Mathematics - 2012-10-26 (page 2 of 3)

9:00 AM

@anon Of course one could say that without they bracket it would not be possible to write $\mathfrak{sl}_n = \mathfrak{h} \oplus (\bigoplus g_\alpha) $

anon

without the bracket one would not be able to define either side no?

BenjaLim

yes

@anon You are writing this up as some kind of project/book?

@anon In that cornell document there is no mention of the lie algebra

anon

@BenjaLim that's where the geometric viewpoint comes into play. the lie bracket of vector fields measures the failure of parallelograms to form when one takes a commutator in the group of diffeomorphisms; alternatively, when one differentiates conjugation by exponentials (which is a way to measure how much conjugation 'changes our perspective' on the potential group actions).

BenjaLim

right.

@anon When would you want me to input material for this thing?

anon

I am writing it up as a note to pass around and use as a guide in an informal meeting (where I will be presenting) at school with prof and other students this coming Wednesday. Good catch about the document, but lie algebras are of course lurking in the background and I will make it explicit where it doesn't.

BenjaLim

9:05 AM

right. What would you like me to write about?

anon

I just wondered if you had anything to add that came to mind. My differential geometry is very weak, so I may want what I write to be fact-checked or whatever too.

BenjaLim

@anon I have to admit that I have not studied differential geometry before.

I am only studying it in a course next semester

Though I understand a little about the tangent space

For example I can say why the two equivalent definitions of the lie algebra

1 as the tangent space at the identity

2 as the set of all matrices $X$ such that $e^{tX}$ is in the Lie group for all $t$

why these two are equivalent

user19161

@BenjaLim Would it be only curves and surfaces?

BenjaLim

@anon But I would be happy to read what you write and give some comments

@JasperLoy No it is one of those crazy super fast paced courses that starts with smooth manifolds and ends with connections and riemmanian geometry

user19161

@BenjaLim Well done! You know what book I will ask you to read.

BenjaLim

9:08 AM

john lee

anon

you don't perchance know how to make the ToC come out single-spaced do you? the rest of my doc is single-spaced, and I thought the toc was default 1-spaced but apparently not

user19161

@BenjaLim Notice how we share the same initials. =)

Jim_CS

Guys can anyone help me with the cosets of N in this question? math.stackexchange.com/questions/220487/…

N3buchadnezzar

@anon Try tocloft

BenjaLim

@anon isn't that automatic?

user19161

9:10 AM

@BenjaLim He wants to alter the defaults.

BenjaLim

@anon

\setcounter{tocdepth}{1}
\tableofcontents

anon

@Jim_CS the cosets of $N=G\times\{1_H\}$ are $G\times \{h\}$ for each $h \in H$

@BenjaLim isn't what automatic?

BenjaLim

yes.

user19161

@anon Spacing.

BenjaLim

@anon wait what is?

user19161

9:12 AM

So I gave two answers, went to take a nap and after I woke up they are still at zero, sigh.

N3buchadnezzar

\begin{spacing}{1.5}
\tableofcontents
\end{spacing}

anon

Do I have to throw my shoe at someone? I'll do it, believe you me. Internet and oceans or no.

Jim_CS

@anon: Thanks mate

user19161

Why am I answering @ben and @anon questions for them?

N3buchadnezzar

\usepackage{setspace}

anon

9:13 AM

@BenjaLim didn't work

user19161

From my perspective, I see it all, while Ben and anon are confused. =)

anon

I am confounded at how Benja could possibly be confused.

user19161

That's why we should use the arrows kids, use the arrows.

anon

Also, I don't have the setspace package and my guilty secret is I've never learned to install packages yet...

BenjaLim

@anon

N3buchadnezzar

9:15 AM

@anon You just add them in the preamble...

user19161

And some people after learning about the arrows forget what they mean.

BenjaLim

@anon \documentclass[11pt, letterpaper]{amsart}
\setlength{\topmargin}{0.0in}
\setlength{\textheight}{21.5cm}
\setlength{\evensidemargin}{0.25in}
\setlength{\oddsidemargin}{0.25in}
\setlength{\headsep}{0.1cm}
\setlength{\textwidth}{16.0cm}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{youngtab}
\pagestyle{plain}

\newtheorem{irrepsofSd}{Theorem}[section]
\newtheorem{examplesofyoungdiagrams}{Examples}[section]
\newtheorem{standardrep}{Proposition}[section]
\newcommand{\GA}{\Bbb{C}[\mathfrak{S}_d]}

anon

@N3buchadnezzar and the .sty file goes where on my computer (and where do I get the sty file from)?

user19161

Again, the famous ping without words @ben. You should abolish them.

BenjaLim

@anon \maketitle
\setcounter{tocdepth}{1}
\tableofcontents

That's my preamble

Jayesh Badwaik

9:16 AM

@BenjaLim what does youngtab do?

anon

the ping:post ratio for Benja is paradoxically greater than 1

user19161

@anon Same for Pedro.

BenjaLim

@anon

@JayeshBadwaik Young diagrams

user19161

Geezis!

skullpatrol

1 min ago, by BenjaLim

@anon

user19161

9:17 AM

@ben You are killing me with your pings without words.

