QED

[Anonymous No.9487261]

>>9487180
[math]\frac{1}{n}[/math] goes to zero as n goes to infinity, and since every term of your serie must be sandwiched between [math]\frac{1}{n}[/math] and 0, your series' terms must also go to zero. Since it's an alternate serie, it will necessarily converge. Oh also your serie's terms must be monotonocally decreasing, so maybe play with that.

https://en.m.wikipedia.org/wiki/Alternating_series_test

[Anonymous No.9487263]

>>9487240
This is a well-known result. An interesting consequence of it can be seen by noticing that [math]\bot \rightarrow \bot[/math] holds classically, where [math]\bot[/math] is the initial object in the category of propositions, therefore by applying the theorem we get that [math]\neg \neg \bot[/math] holds intuitionistically.

[Anonymous No.9487274]

>>9475426
Mathematics is just about bringing the definitions of the definitions of the symbols in a new form, better understandable form, which has the same meaning. So mathematics is not real, but just a way to simplify language.

[Anonymous No.9487279]

>>9487180
>What's an example of a series (an) such that 0≤an≤1n but ∑(−1)n+1an diverges?
Why do you think there exists such a series?

[Anonymous No.9487283]

>log in to /mg/
>no new (you)'s
>log off

[Anonymous No.9487284]

>>9476932
>Ask well informed questions during lecture (because you read ahead and already know what is to come, but maybe there is something you didn't understand that the professor can clarify)
I think it depends on the prof, some are annoyed by this behaviour

[Anonymous No.9487333]

>>9475076

I live near this statue, dubs gets a selfie with a requested message

[Anonymous No.9487344]

>>9487333
So what about an `infinite set'? Well, to begin with, you should say precisely what the term means. Okay, if you don't, at least someone should. Putting an adjective in front of a noun does not in itself make a mathematical concept. Cantor declared that an `infinite set' is a set which is not finite. Surely that is unsatisfactory, as Cantor no doubt suspected himself. It's like declaring that an `all-seeing Leprechaun' is a Leprechaun which can see everything. Or an `unstoppable mouse' is a mouse which cannot be stopped. These grammatical constructions do not create concepts, except perhaps in a literary or poetic sense. It is not clear that there are any sets that are not finite, just as it is not clear that there are any Leprechauns which can see everything, or that there are mice that cannot be stopped. Certainly in science there is no reason to suppose that `infinite sets' exist. Are there an infinite number of quarks or electrons in the universe? If physicists had to hazard a guess, I am confident the majority would say: No. But even if there were an infinite number of electrons, it is unreasonable to suppose that you can get an infinite number of them all together as a single `data object'.

[Anonymous No.9487345]

>>9487344
>>9487344
W E W
L A D

[Anonymous No.9487349]

>If the right hand side of a differential equation is sin(x) you can use A*sin(x) + B*sin(x) as an approach to differentiate and put into the left hand side.

But what if the right hand side is x + sin(x)?

I tried using Ax + B*sin(x) + C*sin(x) but it always leads to a paradox

[Anonymous No.9487354]

>>9487349
Wait, is it simply A + Bx + C*sin(x) + D*sin(x) ?

[Anonymous No.9487374]

"mathematics" is a spook

[Anonymous No.9487656]

>take an honest proof of Theorem X
>erase some specific details and replace them with abstract details
>"we present a proof of a new Theorem Y from which Theorem X follows as a trivial corollary"
category theorists WILL defend this

[Anonymous No.9487665]

>>9487656
>unironically considering what category theorists are doing
Why are you doing this? Let them circlejerk on their own and use your own time on math instead of complaining about their shit.

[Anonymous No.9487714]

>>9487656
>Take Helly's theorem
>Erase some terms and replace with another shit
>Do the same in the proof
>A new Helly type therem has been found
>New paper guaranteed
>Uroboros

[Anonymous No.9487730]

>that one paper from the 1940's that basically started the field so every paper published in that field has to cite it for some reason

[Anonymous No.9488418]

>>9487730
what?

[Anonymous No.9488873]

>>9487656
>abstract details
Fuck "Abstract" details and unphysical garbage not based on empiricism.

