Mathematics - 2017-03-30 (page 2 of 2)

BAYMAX

17:00

hey @TedShifrin actually i am not overriding your suggestion but can you tell me some book based on a question?..this may be cheating but ..

Danu

@TedShifrin But if you reverse the SES I just had to get $0 \to \Bbb Z_2 \to X \to \Bbb Z \to 0$, then it's $\Bbb Z\oplus \Bbb Z_2$ because you can split it (it ends in a free thing) right?

PVAL-inactive

@Mike For the paper I am talking about, one of your colleagues told me she had to read it and complained that there wasn't enough details in it.

BAYMAX

that is if I give the question here.

?

Ted Shifrin

Well, yeah, @Danu, but what right do you have to reverse it?

Danu

Oh, nothing. The reversed one also occurs in the LES I'm looking at ;-)

Ted Shifrin

17:02

Oh.

Mike Miller

@Danu What are we working on?

Danu

Cohomology of an $S^2$-bundle over $G_2/SO(4)$

Ted Shifrin

@BAYMAX: I may or may not have a suggestion, but is there a link to the question?

Danu

(assuming the result for $G_2/SO(4)$)

Mike Miller

@PVAL I can't guess who. Not many female colleagues at UCLA. Maybe KH.

So you're running Gysin?

Danu

17:03

Exactly

PVAL-inactive

@MikeMiller KH's student.

Danu

And I get something uniquely determined by the Euler class except for that one thing I just asked Ted about

the $0\to \Bbb Z \to X \to \Bbb Z_2 \to 0$

Ted Shifrin

@Danu: In general, there is a group called Ext which tells you all the things that $X$ can be.

Yea, homological algebra.

Mike Miller

an undergrad?

Danu

@TedShifrin I have used Ext before (in universal coefficients). How can I use it here?

Ted Shifrin

17:05

@Danu: See here.

BAYMAX

@TedShifrin , yeah please have a look at this - actually I am practicing concepts by seeing question -

1

Q: A question regarding existence and uniqueness in IVP

Consider the IVP $$y'(t)=f(y(t)), \ \ \ \ y(0)=a \in \mathbb{R}$$ $$f : \mathbb{R} \rightarrow \mathbb{R}$$ Which of the following is/are true $(A)$ There exists a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ and $a \in \mathbb{R}$ such that the above problem doe...

differential-equations

PVAL-inactive

@MikeMiller KH's student graduating this year.

Mike Miller

Hahaha, I've been thinking of a different KH, postdoc here now at MSU

I understand now

Ted Shifrin

@BAYMAX, it seems to me this requires a good understanding of (a) the theorems, and knowing their hypotheses carefully and (b) standard examples/counterexamples.

@PVAL @MikeM: frustrating when initials aren't an injection ...

PVAL-inactive

I only know one person with the initials KH

and that person is a professor at UCLA

Ted Shifrin

17:08

LOL, so do I offhand

Danu

After Googling, so do I! :D

BAYMAX

ok for that as per your suggestion i will refer Smales book on Dynamical systems.@TedShifrin

Ted Shifrin

@PVAL @MikeM: In my UGA email today was an announcement for the 8th yuge every-8-years topology conference this summer.

Mike Miller

@PVAL Look at MSU faculty.

PVAL-inactive

@Ted I have already been told to go to this.

Mike Miller

17:12

Me too.

Ted Shifrin

I would concur.

If I were still there, I would volunteer to be helpful. But ...

Adeek

17:23

hi

@MikeMiller I was wondering why is the case that if we have $p : E \rightarrow B$ covering space then cardinality of $p^{-1}(x)$ is locally constant ?

Balarka Sen

Hi

@Adeek Prove it. (Recall the defn of covering space)

Danu

@TedShifrin Which one? :P

Adeek

hm

PVAL-inactive

@Adeek That's somehow the point of the sort of annoying nature of the definition of covering space.

Danu

Silly question: Why is $G_2$ called that way?

PVAL-inactive

17:28

i.e its why the definition of a covering space takes in open nbhds of points as data.

@Danu It's some perversion of the Dynkin diagram list.

Adeek

oh @PVAL-inactive

PVAL-inactive

According to wikipedia it was called E_2 originally

I don't know why it was changed from E to G

Danu

Hmm, funny

and why the 2? I remember that for $E_{6,7,8}$, the Cartan matrix has this dimension...

