Mathematics - 2017-12-10 (page 2 of 2)

@MatheinBoulomenos What exactly is the difference between sheaf cohomology and Cech cohomology?

The definitions looks similar at a glance

Hello

an anyone explain me how is this possible?

I tried assuming sina=m and cosa =n then substitute for tan (a+b) but I am not getting 1

@BalarkaSen sheaf cohomology cares for global sections. Cech cohomlogy looks at open covers (and you take the direct limit over all open covers)

Anyone? Please help me

they are isomorphic in degree $\leq 1$

and for paracompact Hausdorff spaces they're always isomorphic iirc

1:44 PM

@MatheinBoulomenos No I mean for example Forster defines sheaf cohomology to be the cohomology arising from the chain complex $0 \to C^0(\mathcal{U}; \mathscr{F}) \to C^1(\mathcal{U}; \mathscr{F}) \to C^2(\mathcal{U}; \mathscr{F}) \to 0$ where the cochain groups are defined for a given sheaf $\mathscr{F}$ defined on an open cover $\mathcal{U}$ of the base space $X$ as $C^q := \prod \mathscr{F}(U_{i_1} \cap \cdots \cap U_{i_q})$

And the boundary maps are defined using the same alternating sum game

This looks like Cech to me

huh, that's not the definition I'm used to

Yeah I thought you guys used derived functors

but Forster considers only paracompact Hausdorff spaces, right?

We definitely use derived functors

it doesn't impose any hypothesis on $X$ but it certainly just cares about paracompact Hausdorff spaces (Riemann surfaces are always those)

@LeakyNun hi . Can you help me if you are free

1:46 PM

yeah that definition is really Cech cohomology

mmkay

so i suspected

Thanks

Dogatemy hhvvjhvjvjyvjyvuyvjyvjyvukcyjtdyjyitstddthxmhkdty

@Fawad Don't spam.

If someone is willing to answer, they will.

@Fawad try computing tan A + tan B and tan A - tan B, then use tangent compound angle formula and then substittute the tan with the mn expressions, it should pop up

Got.

Thanks

2:03 PM

$1=1+\frac{1}{2}$ By analytic continuation.

????????????? but that is not even a countable series

@Secret It is true.

how does one use analytic continuation on a finite series?

Not a finite one, an infinite one where all the other terms cancel out and left is a single 1, but the analytic continuation says that the sum should be 1+1/2.

hmmm... a divergent series where almost all terms cancel out...

$\sum_{n=1}^{\infty}(-1)^n$

2:08 PM

Something like that yes. But several similar kind of sums.

Q: ζ(-n) and "powers" of Grandi's series

1 + 1 + 1 + 1 + ⋯

In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑ n = 1 ∞ n 0 {\displaystyle \sum _{n=1}^{\infty }n^{0}} , ∑ n = 1 ∞ 1 n {\displaystyle \sum _{n=1}^{\infty }1^{n}} , or...

Riemannian zero

9

For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+\cdots$ A value can also be assigned to the related series ${n+0 \choose n}+{n+1 \choose n}+{n+2 \choose n}+{n+3 \choose n}+\cdots$ by comparing its value with that of "powers...

divergent-series

2:25 PM

I am in desperate need of a graduate level diff eq text. Does anybody have any recommendations?

Vrouvrou

@Secret hello,

please how we write $f(u_n)\to+\infty$ with the $\varepsilon$ definition

knowing that $u_n\to+\infty$ when $n\to\infty$

brot

2:48 PM

In Riemann geometry, what does $dx^i$ denote?, if $(x^1,...,x^n)$ are my coordinate functions?

