Mathematics - 2015-12-22 (page 3 of 3)

Mike Miller

19:00

'orning.

DanielSank

@DanielFischer Indeed. It does work though, which is sort of the point of the limit procedure :D

Daniel Fischer

@DanielSank Yes, distributions tend to work as they should, they are nice in that way.

Morning @Mike.

Danu

Hey @Mike

Huy

@Mike: Do you know how to prove that a hyperbolic metric with totally geodesic boundary on $\Sigma_{0,3}$ is uniquely determined by the boundary components?

Mike Miller

Don't think so, no.

Huy

19:04

my prof gave it as an "Exercise: (hard)"

=(

Ted Shifrin

g'night @Mike @Huy

Mike Miller

Maybe knowing that each of the boundary curves is a geodesic determines the exponential map in a small nbhd of each boundary point? And then you bootstrap from there?

Huy

morning @Ted

Daniel Fischer

Evening @Ted.

Ted Shifrin

evening, @DanielF

Frank Science

19:05

Pretty frustrated.

Mike Miller

Yeah, I think it's something like that. Knowing that the boundary is geodesic plus the curvature should tell you a perpendicular geodesic, at least locally.

Danu

Evening? Morning? What is this?! I thought you guys were all US.

Huy

not me.

DanielSank

@Danu Scotland, presently.

Come up and have a beer with me.

Ted Shifrin

@Frank's in France, @DanielF is in Germany, ... @Danu, you just want to be special? :P

Huy

19:06

I've been to Glasgow once. Not a very beautiful city, in my opinion.

Danu

@TedShifrin Ya.

@DanielSank Cool, that ain't so far.

How about you come over to Amsterdam?

I'll be back home in the afternoon :3

DanielSank

@Danu Can't. Busy.

Also no transportation.

Frank Science

I'll go to Germany for Christmas.

Usually mathematics is frustrating.

Mike Miller

I agree with that

Frank Science

Especially when it's claimed easy on the book but I don't have any idea on that.

dorothy

19:14

@Clarinetist ok so the tricky question seems to be, can we actually prove convergence to multivariate Gaussian? It seems to depend on the ratio of the number of columns to rows in a way I don't understand.

Clarinetist

@dorothy I'm not sure how this ratio comes into play, unfortunately. Wish I knew this material better, ugh

Kinson Chan

using double angle formula show that sin^2(x) = 0.5 - 0.5 * cos2(x) are equal

help pls ^

Frank Science

What's the double angle formula?

Kinson Chan

cos2x = cos^2(x) - sin^2(x) , 1 - 2sin^2(x) , 2cos^2(x) - 1

sin2x = 2sinxcosx

Frank Science

Okay, so did you try to plug these into the right hand side?

Kinson Chan

19:22

yes

Frank Science

What did you get?

Kinson Chan

i just wont get seomthing like sin^2(x) = sin^2(x)

the things i get just goes back to the original equation

Frank Science

Plug $\cos2x=2\cos^2x-1$ into $(1-\cos2x)/2$, what did you get?

Danu

@KinsonChan Don't just come here to make people solve your homework questions.

Kinson Chan

@Danu im revising for a test and im only asking ones that im stuck on

Frank Science

19:26

The problem is that what we think easy may not be easy for those beginners.

Kinson Chan

^^^

Frank Science

So maybe he worked hard but still didn't get what's involved.

Danu

Maybe. I don't see a lot of attempts, though.

Frank Science

So I didn't give the complete solution, but indications to see what he really did.

*he/she

Kinson Chan

1/2 + sin^2x -1

ive tried this one

wait nvm

user153330

19:30

@Anubhav.K hi hi

Danu

One easy way to show all trig identities: Use exponentials.

3

user153330

@Danu +1

Kinson Chan

@FrankScience ive gotten sin^2(x) = 0.5 + sin^2(x)

idonutunderstand

@Danu how does that work?

user153330

@KinsonChan that's not true since that would be equivalent to 0 = 0.5

Frank Science

19:37

@KinsonChan Write out the procedure step by step, please.

user153330

please, please, write the procedures step by step please

Kinson Chan

sin^2(x) = 0.5 - (1-cos2(x)/2

user153330

@idonutunderstand using exponential definition of cosine, sine, tangent, like you know cos x = 1/2(e^(x)-e^(-x))

Kinson Chan

sin^2(x) = 0.5 - (1+1-2cos^2(x))/2

sin^2(x) = 0.5 - (2-2cos^2(x))/2

idonutunderstand

@user153330 nice!