Jayesh Badwaik

@BenjaLim ohh. nice!

anon

UPPING THE ANTE TO TWO SHOES. BOTH OF THEM.

BenjaLim

@JayeshBadwaik I forgot to copy the bit that allows me to put comutative diagrams

N3buchadnezzar

@anon It should either automatically lie on your computer, or be downloaded automatically

anon

@N3buchadnezzar by TeXworks?

BenjaLim

9:17 AM

@anon what are you using, LYx?

user19161

@anon TeXworks: it just works!

BenjaLim

I'm on gedit

anon

all TeXwork tells me is "error file not found: setspace.sty"

user19161

@N3buchadnezzar Wrong assumption: He might not be using MikTeX.

skullpatrol

@anon I call your shoes and raise you the kitchen sink :-D

user19161

9:18 AM

@anon Install the package. What TeX distro are you using?

BenjaLim

@anon Are you on windows ME

user19161

@BenjaLim Not relevant.

Jim_CS

@anon: Are the cosets of N just the quotient D/N? We haven't really covered quotients yet so I am unsure of what they represent...are they just cosets?

BenjaLim

@Jim_CS Ask me

N3buchadnezzar

@JasperLoy Then they should be on his computer, assuming a full install.

anon

9:19 AM

@Jim_CS the quotient is the set of cosets, yes, but you are also able to multiply and invert the cosets. you should look it up instead of aimlessly trying to do an exercise that uses terms you don't understand..

user19161

@N3buchadnezzar A full install is a wrong assumption.

Jim_CS

@BenjaLim: Ill ask u next time mate hehe

N3buchadnezzar

If you are using TexLive2012 and not a full install, then... sigh

anon

I have mentioned what I am using twice now. Also there is \usepackage[pdftex]{graphicx} at the top so I assume my "distro" (distribution) is pdftex? also, windows 7.

user19161

@anon I recommend a serious mathematician like you install the full TeX Live. You do it just once a year.

anon

9:21 AM

does create pdf previews quickly?

user19161

@anon I was asking about MikTeX or TeX Live from TUG or TeX Live from Linux distro's repositories, a little different from what you have in mind.

Jim_CS

Guy does a quotient in group theory have to have a normal subgroup in the denominator or can you have some other subgroup?

user19161

@anon Not related. If you use TeXworks, the preview is in the editor. Speeds no different.

anon

I am so tech-emasculated.

user19161

I would love to help anon do an install. I miss him so much.

user19161

9:23 AM

=)

anon

@Jim_CS you need the subgroup in the denominator to be normal for the cosets to have well-defined multiplication and inversion operations. however, you can still talk about the space of cosets of a not-normal subgroup.

user19161

@anon I assume you are using MikTeX then, in which case if you have an internet connection and set it so it would install missing packages on the fly.

Jayesh Badwaik

@JasperLoy " in which case if you have an internet connection ", LOL!!!

user19161

@JayeshBadwaik Precise, Jay, precise!

anon

my TeXworks is in a MikTeX folder, so I assume so. how do I set the install-on-the-fly option?

Jim_CS

9:25 AM

@anon: Cheers, I think alot of this group theory stuff is on the verge of clicking for me.

For that question I linked above I have said that D/N is isomorphic to H because we can define a mapping \alpha D/N -> H such that \alpha maps each G X h} to {h}. Does that look ok.

user19161

@anon Go to the MikTeX configuration GUI, the last I recall.

anon

I went to MikTeX 2.9 >> Maintenance (Admin) and set install packages on the fly to 'Yes,' closed and reopened my document in editing mode and still the "file not found" error

user19161

@anon You need to compile the document now.

user19161

It will install when you next compile it assuming there are no other erros in the source file.

anon

that's the green arrow I use to preview the document (which automatically updates the corresponding saved pdf file), no?

user19161

9:30 AM

Yes, click the gree arrow. No effect?

anon

yes, the effect is the error I referenced

this is one aspect of latex that bugs me: so many possible complications in finding out how to do just one small thing (single space a ToC)

user19161

That means the setting has somehow not been set properly, might be related to various OS settings.

user19161

I suggest you do a manual install of the package for the time being.

user19161

Go to the package manager now.

user19161

Look for the package name and install it from there by checking the box.

user19161

9:33 AM

The error might be related to a permissions setting. When I used MikTeX in the past I had to set the entire folder to read and write for any user so that MikTeX can install new packages.

user19161

So when you use TeX, you need to figure out how the OS works, how the TeX distro works, how the TeX editor works, and finally how the source file works.

user19161

Also after changing permissions one might need to restart for the changes to be effected. So a million and one things could go wrong here.

anon

well, now I have the package and there is no error message. however, even with
\setcounter{tocdepth}{1}
\tableofcontents
the output still has a double spaced ToC (note that I use only \section{}s in my doc)

Jim_CS

I was told that the Symmetric groups of order 5 has subgroups isomorphic to D5 and D6 but looking here I don't see them listed as subgroups. groupprops.subwiki.org/wiki/Symmetric_group:S5 So does S5 have subgroups isomorphic to D5 and D6 or not?

anon

naming conventions for the dihedral groups are different: some people use D_n to refer to what others call D_2n

user19161

9:40 AM

@anon I don't know about this. Did you try all their suggestions? If it still does not work ask in the TeX room.

anon

let me try nebu's

okay, nebu's doesn't change the double spacing in the ToC itself, what it does change is the space between the ToC and the header of the first section

user19161

Haha, why do you want to change it? Anyway, ask in the TeX room.

anon

I think that and \setcounter were the only two suggestions.