Eric Silva

3:33 PM

@DavidReed what kind of diff eq. It is a massive area of math

Hi eric! One that would be good for quantum mechanical solutions to PDE's

Ideally one that would provide proofs for theorems stated in an undergrad one

Similar to the analogy between a calculus text and a real analysis text

Eric Silva

Idk of any specifically with applications to quantum but whenever I need to find info on a particular aspect of pde the first place I usually go is Evans book or Michael Taylor's 3 volumes

I would suggest looking in either one, I know Evans includes other references at the end of every chapter but I don't remember if Taylor does that

Thank you. I very much appreciate your reccomendation

3:49 PM

What does $Im$ mean in the context of tangent spaces. $T_p S = Im dx_q$?

in this case it just means image

Ah ok, that is also what I was thinking. Thanks.

llama

Hi
can somebody plz help
https://math.stackexchange.com/questions/2560159/%D0%A1alculate-the-curvilinear-integral

4:19 PM

llama I'm showing that question to be deleted

lattice

4:39 PM

Hey chat

Universal properties always depend on the category in which they are stated, correct?

Semiclassical

5:07 PM

@DavidReed I've heard good things about Simon and Reed re: rigorous functional analysis / applications to QM

Great! Thank you, I will look it up

4^(x-1) = 1

How do i solve for x without using log or ln? Using Trial and Error?

@Geocrafter Without using log, yes you just have to know the answer. It's helpful to remember that x^0 = 1 for any real number x.

vanaghka

to show that every euclidean domain is a unique factorisation domain, can i show that every ED is a PID and therefore a UFD because every PID is a UFD

Yes, but for example this question: 7^(w-2) = 49

I could do 7^(w-2) = 7^2

w-2=2

w=4

But since it is 1, i can't make it the same base as 4, how would I do that

5:15 PM

4^(x-1) = 4^0

x-1 = 0

x = 1

this is just using logs, you're just sort of hiding that by not writing it down

Q: Let $A$ be a set of all continuous functions $f:[0,1] \to \mathbb R^+ \cup \{0\}$ that satisfies the following condition.

ah i see now! thanks

Maneesh Narayanan

0

Let $A$ be a set of all continuous functions $f:[0,1] \to \mathbb R^+ \cup \{0\}$ that satisfies the following condition: $$\int_{0}^{x}f(t)dt \ge f(x), \forall x\in [0,1].$$ Which of the following statements is true? (A) $A$ has cardinality $1$. (B) $A$ has cardinality $2...

real-analysis integration proof-verification continuity cardinals

How would you go about this one? (-1)^(2x)=1

Please don't just copy your questions to chat minutes after you have posted them on Main. This is what Main exists for.

@geocrafter 1 = (-1)^??

what can you do to -1 to get 1?

ah i got it now, gotta think about it more before posting.

5:21 PM

Let me warn you though: this one has multiple answers

mhm, it could be anything

Maneesh Narayanan

@KevinDriscoll sorry

I won't repeat again

Vrouvrou

5:41 PM

hello @AkivaWeinberger

Bonjour

Vrouvrou

please can you see my question :math.stackexchange.com/questions/2559858/…

i found in a book this answer :
for all $k\in\mathbb{N},\exists n_k\in\mathbb{N}$ such that $$s_n\geq \phi(s_k),~ n\geq n_k$$

as $\tilde{\phi}$ is non decrasing we have that $\tilde{\phi}(s_n)\geq \tilde{\phi}(\phi(s_k))$ then we obtain the result

i don't anderstand from where we have

$\forall k\in\mathbb{N},\exists n_k\in\mathbb{N}$ such that $$s_n\geq \phi(s_k),~ n\geq n_k$$

have you an idea @AkivaWeinberger ?

6:14 PM

@Akiva a few days ago you were wondering whether there exist $X\subset \Bbb R$ such that $\lambda(X\cap[a,b])=(b-a)/2$, what were the requests on $X$?

Gabriel Romon

6:27 PM

@AntonioVargas math.stackexchange.com/questions/2560334/…

Funny how adding $\log k$ into the power series coefficients results in an additional $\log x$ in the asymptotic expansion

7:19 PM

I don't understand a step in a proof I wrote down months ago, can you guys help me fgiure it out? here it is

thm, for every (let's say positive to make the proof easier), real number $r$, there exists a sequence $(a_k)$ of numbers in $[0,1,2,\ldots, b-1]$ where $r = \sum_{k=0}^\infty \frac{a_k}{b^k}$

so I inductively defined a seqeuence

hi guys didn't use lagrange multipliers for a long time...looking for a heads up..if we find critical points of the constructed lagrange function do these points already satisfy the constraints? or are they just possible candidates?