Kinson Chan

19:39

sin^2(x) = 0.5 - (1-cos^2(x))

Mike Miller

@user153330: As written that is not cos on the RHS.

Kinson Chan

sin^2(x) = 0.5 - 1 - cos^2(x)

sin^2(x) = 0.5 - sin^2(x)

@FrankScience ^

user153330

@MikeMiller oh my god, yes you are right, cos x = 1/2(e^(ix)+e^(-ix))

Kinson Chan

what?

user153330

@MikeMiller that was the definition of sinh(x)

Mike Miller

19:42

I think what you had is still cosh.

Frank Science

@KinsonChan Check this.

Kinson Chan

my mathjax isnt working rip

user153330

@KinsonChan sniff :'(

Kinson Chan

i know cosx = 2cos^2(x) - 1

but her it is -cosx

Frank Science

So?

Kinson Chan

19:44

so shouldnt -cosx = 1 - 2cos^2(x) ?

Frank Science

LHS should be $-\cos2x$.

user153330

@KinsonChan try x=pi/2

Mike Miller

@Danu: So what are you up to today?

Danu

@MikeMiller Just finished writing up my topology notes.

Tonight I'll pack my bags and get ready to go to Amsterdam tomorrow :)

Kinson Chan

@FrankScience it is same as cosx

Danu

19:47

I'm also reading the CFT textbook.

Frank Science

@KinsonChan No, and please learn how to write math in TeX.

Danu

Furthermore, I'm spending a lot of time debating about what to do for my (MSc.) thesis project

Frank Science

CFT?

Danu

Conformal Field Theory (physics)

dorothy

@Clarinetist hi

@Clarinetist are you about?

Danu

19:48

With the amount of effort I'm putting into typing up these (algebraic) topology notes, I am thinking about what I can do with them once I'm done; It'll be a ~200-250 page manuscript with very high quality pictures etc.

dorothy

@Clarinetist the two issues are a) the correction factor because differential entropy is not scale invariant and b) does it actually converge?

b) is more interesting :)

Frank Science

Ah, physics is all Greek to me. Even quantum mechanics is difficult to me.

user276387

A simple question I've on this paper: ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689471-1/… -- what are the primed x's in the second page, and why is their sum equal to 1?

Danu

@FrankScience "even" haha

Frank Science

What's wrong?

Danu

19:50

QM is not the simplest thing.

dorothy

@Clarinetist I think that the literature on multivariate CLT should tell us but I don't understand it properly sadly

Agawa001

@FrankScience can u remind me how s a "developpement limité" translated in english ?

limited development ?

Frank Science

Taylor expansion?

Mike Miller

@Danu: There's only one CFT book?

user153330

@Agawa001 developpement limité as in finite expansion

Carl Mummert

19:51

@user276387 : aren't they dx/du

Agawa001

@FrankScience yes thank u

user153330

@Agawa001 no thank u for me?

Agawa001

@user153330 thk u , thats also said

Danu

@MikeMiller No, but there aren't many.

In my case, it's Blumenhagen's book (he's doing the lecture I'm attending, yay!)

Frank Science

Bell's inequality, etc.

I'm confused

Danu

19:53

and Bell's inequality is not basic QM either.

Frank Science

What do you mean by basic QM? Fundamental postulates of QM?

user276387

@CarlMummert thanks for the reply. How come that's dx/du?

Danu

No, solving Schroedinger's equation for a simple potential, for instance.

Frank Science

It's not how QM is taught here.

Carl Mummert

Well, the prime usually means a derivative, and dx / du is what you need if you want to do a change of variables from x to u in the integral, and it gives the correct equation (equal to 1) based on the previous equation in the paper @user276387

Frank Science

19:55

All reference books here start with fundamental postulates.

Danu

That's one approach.