PVAL-inactive

The 2 is the number of nodes in the Dynkin diagram

Mary Star

@TimTheEnchanter Ok!! Thanks!! :-)

PVAL-inactive

17:40

Looking at his original paper he says G_2

mathnet.ru/links/0ad19c8f91855b537085d28e9e277f89/sm5435.pdf

page 7

Danu

Well-spotted.

user193319

Here is the question I am working on: If g is a function defined on the open interval (a,b) such that a < g(x) < x for all x in (a,b), then g is nonconstant. Would the following proof work? Suppose that g(x) = c for all x in (a,b). Then a < c < x < b for every x in (a,b), implying that c in (a,b). Hence, x= c yields a contradiction, from which it follows that f is nonconstant.

Zophikel

17:55

Does anyone know what the "corridor" of a toy contour is in the context of complex analysis

having trouble with understanding the choice of words used withen the proof of Cauchy's Integral Formula

Astyx

18:32

Hi chat

user193319

@Astyx Hello. Would you like to critique the proof I gave above!? :)

I've been trying to get someone to do it for several hours.

Astyx

It seems fine to me

user193319

@Astyx Thanks!

Astyx

My pleasure

Alessandro Codenotti

18:48

hi chat

Astyx

Hi @Alessandro

Balarka Sen

heya

Astyx

Hi @Balarka

Danu

Hi Balarka

SoumyoB

hello everyone

Balarka Sen

18:57

this is hella amazing poem

Ali Caglayan

Hi @BalarkaSen

Balarka Sen

"No poetry before ours / with our wireless imagination words / in freedom longggGGG live FUTURISM finally finally finally finally finally FINALLY / poetry being BORN"

Hi @AliCaglayan

Danu

@BalarkaSen It does not... mean anything much to me. Care to explain?

Mike Miller

I find that flavor of impressionistic writing a tad tedious

House of Leaves is popular in a similar flagor

Balarka Sen

@Danu It's from Zang Tumb Tumb, apparently one of the earliest modernist works in the 1st world war. It doesn't mean much; he's just putting nouns after one another

@MikeMiller I laughed a lot reading that

this poem, I mean, not what you mentioned

Ali Caglayan

19:08

It's ok @Mike I find you hilarious :P

Danu

@BalarkaSen You laughed? Hmm...

Ali Caglayan

So @BalarkaSen I feel like I am really missing something obvious with Ex6 in Rolfsen

I have been staring at it for a while today and looking back through but I still can't see what I need to do

Cows

What is the monster group about?

Danu

String theory

Cows

What is it symmetries of?

Balarka Sen

19:12

@AliCaglayan Have you thought about $A \cup B$?

Ali Caglayan

@BalarkaSen I thought about it but what happens at $h_1$

Danu

@Cows A string theory compactified on the $E_8$ lattice*, if I recall correctly.

Ali Caglayan

then how is it a knot?

because its basically a homeo $B\to B$

Forever Mozart

its not a knot youtube.com/watch?v=SFdq-8wwNeM

Balarka Sen

What's $h_1$?

Ali Caglayan

19:13

Thats the ambient isotopy

Forever Mozart

oops i meant this one youtube.com/watch?v=xm7jp5A9Vks

Ali Caglayan

@BalarkaSen lets move to the other room

Balarka Sen

OK

Cows

@Danu so I get a bit confused about what a lattice in math is. Just so I am painting the right picture, can you tell me what a lattice is,

Danu

Lattice (discrete subgroup)

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth...

Cows

19:15

so quotient of my favorite manifold?

Danu

@ForeverMozart lel

@Cows No, a discrete subgroup with some properties of a bigger group

Cows

@Danu ok i see, It must be discrete

Balarka Sen

@Danu I did read some impressionistic stuff which actually felt serious and I was actually moved (disturbed/depressed) by it. But while I understand this sound poem is a textural account of Battle of Adrianople, I can't really take it seriously.

Danu

@Cows No, not just that. It must be a subgroup of a bigger group. Manifolds are not typically groups.

Cows

@Danu so like wikipedia says "In the study of discrete subgroups of Lie groups, the quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.

For example, modular curves are compactified by the addition of single points for each cusp, making them Riemann surfaces (and so, since they are compact, algebraic curves). Here the cusps are there for a good reason: the curves parametrize a space of lattices, and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account som…

Danu

19:23

The compactifications in those paragraphs are not the same thing as in string theory

Cows

@Danu so just so I am painting the right picture here, you have a lie group. Then you find the discrete subgroups of the lie group. This forms a lattice

@Danu oh and properties carry over

Danu

In string theory, you just work with $\Bbb R^n$, and lattices in it.

Cows

@Danu ok nice

@Danu so i am a bit confused about the term compactification

@Danu generally does this mean encoding all the stuff in a d dim, a few dims lower?

Danu

In string theory?