$a_{n} = \operatorname{max}_{m} (a_0 + a_1/b + a_2/b^2 + \cdots + a_{n}/b_{n} + m/b_{n+1} < x)$

and $a_0$ is the floor of $x$

and I wrote down that

$\sum_{k=0}^n a_k/b^k + 1 / b^n \ge x$

and I dont see why that is

Felix, you mean, are they automatically extrema?

not always, you have to check to see

@GFauxPas from my understanding, these points are just those where the gradients of our "actual" function aswell as those of our contraints are in parallel..at least on first look I don't see why those points already satisfy our constraints, altough agreeing that the points we are looking for must of course have parallel gradients..so my question is if it might happen that one of the points we find does not satisfy the contraints we want to impose

7:35 PM

well isn't part of solving for lagrange multipliers solving for points that satisfy the contraints? when I use them last, it was a whiel ago

we had a set of equations, as well as an equation defining the constraint

and we wanted to solve them simutaneously

the issue is not whether it will satisfy the constraints, the issue is are you going to find points that arent really extrema

Thats what I remember too, the question is then, if we found all critical points of the lagrangian function, and assume it to be easy to find compute the value of our function at that point, but hard to prove if that point rly satisfies the constraints, can we just compare the results and take the greatest/smallest being sure that it satisfies the constraints or do we have to first filter out all "bad" points and the check for the smallest/largest?
anyway reading through wikipedia now...thought this to be a everyone-except-me-knows-it question

not sure I understand

when i did it i checked all the points as well as the points where the function was not diffable, particularly if the area being considered is a polygon

so i think you have to check each one

FreeMind

Guys do you know any good online course for Ring Theory?

7:50 PM

Well let $\{x_1,..,x_n\}$ be the set all of all points such that $DL(x_i) = 0, \forall i \in \{1,..,n\}$ where $L :=f(x) - \lambda_1 g_1(x) - \ldots -\lambda_k g_k(x)$. Assume that we found all of them. Now the question is whether $g(x_i) = 0, \forall i \in \{1,..,n\}$

That's very precisely my question :)

8:06 PM

I just remarked that to find these points we use the contraints, otherwise we can't solve the lin. eq. system so obviously all these points satisfy the constraints..thx a lot for the help any @GFauxPas I hope for you that you'll find an answer soon too :)

ty i think i figured it out

8:21 PM

@Vrouvrou We know $s_n\to\infty$

@AlessandroCodenotti Any

Just that $X$ is independent of $a$ and $b$

Hm, ok

If I had to bet I'd put my money on the fact that there is such a set so let's see if I can build one

user76284

What is the significance of the eigenvalues of a second-derivative matrix?

I guess this is equivalent to asking for a function from $\Bbb R$ to $\{0,1\}$ such that $\int_a^bf(x)dx=\int_a^b\frac12dx$

Hm, can you have a function such that the integral equals zero on every interval, and such that it's not simply $0$ almost everywhere?

Ok so if such a set exists it must be a Bernstein set

which makes me already much more doubtful about the bet I was placing a moment ago

@AkivaWeinberger No. Suppose that $f$ is nonzero on $A$, with $\lambda(A)>0$, then either $A$ is bounded (contained in an interval) or you can intersect $A$ with an interval such that the intersection has positive measure and integrate on that interval

brot

What's the intuition behind defining the tangential space with derivations?

Q: Assosciativity in forming Disjoint Unions

8:33 PM

0

Let $X'$ be a set and let $\{X_{\alpha}\}_{\alpha \in A}$ be an indexed collection of sets. We can form the disjoint union space $$\bigsqcup_{\alpha \in A} X_{\alpha} = \left\{(x, \alpha) \ | \ \alpha \in A \ \text{and } \ x \in X_{\alpha}\right\}.$$ Now my question is what does $X' \sqcup \left...

general-topology elementary-set-theory

Do you want a measurable $X$ or are you working with the outer measure?