[Anonymous No.9489154]

>>9475076
I've recently gotten into the habit of LaTeX'ing my assignments (they look loads nicer imo), but I'm not sure on a couple of things:
Is there a better way of showing the steps to the derivation of an equation than just
"
\[
\begin{aligned}
Step 1\\
Step 2\\
...
\end{aligned}
\]"?
I was also curious if some of the proofs were at least decent (from a non-rigorous standpoint). This class is the first class in the math department I'm taking (and it's a seminar for non-math people on capillary surfaces), so i just wanted to make sure the proofs could at least be followed.
Any advice?
PDF: https://www.docdroid.net/K2FMJVo/hw2.pdf

[Anonymous No.9489230]

>>9489154
Most of the time this is avoided by skipping the trivial computations. This may not fly for your undergraduate courses, though. I'd try to create a bit of a narrative between lines if possible. Also, at least in my opinion, it looks better to not have your text (i.e. words) centered during your derivations.

[Anonymous No.9489239]

>>9489154
>decent from a non-rigorous standpoint
No such thing.

[Anonymous No.9489514]

>>9487284
depends on how much of a cheese dick you are about it

[Anonymous No.9489778]

what’s a decent book for set theory? i don’t expect to “””self learn””” the damned thing as i don’t have the time to spare as is, i just want to be prepared for the next semester.

[Anonymous No.9489785]

>>9489778
http://4chan-science.wikia.com/wiki/Mathematics#Introductory_Set_Theory

Unless you mean intro to proofs set theory
http://4chan-science.wikia.com/wiki/Mathematics#Proofs_and_Mathematical_Reasoning
Naive Set Theory (Dover Books on Mathematics) by Halmos

[Anonymous No.9489802]

>>9489778
>what’s a decent book for set theory? i don’t expect to “””self learn””” the damned thing as i don’t have the time to spare as is, i just want to be prepared for the next semester.
Which school for brainlets do you go to that unironically uses sets?

[Anonymous No.9489846]

>>9483948

[Anonymous No.9489884]

What's the intuition behind the direct sum of two vector spaces? That is, what is it doing geometrically (if anything)?

[Anonymous No.9489903]

>>9489884
It's isomorphic to their geometrical tensor product.

[Anonymous No.9489911]

>>9489884
Take the direct sum of two vector speces generated by single l.i. vectors in R^2. What do you get? Now try a plane and a line in R^3.

[Anonymous No.9489921]

>>9489903
what's a tensor product

[Anonymous No.9489925]

>>9489921
I said "geometrical" tensor product. It's like a tensor product, but in a geometrical way.

[Anonymous No.9489929]

>>9489884
The process of construction is the same as a regular sum. The condition that makes it a direct sum is bound to what subspaces you are working with. Basically it is a direct sum iff the intersection of both is the neutral element 0. But don't mistake it for orthogonality. It is weaker.

[Anonymous No.9489931]

>>9489925
what's a geometrical tensor product

[Anonymous No.9490035]

>>9489931
A cute version of the direct sum.

[Anonymous No.9490174]

>>9475076
We bought a Gomboc to our maths teacher last year. Was pretty cool to see him play with it.

[Anonymous No.9490250]

What's a good source for learning more about symmetric bilinear forms especially with regards to the Hessian?

[Anonymous No.9490276]

>>9489884
There's this set of vectors that go one way, and another set of vectors that go some other way. Now you take both sets of vectors, and create a bigger space that goes that one way AND that other way. Sometimes both sets of vectors can go in the same way and that's OK!

[Anonymous No.9491808]

>>9475076
I need some help with double checking a problem my sister asked me to solve.

If x + y is 5 and x - y is 5 what is x and y?

I said x is 5 and y is 0. Are there more answers than this one because it seems too easy.

[Anonymous No.9492017]

to this day i have been ridiculing musk for being a retard conman.
but now, he proved himself to be legit.
how can i do maths anymore after this?
goodbye mathematics, i had fun times with you,
but i must leave you for physics.