Cows

@Danu well in string theory, then generally

Danu

19:30

In the standard/simplest form, it just means that you assume the spacetime your theory is formulated on is of the form $\Bbb R^{4}\times M^{d-4}$ where $d$ is the number of dimensions of your spacetime (26 for the bosonic string, 10 for the superstring, 11 for M-theory, etc). Here, $M^{d-4}$ is supposed to be a compact manifold, so that you can argue you don't observe it in low energy experiments.

It's a way to get 4-dimensional theories out of 10 or other dimensional theories.

In general, there are many different notions of compactification. I can't give you a good summary.

Cows

@Danu ok nice

@Danu so now I have trouble understanding what it means to compactify to a lattice.

Danu

I don't have the energy to give more explanations, sorry.

I need to focus on my own problems right now :P

Cows

@Danu lolz

@Danu but I am so close lol

I meant "A string theory compactified on the $E_8$ lattice*" lol

what does this mean lol . A link can help :P

Danu

link.springer.com/book/10.1007%2F978-3-642-29497-6

Cows

@Danu awesome thanks :~)

Alucard

19:49

I visited my mom, I got steak, it was very good. she is so good at cooking.

Alessandro Codenotti

I posted my question about $\Bbb R^2\setminus \Bbb Q^2$ on main if you suddenly get interested in weird spaces @Balarka

Balarka Sen

I upvoted, @Alessandro. I'd like to see an answer, but I won't like to think about it :)

Alessandro Codenotti

fair enough

Balarka Sen

funny: it seems i am so sleep deprived i am seeing every font as lucida sans

3

Alucard

so a quotient space with (0,0), otherwise it would be no space...

Alessandro Codenotti

19:54

that's a sign you should go to sleep

SteamyRoot

At least it's not comic sans

Alucard

@SteamyRoot you bring me on good ideas :D

Meow Mix

Hi

Alucard

a website in comic sans XD

SteamyRoot

Hmmm

I'm in charge of my research group's website...

Maybe I should program it so the font switches to comic sans on april 1st

Balarka Sen

20:01

yikes lol

Alucard

@MeowMix Where we well-behaved today?

Meow Mix

What?

Alucard

Shoot it, what you got for us? :D

Meow Mix

Huh?

Mike Miller

@Alucard Man, I'm going to use the ignore feature for the first time in a year. I don't think anybody enjoys the non-sequiturs.

4

Alucard

20:03

do what you wanna do

Balarka Sen

@Alessandro For a minute I though Omnomnomnom was talking about one point compactification of $\Bbb R^2 - \Bbb Q^2$. My heart skipped up to my mouth.

Cows

who is Omnomnomnom

Alessandro Codenotti

I'm not familiar with compactifications, but that seems scary indeed

(Also I'm glad Omnomnomnom came up with a simple path, I was already looking for an elliptic curve with a single rational point)

Cows

who is Omnomnomnom?

Balarka Sen

actually it's not locally compact so I am not even sure if it can be done

Alessandro Codenotti

20:07

@Cows An MSE user who commented on a question I asked

Balarka Sen

Meh, whatever.

Meow Mix

@Cows Why do you feel the need to ask it twicE?

Cows

hehe

@MeowMix i dunno lol

@AlessandroCodenotti oh cool

Mike Miller

Maybe an easier question: what are the compact subsets of R^2 \ Q^2?

Alucard

@Cows yeah baby, show them

Alessandro Codenotti

20:23

@MikeMiller they need to be complete and totally bounded, so completeness alone removes a lot of possibilities

Semiclassical

hi chat

skill patrol

hi pal

Daminark

Hey everyone!

skill patrol

Hey buddy

Alessandro Codenotti

Hi @semi @skill @dami

Daminark

20:34

How's everything going?

Maks

Hey ! So long

skill patrol

Hi @aless

Maks

I have a question, is the sum of all integers
$ 1 + 2 + 3 + 4 + 5 + ... = -\dfrac{1}{12}$ ?

Semiclassical

Depends what you mean by 'sum.'

Daminark

@Maks So that meme basically comes from abuse of analytic continuation

skill patrol

20:38

What does "..." mean?

Maks

@skillpatrol To infinity

But it appears to be true

Daminark

So you can define $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$

This converges for complex numbers with real part greater than $1$

Balarka Sen

@Maks No.

Maks

@Daminark Wolfram gives a similar formula with that result

https://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B5%2B6%2B...
At the bottom of the page

Balarka Sen

It's often used as a notation for a formula which makes sense (which Daminark is about to tell you). It's not an equality or a formula or anything in a literal sense, contrary to popular belief.