@Akiva

If I have a map $x(u,v)=(u,v,uv)$ how do I show that it is a $C^1$ map (continuously differentiable)? Do I just differentiate each the three components with respect to both $u$ and $v$ and show that the result is a continuous map?

Uhm actually I was wrong, your set isn't necessarily a Bernstein set, but it's very close and Bernstein sets are all nonmeasurable

brot

@berrygreen It follows from the fact that the components are $C^\infty$

($C^1$ would also be sufficient)

Yes I see. Thanks

Overflow2341313

8:54 PM

How would I go about using the fact that having a cut-point is a topological property to show that R,U -> R,C is not a homeomorphism where U is the usual topology on R, given by U = {U | x in U -> x in (a,b) \subseteq U} and C = {emptyset} U {(a,infinity) | a in R} U {R} is the open half-line topology.

Sets are easier to see here with a screenshot if that wasn't clear: For U: postimg.org/image/8qh2anhiz. For C: postimg.org/image/erer7qraj

@AlessandroCodenotti I don't understand

What's wrong with $A$ being bounded

nothing, but then $A\subseteq [a,b]$ and $\int\limits_a^bf \mathrm{d}\lambda>0$

@AlessandroCodenotti $f$ is not nonnegative here

What kind of metric spaces have Bolzano-Weierstrass? Any complete metric space?

Doesn't $f$ take values in $\{0,1\}$ @Akiva?

9:05 PM

Erm with an ordering I mean

43 mins ago, by Akiva Weinberger

I guess this is equivalent to asking for a function from $\Bbb R$ to $\{0,1\}$ such that $\int_a^bf(x)dx=\int_a^b\frac12dx$

42 mins ago, by Akiva Weinberger

Hm, can you have a function such that the integral equals zero on every interval, and such that it's not simply $0$ almost everywhere?

^Those are two different functions

Howdy, DogAteMy, @Alessandro, @GFauxPas.

Hello

Ah, I didn't read carefully, sorry @Akiva

Hi @Ted

Hyello

9:08 PM

@GFauxPas that's false in $L^p$ spaces so completeness is not enough

@AlessandroCodenotti Specifically, the second function would be $f(x)-\frac12$

Hmm something to think about Alessandro

Bolzano-Weierstraß never holds in infinite-dimensional normed spaces

I never even remember what B-W means.

Which one is Bolzano–Weierstrass again?

9:10 PM

Bolzano-Weierstraß is a very Euclidean thing, I think

See, DogAteMy agrees with me :P

Every bounded sequence has a convergent subsequence

Yeß I do

Is this gonna require something like, bounded implies completely bounded?

First of all, you can keep the topology the same and make every set bounded, so then you need compactness. And then it's true in any metric space that compactness implies sequential compactness. :)

(Is that the right word? Just a moment)

9:12 PM

Yeah, that's right, DogAteMy.

Actually, I got to go

There's a theorem in functional analysis that every bounded sequence has a weakly convergent subsequence

Well, then ... Bye.

(this is equivalent to Banach-Alaoglu)

But there are some hypotheses?

9:13 PM

normed vector space

But the weak topology is rarely metrizable

so this is strictly speaking not an answer to the question

If you have a complete Riemannian manifold then every bounded sequence has a convergent subsequence (as closed balls are compact by Hopf-Rinow)

Hey everyone again!

Heya @TedShifrin, @MatheinBoulomenos

Hi @Perturbative

Hoooowdy folks

Hey @Kevin

Today's a big day. I'm going to try and compute De Rham cohomologies for the complex projective space

9:18 PM

@TedShifrin Quick question, do you know the quickest way to learn Riemannian metrics and Pseudo-Riemannian metrics

hi Perturbative

The answer to that question is going to depend on background, but Ted did write a whole book on this

@Kevin: Using what as tools?

so weakly convergent subsequences in infinite dimensional normed space and complete Riemannian manifolds are the only examples I can think of for generalizations of Bolzano-Weierstraß (though the first one is not really convergence in the metric) @GFauxPas

No, Perturbative, Ted didn't. Wrong level.

9:19 PM

Thanks

Perturbative: You're in way too much of a hurry.