[Anonymous No.9492088]

>log into funnyjunk.com/math
>no new (you)'s
>log off

[Anonymous No.9492734]

>>9491808
x+y=5 and x-y=5 imply (x+y)+(x-y)=2x=10 <-> x=5 -> y=0

[Anonymous No.9492743]

Come and lend a hand here: >>>/a/168129470
We must show people how beautiful math is.

[Anonymous No.9492784]

>>9492743
Simply fuck off.
>beautiful
>>>/lgbt/

[Anonymous No.9493047]

>>9475076
So I was fucking around with primes and wrote a program that maps them onto pyramids. There are two options that allow for four different graphs to be made, as well as any arbitrary color. In the ones I made, black is non-prime and white is prime.
One option is what I call 'snakiness', which means when it goes to a new row on the pyramid, does it go from left to right or does it snake back and forth. The other just controls whether it makes a right triangle or a pyramid.
Pic related is the output of a non-snaked right triangle.
My question is, what the fuck can I do with this? Can I learn something from what I've done?

[Anonymous No.9493053]

>>9492784
Fag.

[Anonymous No.9493066]

>>9493053
>Fag.
Why the homophobia?

[Anonymous No.9493114]

>>9493053
Pls just die.

[Anonymous No.9493431]

>>9493114
Why do you hate fun?

[Anonymous No.9493448]

>>9493431
Citation needed.

[Anonymous No.9493478]

>>9493448
It has already been provided. Please check your notes.

[Anonymous No.9493502]

>>9493478
Show it to me explicitly.

[Anonymous No.9493522]

>>9493502
But it cannot be expressed using standard functions!

[Anonymous No.9493558]

>>9493522

[Anonymous No.9495127]

Anybody have a good topic for me to write a paper on related to differential geometry? Please don't say GR

[Anonymous No.9495145]

>>9495127
Help me out here >>9492743 and I'll lend you a hand in return.

[Anonymous No.9495160]

All weebs are cancer and should kill themselves.

[Anonymous No.9495165]

>>9495160
>>>/reddit/

[Anonymous No.9495184]

>>9495165
Wow, that changes everything, I have a new fond appreciation of neckbeard, pedophile, autistic mororns that feel unique and hilarious for spamming the same shit every day.

[Anonymous No.9495202]

>>9495184
>I have a new fond appreciation of neckbeard, pedophile, autistic mororns that feel unique and hilarious for spamming the same shit every day.
Good.

[Anonymous No.9495256]

>>9495202
Glad I could make it clear.

[Anonymous No.9495314]

>>9495184
What's a morom?

[Anonymous No.9495365]

Can someone explain countably/uncountably infinite to me? I can't grasp the concept very well

[Anonymous No.9495394]

>>9495365
sets in bijection with natural numbers have countably infinite cardinality
infinite sets not in bijection with natural numbers have uncountable infinite cardinality (simplest example being the real numbers by https://en.wikipedia.org/wiki/Cantor's_diagonal_argument))

[Anonymous No.9495565]

>>9475426
>mathematical objects
>not somehow just a special case of formulae over a language in a specific type/logic system
the only things that exist are types and the various systems they determine

[Anonymous No.9495854]

>>9495394
but why can't i biject something like 0.000000001 with 1, 0.000000002 with 2, and so on?

[Anonymous No.9495855]

>>9495565
Why do types "exist"?

[Anonymous No.9495861]

>>9495854
>but why can't i biject something like 0.000000001 with 1, 0.000000002 with 2, and so on?
What do you mean by "and so on"? You need to define this alleged bijection for all real numbers.

[Anonymous No.9495867]

>>9495854
>but why can't i biject something like 0.000000001 with 1, 0.000000002 with 2, and so on?
Because using the diagonal argument, you simply construct a real number not in this list by making it differ with every single real number you list in this fashion.

[Anonymous No.9496021]

>>9475076
Brainlet-tier question.
I have two equations and two unknowns:
[math]
u = a_1 \cos(\phi_1)
v = a_1 \sin(\phi_1)
[/math]
If I'm given u and v, how do I find a and [math] \phi[/math]

I'm trying to rack my brain, but nothing's coming up

[Anonymous No.9496343]

>>9495867
>you simply construct a real number
That's not possible in general.