Frank Science

Such as Tannoudji, or Le Bellac.

user153330

@FrankScience or Zettili, or Griffiths

Danu

Griffiths does not start with the fundamental postulates.

user153330

@Danu yeah but he just gives you like a very small discussion and then immediately provide you with schrodinger equation, i don't remember when he talks about the fundamental postulates

but because that's what i hate about those textbooks, they provide those postulates without giving any motivation

Danu

19:58

Sakurai is a classic (and does provide motivation).

user153330

@FrankScience do you know a good intuitive (ie gives motivation) introductory QM textbook with full historical context?

Frank Science

I am math-oriented and I come directly into formalisms.

user153330

@FrankScience yeah but even with formalisms one needs a bit of intuition and motivation, coming up with it on ones own costs tons of time and most of the time your intuition is flawed

Frank Science

What I was confused about is the composite system.

user276387

Thanks @CarlMummert If I may trouble you more, a bit above that when they say "equation (3) reduces to an nth degree equation for x of the form..." how did they reduce it?

Frank Science

20:01

For example, if $H_1$ is the state space of one system and $H_2$ is that of the other.

Danu

@user153330 Like I said, Sakurai does this stuff.

user153330

@Danu okay thanks i'll look into it

Frank Science

$H_1\otimes H_2$ gives the state space of the composite system.

Carl Mummert

I believe it comes from combining the fractions to get a common denominator, then multiplying that by both sides, then rearranging the resulting polynomial equation to put 0 on one side. Try it out in a specific case (e.g. n = 3 or 4 and make up values for the constants) and see what happens. @user276387

Danu

@FrankScience so...

AngusTheMan

20:19

Hey everyone, what is the correct way to take the exterior derivative of the product of two functions? i.e $d(fg)=?$ Here is my attempt, but I am left with a result that doesn't look too good imo... $d(fg)=d(f\wedge g)=dg\wedge f+g\wedge df $

user276387

@CarlMummert Wonderful, thanks again.

Daniel Fischer

20:34

@AngusTheMan Since functions are $0$-forms, hence of even degree, your result turns out to be correct. Generally, one has $d(\alpha \wedge \beta) = (d\alpha) \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge (d\beta)$. So $d(fg) = d(f \wedge g) = (df)\wedge g + (-1)^0 f\wedge dg$. But since $f,g$ have degree $0$, that is equal to $dg \wedge f + g \wedge df$.

Mike Miller

(This is usually taught in first-year calculus courses with the name "the product rule".)

Huy

ö_ö

Mike Miller

Don't give me that look. :(

Huy

.___.'

AngusTheMan

@DanielFischer Thank you for the reassurance! Sorry if it is obvious, I am a chemist after all :p

OFFSHARING

20:44

$$\huge{\text{What an amazing result I got!!!}}$$

Where is @robjohn?

Mary Star

Could someone of you take a look at my question:

0

Q: Reparametrization - First fundamental form

Suppose that a surface patch $\tilde{\sigma}(\tilde{u}, \tilde{v})$ is a reparametrization of a surface patch $\sigma (u, v)$, and let $\tilde{E}d\tilde{u}^2+2\tilde{F}d\tilde{u}d\tilde{v}+\tilde{G}d\tilde{v}^2$ and $Edu^2 + 2F dudv + Gdv^2$ be their first fundamental forms. I want to show that...

?

Clarinetist

@dorothy

Semiclassical

@offsharing what kind've result?

OFFSHARING

@Semiclassical A series involving the reciprocal of Fibonacci numbers. Most probably unknown to this date.

Semiclassical

well, here's hoping :)

Forever Mozart

20:56

look my hat

Agawa001

had same panador hat, looks awsome on ma mountain

user276387

Me and this paper again > ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689471-1/… -- at the very end of page one, in the sum can we have Cj = aj?

Clarinetist

21:25

So I'm a complete noob to Topology. Define the ball of radius $\delta$ centered at $c \in \mathbb{R}$ by $$B(a, \delta) = \{x \in \mathbb{R} : |x-a| < \delta\}$$

A set $U \subset \mathbb{R}$ is open if for all $c \in U$ there is a $\delta > 0$ such that $B(a, \delta) \subset U$.