My bad, shouldn't answer questions not directed at me

@TedShifrin I wouldn't usually rush like this, but I need to understand Riemannian metrics and Pseudo-Riemannian metrics for a research group that I'm about to enter

Kevin: You obviously haven't learned sheaf cohomology. You probably haven't learned invariant differential forms and how to compute cohomology of symmetric spaces using them. And do you know about the Kähler form on $\Bbb CP^n$?

@Perturbative: What does "understand" mean? Know the definitions? Know a whole Riemannian or pseudo-Riemannian geometry course? I mean, that's what those courses are all about —

@TedShifrin Enough to understand examples like the one above

9:23 PM

Again, what does "understand" mean?

@TedShifrin I have no idea what tools I'm going to need yet. I'm just going to try and figure it out. So far I've done $S^n$, $T^n$, $\mathbb{RP}^2$, and the mobius strip. We talked about $\mathbb{CP}^n$ a good bit during the course and I don't want to get caught flat-footed on the exam regarding cohomology classes there

And you're right, I don't know about any of those things

@Kevin: This is not a reasonable exercise for you, other than $\Bbb CP^1$.

@Ted Okay, thanks for the advice. I'll try $\mathbb{C}P^1$ then and then move on to something else

Understanding my exam question about degrees $S^n\to X\times Y$ in both ways seems a much more useful mission.

I'll move it to the top of my list then. We just didnt have any homework problems on using M-V and I didn't really understand it before yesterday, so I didn't want to get caught out

9:28 PM

@TedShifrin Just to know the definitions, and be able to work with flat metrics, induced metrics, and understand how transformations between co-ordinate systems affect metrics, and I need to know how to me able to work with geodesics

Unless you expect John to be brutally unfair on his exams, concentrate on what he's prepared you to do.

@KevinDriscoll did you do connected sums?

You need to start by learning how to pull back differential forms and tensors, Perturbative.

@MatheinBoulomenos No, we didn't cover that

@KevinDriscoll okay. These would provide examples for MV, that's why I asked

9:33 PM

Hello everyone!

Hi Demonark

Hi @Daminark

Hi @Dam

How's it going?

@Ted Of course. I felt like applying our cohomology tools to something like $\mathbb{C}P^n$ might be reasonable given that ~10% of homework questions tht we turned in were about that space, but clearly that's just my ignorance talking. I am expecting the exam to be quite difficult though. He said it would be "comprehensive level": 8 questions, choose 5, 3perfect answers + $\epsilon$ will be a high A.

Certainly not unfair though

9:37 PM

@TedShifrin Hmmm I'm working off Lee's book and that will take me a good few months to learn properly (and even in that case I'll be learning it very sketchily), the problem is that I only have a couple months to learn that stuff and my background in DG is almost non-existent

Yo! @Daminark

I think I may have to opt out of the research group

Hey @Perturbative, what's up? And :(

@Perturbative DId the person leading the research group talk to you about your bachground before you started?

@Perturbative I think Lee is not the best option if your time is limited. He's very thorough with the basics which means that it will take long to get to stuff like Riemannian metrics. He's only doing proper differential geoemtry in the third volume of his trilogy and spends the first volume on point-set topology and topological manifolds and the second on differential topology

@Daminark Currently contemplating whether to fight my way through Lee or just drop the research group and focus all my time on Algebraic Topology and Differential Topology

@KevinDriscoll Well the research group would be a PhD student, a professor and me, all working on this paper which is geared at analyzing singularities on pseudo-riemannian manifolds, the first half of which deals with a topological construction on smooth manifolds, the second half specializes to pseudo-riemannian manifolds and the consequences of the topological construction on it

I can understand the first half of the paper just fine, (anyone who went through a few chapters in Munkres' point set book can), like the topological stuff is really simple, the DG stuff should be simple too, but because I have no background on DG it's unreadable to me atm

So the PhD student knows my background in topology (which is my strength) and he knows that I have minimal background in DG, (I was hoping to fill that gap in my holiday period), but now that doesn't seem realistic

The rest of my group would be looking to publish a paper (although it would be published in a Applied Maths/Physics journal given the contents of it is mostly geared towards Pseudo-Riemannian stuff) sometime in June next year, and best case scenario I make some contribution to it and get my name as co-author

9:55 PM

You should look at a more efficient book like Boothby, @Perturbative. But you definitely need to learn how to play with tensors and pullbacks. You aren't going to learn a year's long graduate course in a few weeks of self-study.