[Anonymous No.9496676]

>>9495855
Because I postulate their existence.

[Anonymous No.9496727]

>>9496343
>it's not possible to construct a subset of the natural numbers
WDHMBT?????

[Anonymous No.9496749]

>>9496727
Ignore him

[Anonymous No.9497141]

>>9496749
>Ignore him
I'm not a "him".

[Anonymous No.9497143]

t-teach me vector calculus..senpai...

[Anonymous No.9497254]

>>9497143
GEE WHIZ, HOW ON TOPIC.

[Anonymous No.9497323]

>>9481186
So basically mental objects exist in the human brain or "spirit" and abstract objects are independent of that.

[Anonymous No.9497332]

>>9475076
what does "not" as in this is NOT that, actually mean? why is it actually hurting my brain that I can't come up with a satisfying answer.

[Anonymous No.9497373]

>>9497254
what is your problem sir?

[Anonymous No.9497379]

>>9497254
Stop evading bans.

[Anonymous No.9497407]

>>9478442
I would agree with this.

[Anonymous No.9497412]

>>9486299
love weyl. Atiyah expressed something similar.

[Anonymous No.9497418]

>>9497332
It encompasses all situations in which a condition is false.
Take the phrase "it rains every day." The negation is not "it never rains," but more like "there is at least one day when it does not rain."

[Anonymous No.9497422]

>>9497254
Reddit trash is not welcome here.
>>>/r/eddit/

[Anonymous No.9497423]

>>9497418
But what does it mean for a condition to be false?

[Anonymous No.9497433]

>>9497423
The condition being true entails a contradiction.

[Anonymous No.9497437]

>>9495365
A countably infinite set is a set in which you can relate every element from it with a natural number (this process is called counting, and, formally speaking, is creating a bijection between the set and the set of natural numbers). It's called countably infinite because you can make this process of counting without getting any contradiction. And since bijections only occur with sets of the same cardinality, you conclude that all of the countably infinite sets have the same cardinality as the natural numbers.

For example, consider the set of all positive even numbers. One could say it has half the cardinality of N, but you can still create the bijection B(even) = even/2 and thus you can 'count' that set and it has the same cardinality. Other examples of this kind of sets are: The integers, the rationals, the set of all possible words (with or without meaning)

Uncountably infinite sets are sets in which you can't count the elements, because supposing you can do so follows a contradiction since you can always 'construct' one different from the others. See Cantor's diagonal proof on it.

[Anonymous No.9497449]

>>9497433
but what does it mean to contradict?

[Anonymous No.9497501]

is is is?

>logical:
is is is
not is not is
is is not not
not is not not not

>illogical
not is is
is is not is
not is not not is
is is not not not

[Anonymous No.9497525]

>You forgot to see what happened if the set is emptyXD, but it's just a couple of decimals sweetie
FUCK YOU MATH AUTISTS.

[Anonymous No.9497558]

>>9475076
who's smarter mathematicians or physicist? no
>they're both smart in their own way

[Anonymous No.9497567]

>>9497558
From smarter to dumber: mathematician, physician, physicist.

[Anonymous No.9497573]

>>9497558
One needs to know scientific principles, have intuition to model, and know a shiton of math. The other one needs to know a shitton of math. No, undergrads don't apply.

[Anonymous No.9497588]

>>9497567
>physician
wtf how are they smarter than physicist?

[Anonymous No.9497614]

>>9497588
They are not geniuses like mathematicians, but they actually do something useful even with their limited intellect.

[Anonymous No.9498517]

why the fug are metal gombocs $300?

I'm gonna crank some out with our 4-axis cnc at our school shop and sell em for fiddy bucks. Anyone want one?

[Anonymous No.9499319]

I've read the wiki, but I'm looking for a specific answer here I guess. Can you guys layout the progression of math topics I should study that would cover an undergraduate education? I'm an engineering major and I hate how we just skip over the fundamentals.


Something like Calculus->Discrete Math->Analysis, etc etc idk