I would like to show that an open interval $(a, b)$ is open. I understand what I'm reading, but I'm wondering (like with $\delta$-$\epsilon$ proofs) if there's a brute-force way to find $\delta$.

Carl Mummert

@user276387: according the the text at the top of the second page, yes

@Clarinetist: for a given point x, the delta to take is the distance from x to the closer end of the open interval

Clarinetist

(sorry, the definition above is wrong. It should be the ball centered at $a$)

@Carl Is that always going to be the case?

I mean, with an open rectangle in $\mathbb{R}^n$, sure, that's easy

But I'm wondering if there's a brute-force way to find a $\delta$ that will satisfy the above

Carl Mummert

what do you mean by "always". In the case of an open interval, certainly. In general, the largest delta you can take, given x, is the distance from x to the complement of the open set (which will be positive if the set is open - basically, you are trying to show it is positive if you are trying to show that the set is open)

Mike Miller

@Clarinetist: By definition, yes. Set delta to be the infimum of the distances to points outside the set. Then by definition, points within delta of x are inside your open. The question is whether this is not zero.

Clarinetist

Thanks @Carl, @Mike

Carl Mummert

21:31

In the case of an open interval, the distance is the minimum of two positive numbers, which will always be positive. One number for each end.

user276387

@CarlMummert thanks.

dorothy

21:45

@Clarinetist Thank you. where is that from?

@Clarinetist I plotted a graph which might show some of the problem. imgur.com/PdYUzb4 is a graph of 2^17 6 by 12 matrices whose entries are 0 or 1. The x-axis is the entropy with random $\pm 1$ vector $x$ and the $y$ axis is the log of the determinant. Sorry about mixing M and B as the matrix names

Jake1234

Guys, consider an open square say $A= (-1,1) \times (-1,1)$, and $A \setminus \{(0,0)\}$ - I'd guess they are not homeomorphic, but how should I show this? Didn't know how to search for this, so a hint on what to search for would be good too.

Danu

Fundamental group.

@TedShifrin Now that you're freshly retired, how do you spend most of your time? :)

Jake1234

Thanks Danu, haven't learned that in school yet, I'll look into it.

Danu

You're in high school? Or do you mean university?

Mike Miller

@Jake: I don't know any other effectively different approach here unless you know some machinery. What have you guys learned?

Carl Mummert

21:54

For this one, can't you show that the homeomorphism would have to swap 0 and \infty, so to speak, and go about it that way?

Mike Miller

Yes, I suppose one could show that one has one and and the other has two ends. But this seems like the kind of idea that wouldn't be covered in a first topology course.

Carl Mummert

I don't think any proof of that question is included in a first point-set course :) Definitely not mine.

Mike Miller

My first had the fundamental group in it... but I spose whether or not that's true is the taste of the instructor.

Danu

Actually, I think you can do this:

A homeo will induce a homeo between space- points

finitely many points

and somehow reductio ad absurdum should work

or maybe there is nothing more absurd one can do :P

Mike Miller

I can't take what you said and make an argument out of it.

Daniel Fischer

22:05

On $A\setminus \{(0,0)\}$, there is a harmonic function without a harmonic conjugate ;)

Mike Miller

Now prove that's a topological invariant, @DanielF :)

Daniel Fischer

@MikeMiller Well, we have the Riemann mapping theorem after all. Another approach: $A\setminus \{(0,0)\}$ has a two-sheeted (unbranched unlimited) covering.

Danu

hehe

I like this... I wanted to study some complex analysis next semester break.

Mike Miller

@DanielF: Ok, now let's prove that doesn't happen for the plane...

I guess that's simple enough.

Daniel Fischer

@MikeMiller Once you know your fundamental groups, that one isn't so hard any more ;)

Balarka Sen

22:19

Grumble, had to do a buckload of work today.

Jake1234

@MikeMiller Not that much, first course in point set topology - basic definitions, projectie inductive generated spaces, embedding lemma, separation axioms with urysohn lemma,tietze ext theorem, stone weierstrass theorem, tychonoff theorem, compactifications, uniform spaces.