@TedShifrin did you know that it is impossible to dissect a square into an odd number of triangles of equal area?

@TedShifrin Okay I'll check it out

Also @MatheinBoulomenos thanks for the advice, I was looking at this third volume the past few days to try learn some Riemannian stuff super quick

@Vrouvrou I think you need M instead of varepsilon. Also you need to use whatever the closed form of un is. Either way I don't know enough real analysis to be sure

@Perturbative his third volume assumes that you have read his second volume which does all the bundles & tensor stuff which is necessary

Eric Silva

Idk if you can learn the tensor game for Riemannian geo super quick

You kinda have to spend a lot of time getting it into your bones

LastIronStar

10:03 PM

Could someone recommend a text and/or Lecture Notes on Category Theory that approaches from a discrete math perspective?

I'm specifically looking for ones that draw examples from discrete mathematics. I looked into some of the books that are written from Computer Science perspective and most of the examples there are drawn from Programming theory, etc,. which is of little interest to me.

@MatheinBoulomenos Oh well there goes my plan lol

10:26 PM

@Perturbatie sorry I'm a bit late

But yeah so I can definitely see why it feels like you're missing a great opportunity

The best thing to keep in mind is that doing this stuff too quickly to process right could make the experience with the research group a disaster if that lack of background rears its head

Like, to the point where it will have been better to just not be involved in the research group than to undergo whatever might happen

Not to say it will or something, but it's a risk you'd be taking, so assess that risk

Knowing that learning DG is a monumental task, decide what's worth it. And my guess (as is Ted's) is to let this one slide

Using Gauss Theorem I want to calculate $\iint_{\Sigma}(x^2 + y + z) dA$, where $\Sigma$ is the unit sphere with center at $(0,0,0)$.

According to Gauss Theorem we have that $$\iint_{\Sigma}F\cdot NdA=\iiint_{\Omega}\nabla\cdot FdV$$ So, do we have to calculate $N$ and then to find $F$ such that $F\cdot N=x^2+y+z$ ?

10:53 PM

@Daminark Thanks for the advice, I'm gonna let it slide

Hopefully other opportunities will come your way

(Ideally on the algebra/NT side :P )

11:16 PM

Hello @KevinDriscoll !! Do you maybe have an idea about my question above?

@MaryStar I think what you wrote is correct. Though its unusual in my experience. Usually when I want to sue Gauss theorem, I know $F$ and not $F \cdot N$, but yes I think you're right.

the normal field to the unit sphere has a really simple form

it's almost the "identity"

Hmm, not sure what it is

if you have a point on $\Sigma$ you can interpret it as a vector

that's $N$

@MaryStar Yes it does, its just $\{x, y, z \}$

since $x^2 +y^2 +z^2 =1$

11:24 PM

Oh yeah okay that makes sense, the tangent space to the sphere is orthogonal to it

So, we have that $N=(x,y,z)$. Suppose that $F=(F_1, F_2, F_3)$ then $F\cdot N=(F_1, F_2, F_3)\cdot (x,y,z)=F_1\cdot x+F_2\cdot y+F_3\cdot z$. This must be equal to $x^2 + y + z$. Therefore we have that $F_1=x, F_2=1, F_3=1$, i.e. $F=(x, 1, 1)$.
Is this correct? @KevinDriscoll

yes it's correct

We have that $\nabla\cdot F=1+0+0=1$.

Therefore we get $\iiint_{\Omega}1dV=\text{Volume of unit sphere}=\frac{4\pi}{3}$.

Is this correct? @MatheinBoulomenos

seems correct to me

Thank you!! :-)

Anubhav Mukherjee

11:54 PM

@MikeMiller hey, how are you?

Mike Miller

doing ok