(pro-tip: 1+2+3+... diverges)

Daminark

20:40

Now, as an analytic function, you can extend it, this is called the Riemann-zeta function

Now, $\zeta(-1) = -\frac{1}{12}$

Semiclassical

The way it shows up in physics is a useful example, I think.

Maks

@Semiclassical That's what gave me the doubt

Semiclassical

There's a certain phenomena in quantum field theory called the Casimir effect.

Maks

they use it in string theory and quantum fields

Daminark

At this point the series no longer represents the function, but if you chuckle and pretend it does, you get $1 + 2 + 3 \ldots = -\frac{1}{12}$

Maks

20:42

but then, is the sum of all integers infinity of -1/12 ?

Balarka Sen

en-oh.

Semiclassical

To carry it out, you have to 'add up' the influence of all the possible degrees of freedom in the problem. If you do that naively, you'll get 1+2+3+...

Daminark

Absolutely not, again, to make this work you have to pretend that this series still represent the Riemann-zeta function

Semiclassical

However, what the appearance of 1+2+3+... really signals is that the calculation hasn't been set up with sufficient care.

Daminark

But it only does when it converges, aka when $\Re(s) > 1$

Maks

20:43

I didnt express quite correctly.
What I mean was, is the result true ? Like, its possible ? And why ?

Balarka Sen

The result is not true and not possible. Read what Daminark told you to see why people write it.

Danu

o/

Semiclassical

If you do include certain control parameters which keep the setup from being absurd, then instead of 1+2+3+... you get a more subtle summation which does in fact converge.

If you then take appropriate limits of that, you'll get a finite sum.

(Disclaimer: This is a sloppy explanation.)

skill patrol

\o

Semiclassical

And when you compare the finite result with what you'd get via the naive approach, you find that you need to replace 1+2+3+... with -1/12.

Now, does that mean that the two sums are equal? No.

What it means is that, if you were to rework the calculation in such a way that the sum actually makes sense, then in effect you replace the (divergent) series 1+2+3+... with the finite value of -1/12.

You're replacing the obvious notion of summation (which diverges) with a more subtle one (which converges).

Hence why I say it depends on what one means by 'sum.'

Daminark

20:47

Similarly, $1+1+\ldots = -\frac{1}{2}$, up to memetics

Semiclassical

On the other hand, 'clearly' 1+0+1+0+... = 1/2. @Daminark

Daminark

... did you mean -1 instead of 0?

Semiclassical

Probably.

I was remembering it wrong.

Daminark

Like that idea is basically using the formula for a convergent geometric series

Side note: Differential topology is pretty dank

Semiclassical

Right.

s=1-s -> s=1/2.

Balarka Sen

20:51

@Daminark it's my favorite

Daminark

I definitely intend to read up on Sard's theorem, which we mentioned in class today but didn't prove

Balarka Sen

The proof of Sard's theorem is not very important. The theorem is.

Daminark

Also I love how Neves was explaining the idea of a commuting diagram

He was like "Yeah, so I'm from Portugal, and if I want to visit, I could take a flight from here to London to Lisbon but what I actually do is fly through Madrid since it's cheaper. But I end up in the same place!"

Balarka Sen

:P

Daminark

@Balarka Perhaps, but why not? I heard Milnor's proof is pretty short anyway

Alessandro Codenotti

20:56

There's a proof in an appendix to GP if you want to read it

Balarka Sen

Sure, I am not saying it's not worth knowing the proof of. The proof-technique is never used anywhere.

Perhaps you should read the proof and teach it to me.

(I am not sure if I ever seriously read it either :P I sort of pretend to)

Alessandro Codenotti

@Balarka do you want a question a classmate gave me today to tingle your interest in number theory?

Balarka Sen

You can ask. I can't guarantee I would be able to answer :)

Semiclassical

I'm curious, though I'm rather dubious I'll be able to say anything.

Alessandro Codenotti

The question is to find solutions to $a^p+b^p=p^c$ with $a,b,c,p$ positive integers and $p$ prime

Balarka Sen

21:03

Too hard. Catalan's conjecture is relevant in this direction, probably

Alessandro Codenotti

I know nothing about number theory so I only made some easy observations, there are infinitely many solutions for $p=2$, exactly one for $a=1$ or $b=1$ as a consequence of the Catalan's conjecture and there are solutions in $\Bbb Z/p\Bbb Z$ for every $p$

Alucard

@Astyx Comment allez-vous?

Alucard

21:16

The Weather Girls - It's Raining Men

►

i'm out

Balarka Sen

@Alessandro A naive PARI/GP search gives a bunch of solutions for $p = 3$. I can't find one for $p = 5$ in 10^3's range. I can probably write down a better search algorithm but I'll let someone else do it.