Balarka Sen

@Danu The problem with using exponentials to show the angle summation identity for trig functions, say, is that this might or mightn't be circular depending on your definition of trig functions. To derive the Euler's formula, you need to compute the derivative of $\sin(x)$ and (depending on defn) you'd need angle sum formula to do that.

Danu

@BalarkaSen Circularity? Ain't no physicist worried about that ;D

Balarka Sen

Fair enough :)

Danu

I'd never even seen a proof until about half a year ago.

Balarka Sen

22:23

Proof of what?

Danu

Euler's formula

user159870

How can we calculate the integral $\iint_{\{u^2+v^2<1\}}\sqrt{1+4(u^2+v^2)}dudv$ ?

Mike Miller

Anything

Danu

^

Balarka Sen

:P

Danu

22:25

I didn't know what a topology was about 15 months ago.

Balarka Sen

Me neither.

Mike Miller

I still don't know what a topology is.

Balarka Sen

In what sense?

Carl Mummert

Maybe that is how your first course got to fundamental groups so quickly @Mike :)

Danu

hehe

Mike Miller

22:31

@Balarka: The definition, of course

Danu

BS.

Carl Mummert

I would guess it would be completely possible to do a lot of topology without knowing the definition of a topology, if everything you study is a subset of R^n for some small n

Mike Miller

Who needs anything more than metric spaces? I haven't.

Balarka Sen

One can just study metric spaces.

Mike beat me to it.

Mike Miller

Of course, to take quotients under group actions, one needs a G-invariant metric. But any student of topology knows that if G is compact one can do this and get a metric space homeomorphic to the one you started with.

I guess I lose now, because I have wanted at times to quotient by the action of a noncompact group

Balarka Sen

22:40

While I agree that every reasonable space is metrizable, I am not sure if one can start off with a metric without much trouble. The one you mentioned about coset spaces is one (the solution to which you have also mentioned). I haven't thought about it, but I'll believe you.

Mike Miller

There's also the implicit assumption that I only care about reasonable spaces.

Carl Mummert

from a certain definition of "reasonable". I have my own definition, of course...

Balarka Sen

Mostly, me too.

@CarlMummert The reasonable spaces, by definition, are precisely... the metrizable spaces :D

Mike Miller

@Carl: "A reasonable space is a compact, connected, non-Hausdorff space..."

OFFSHARING

@user159870 Shoot it down by polar coordinates.

Carl Mummert

22:43

I have heard there are plenty of uses for Zariski topologies, although that isn't my area. I have studied completeness properties of spaces to some extent, and we have other spaces there that are of interest in that program

user159870

You mean to set $u=r \sin\theta$, $v=r \cos\theta$ ? @OFFSHARING

Balarka Sen

Zariski topology on varieties aren't quite used to study topology, though. They are there for the sheaf theory.

(although I don't know much about algebraic geometry)

OFFSHARING

@user159870 I only gave you a hint, not telling more.

user159870

Ok. Thanks for the hint! @OFFSHARING

Balarka Sen

I wonder if I have ever seriously thought about spaces which do not have any CW-structure.

Mike Miller

22:46

@Balarka: You claim to like topological manifolds.

OFFSHARING

@user159870 Welcome! ;)

Mike Miller

Non-metric topologies are unavoidable if one wants to do analysis, where occasionally you have mapping spaces that want the point wise convergence topology.

Balarka Sen

@MikeMiller I haven't seriously thought about topological manifolds, but I am very excited to know there are topological manifolds with no CW structure. Is there an easy example?

Mike Miller

@Balarka: No; but rather, it's open that they all do have a CW structure. I would be quite unsurprised if one failed to.

Carl Mummert

In analysis, even "topologies" are not general enough, so no surprise that metric topologies are not enough for the modes of convergence that are topological

Balarka Sen

22:50

This is fun. Thanks. Is it known if every topological manifold is homotopy equivalent to a CW-complex?

Mike Miller

Yes.

@Carl: You're thinking of uniform structures?

Balarka Sen

Oh, fair point, I have given $Map(X, Y)$ some thought. They are certainly not CW-complexes.

Carl Mummert

convergence a.e. is not characterized by any topology

OFFSHARING

National Mathematics Day

In India, the day December 22 has been declared as the National Mathematics Day. The declaration was made by Prime Minister of India, during the inaugural ceremony of the birth of Shaswat Mishra held at the Odisha University Centenary Auditorium on 26 February 2012. Dr Manmohan Singh also announced that the year 2012 CE would be celebrated as the National Mathematics Year. The Indian mathematical genius Srinivasa Ramanujan was born on 22 December 1887 and died on 26 April 1920. It was in recognition of his contribution to mathematics the Government of India decided to celebrate Ramanujan's birthday...

2

Did you celebrate the day? Never forget the famous SRINIVASA RAMANUJAN!!!

Balarka Sen

@MikeMiller Can you refer me to a proof?

Mike Miller

22:54

@Balarka: Why are they not CW complexes?

Balarka Sen

No, the proof that topological manifolds are htpy eq to CW-complexes. I haven't seen the proof, but I think it should be significantly less harder.

OFFSHARING

PERFECTION IN MATHEMATICS ^^^

Mike Miller

No, I'm asking you why Map(X,Y) is not a CW complex.

Anyway, see the appendix of Hatcher.

OFFSHARING

Anyway.

Mike Miller

@Carl: Hm, fair enough.

Balarka Sen

22:59

@MikeMiller Hmm, I probably can prove that for some choice of X, Y, Map(X, Y) never admits a CW-structure with countable # of cells in each dimension by looking at fundamental group, but not sure about uncountable cell structures.

I don't know.

Mike Miller

What are X and Y?

Clarinetist

@dorothy See here, p.6

Balarka Sen

CW-complexes.

Mike Miller

@Balarka: Then that has the homotopy type of a CW complex.

Balarka Sen

@MikeMiller That is helpful, thanks. The relevant proposition seems to tell me that every compact manifold has the homotopy type of a CW complex, though.

Mike Miller

23:08

That was exactly what you asked for?

Oh, you want noncompact too.

Should work by a similar argument if you have a proper emvedding into Euclidean space.

Balarka Sen

Ok, I'd have to look at the argument carefully.

@MikeMiller Yikes.

Mike Miller

??

Balarka Sen

If it has the homotopy type of a CW-complex, I can't expect $\pi_1$ to capture it, can I?

Mike Miller

Huh? What does that mean?

Balarka Sen

I want to prove Map(X, Y) is not a CW-complex. I can't expect to prove this with $\pi_1$ if it's homotopy equivalent to a CW-complex.

So my idea can't really work.

Mike Miller

23:15

You can't have ever expected to prove that with \pi_1 because you can find a CW complex with whatever \pi_1 you like.

Balarka Sen

Oops. I was thinking of CW-structure with countable many cells in each dimension.

I should go to sleep.

Mike Miller

And you can disprove that with \pi_0. Take X to be the integers and Y to be two points.

Balarka Sen

But \pi_0 of that is \{+1, -1\}?

I think it just means it does not admit connected CW-structure. But then I am sleepy.

DanielSank

@Danu Haha, nice. That was the first thing I suggested and someone shot me down. Then you suggest it and it's starred. It's almost like different people have different opinions on the internet.

Balarka Sen

@MikeMiller Yikes, misread. Thought X was the reals. I follow your example, good one. Thanks.

Mike Miller

23:32

If both X and Y are finite CW complexes this has the homotopy type of a countable complex.

Balarka Sen

Fun.

Semiclassical

oh, to completely toot my own horn

i think i mentioned my collaborators and i had a paper accepted for publication

and i only just realized that it shows up online as an accepted paper now :)

journals.aps.org/prl/accepted/…

Mike Miller

Grats!

TomSIR

hi

Carl Mummert

Your first paper? @Semiclassical

Ted Shifrin

23:47

Oh, that was a quick acceptance, @Semiclassic. Way to go!

Mike Miller

Hi @Ted.

user159870

Hello @TedShifrin

I look again at an exercise in differential geometry and need some help.

The exercise is:
Show that the Weingarten map changes sign when the orientation of the surface changes.

The Weingarten map is defined by $W_{p,S}=-D_pG$, where $D_pG$ is the derivative of the Gauss map, the rate at which the unit normal $N$ varies across $S$.

So does the sign changes because of the direction of $N$?

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$Mathematics$ Mathematics