$p=3$ has some really small solutions like 3,6,5 and 9,18,8 etc

Alessandro Codenotti

there's infinitely many for $p=3$ I'm being told by my classmate who reads but doesn't join this chat (yes that's an invitation!), $(a,b,c)=(3^k,2\times 3^k,3k+2)$ is a solution for every $k$

Balarka Sen

Ah cute

Alessandro Codenotti

That's also based on the fact that $2^3+1=3^2$ so it won't generalize because of Catalan's conjecture though

Balarka Sen

Yeah. So I guess it's only really interesting for $p > 3$

I have no idea

Alessandro Codenotti

21:22

He also added that there might have been a coprimality requirement on $a,b,c$ but he isn't sure

I guess I'll go back to think about weird topological spaces, number theory is too hard

Balarka Sen

yeah man. very nice question though

your friend should join the chat and befriend Krijn

Mike Miller

I was trying to get my operators algebra friend to join

but he's asleep again

Balarka Sen

lol. I guess he'd find a slowly rising functional analysis crowd here

Alessandro Codenotti

I'll convince him sooner or later

Mike Miller

@BalarkaSen It's worth knowing the idea for Sard at least

Balarka Sen

21:30

I s'pose that's true. I read the idea once and forgot it... all the more reason @Daminark should teach it to me.

Mike Miller

Actually I forgot it too I just realized...

Balarka Sen

oops

OneRaynyDay

Hi guys :D

Learnin about some interesting probability theory stuff. Not so sure how to google some of the stuff though, so was wondering if you guys knew the name of these identities

$E[exp(\lambda Z)] = exp(\frac{\lambda^2\sigma^2}{2})$, where $Z \sim N(0,\sigma^2)$

I know this is a moment generating function for a gaussian, but that's all I could use to google and I found nothing conclusive.

Alessandro Codenotti

I'm not sure what you mean, if you calculate the moment generating function of a Gaussian distribution you'll get that expression

OneRaynyDay

21:48

oh. So you just plug it into the expectation?

Derp I don't know why I didn't try that. I thought it was something to do with chernoff bounds since I'm reading about that right now.

Alessandro Codenotti

yep, in general if you have a random variable $X$ its moment generating function will be $M_x(t)=\Bbb E[e^{tX}]$, which can be calculated as $\displaystyle\int_\Bbb R e^{tx} f(x) \mathrm{d}x$, where $f(x)$ is a probability density function for $X$

OneRaynyDay

gotcha. Working that out right now, completing the square and everything

$\int e^{\lambda z} \frac{1}{\sqrt{2\pi} \sigma} e^{\frac{-z^2}{2\sigma^2}}$

$\int \frac{1}{\sqrt{2\pi} \sigma} e^{\frac{-z^2}{2\sigma^2} + \lambda z}$

$\int \frac{1}{\sqrt{2\pi} \sigma} e^{\frac{-z^2+2\sigma^2\lambda z}{2\sigma^2}} $

$\int \frac{1}{\sqrt{2\pi} \sigma} e^{\frac{-z^2+2\sigma^2\lambda z + \sigma^4\lambda^2}{2\sigma^2}} + e^{\lambda^2\sigma^2/2}$

$1*e^{\lambda^2\sigma^2/2}$

oh shoot I'm dumb.

Alessandro Codenotti

you have a $+$ that should be a $\times$ I think

OneRaynyDay

Yep you're right :) Caught it just in time

Thank you Alessandro :D

Alessandro Codenotti

you're welcome

you can also derive it a few times and check it gives the right moments if you have too much free time :P

anyway I'm going to sleep now, good luck with probability

OneRaynyDay

22:09

Ah yeah - I could. I think I'll continue onto Hoeffding's lemma and finish that first :) this expository textbook for my ML class was focussing on that

I'll be taking a probability course next quarter that will probably cover this as well. Good night man (Y)

Daminark

22:57

@OneRaynyDay You use quarters as well!

OneRaynyDay

@Daminark yes :) where are you from?

Daminark

Chicago

OneRaynyDay

ah. ucla here :)

Daminark

You?

Nice

OneRaynyDay

All UC's except Berkeley use quarters

Semiclassical

23:02

hi chat

Daminark

Huh, that's weird @One

And hey @Semi!

And @skull!

OneRaynyDay

why weird in particular? :o

skull petrol

Hey there @Daminark

Eric

Hello chat

skull petrol

Hello

Daminark

23:06

That all of the schools in the system except one randomly use quarters

And hey @Eric!

Eric

yo Daminark the third problem on your manifolds pset is a p good one

OneRaynyDay

Oh gotcha

Transcript for

$Mathematics$ Mathematics