@JasperLoy because it takes up too much space; my document is very tight

Jim_CS

ok cheers

anon

anyway I am going to sleep now

user19161

9:44 AM

@anon OK, but are you trying to save paper? I just go with the defaults, not that I know much about TeX. Good night. See you in your dreams!

skullpatrol

later

anon

@Jim_CS "direct product of S3 and Z2" listed in the table is the same as your D6, and "D10" in the table is your D5

BenjaLim

@anon Send me the file when you're done

if I can input anything

ping me

anon

it will be a few more days

BenjaLim

also

if you need anything from me tell me

right

anon

9:46 AM

now see, I waited too long to go to sleep, and now I need to go to the bathroom again

skullpatrol

the silent Ben ping "@"

3

anon

I have some other stuff related to plethysm I could talk to you about (a dimunitive analogoue of what F&H describe as plethysm, transplated from rep thry to pure group thry and perhaps other categories)

but that's for later

user19161

@anon Same as me. Yay! And I use TeXworks too, yay!

skullpatrol

$\Huge\text{"@"}$

2

user19161

@ben Have you seen Varadarajan's book on Lie groups and algebras? Pretty advanced stuff.

N3buchadnezzar

9:50 AM

Yo

skullpatrol

34 mins ago, by skullpatrol

1 min ago, by BenjaLim

@anon

2

nested silence :-D

N3buchadnezzar

What is $A \circ B$ when $A \in M_{m\times r}$ and $B\in M_{m\times n}$ are given by matrices (real or complex)?

anon

http://en.wikipedia.org/wiki/Hadamard_product_(matrices)?

oh, with different dimensions I dunno. (kronecker product is possible, but is associated with the $\otimes$ symbol)

N3buchadnezzar

@anon But they do not have the same number of elements?

Nick Kidman

Hi.

skullpatrol

10:01 AM

hi

Nick Kidman

It appears to me now that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Propably because the subobject classifier doesn't have to be {0,1}. But I wonder, aren't all the multivalued logics also part of set theory? Is there some new logic coming in with topoi which weren't there before? Fuzzy logic etc. are all existent in "conventional set theory mathematics" already, right?

bye

bye

symmetry^

12:14 PM

Hi

sperners lemma

12:27 PM

hello

Jayesh Badwaik

12:52 PM

@spernerslemma hi

wassup?

Jonas Teuwen

1:16 PM

@N3buchadnezzar Composition of the associated operators?

jdoe

i cant change my name back

Rajesh D

2:04 PM

Is O(log(n)^2) faster than O(n) ?

Chris's sister

2:33 PM

I fail to apply Chebyshev here $\int x^{\frac{1}{2}}(1-x)^{\frac{1}{2}} \ dx $

It's the case with $\frac{m+1}{n}+p \in \mathbb{Z}$

I simply obtain a negative value under the radical sign. (something is wrong I suppose)

I used the substitution $x-1=t^2 x$

leo

3:15 PM

take $K\subset \Bbb R^n$ compact, and $\lambda\in(0,1]$, is it true that there exist a $v\in\Bbb R^n$ such that $\lambda K +v\subseteq K$?

@robjohn are you afk?

@BillDubuque are you arround?

Gustavo Bandeira

@leo Better make a question. =)

leo

@GustavoBandeira I'm pretty sure it's true don't know how to prove it.

Gustavo Bandeira

Yep

leo

3:30 PM

the idea is that rescaling do not changes the shape. Oce you rescale, just translate it in $K$

@JonasTeuwen any idea?

Frank Science

How can we observe the infinity?

etc, in Ramsey's theorem.

@RajeshD Nope.

Rajesh D

Hey @Frank nice to see you. Talking about asymptotics and you are back

same answer with any power greater than 2?

Frank Science

@RajeshD Now I'm haunted with stuff of calculus.

@RajeshD Yeah, it's not tough to prove that $(\log n)^m/n\to0$.

Rajesh D

ok

Frank Science

Well, I'll ask a question about the aspect of infinity.

In math.SE

Rajesh D

3:42 PM

?

any more details

Frank Science

Details about what?

robjohn

@leo If I say yes, I don't think you would believe me ;-)

leo

@robjohn any idea to prove the above statement?

this one:

32 mins ago, by leo

take $K\subset \Bbb R^n$ compact, and $\lambda\in(0,1]$, is it true that there exist a $v\in\Bbb R^n$ such that $\lambda K +v\subseteq K$?

robjohn

@leo Let $K$ be the unit circle, and $\lambda=\frac12$ in $\mathbb{R}^2$

Rajesh D

@FrankScience about your question about infinity

leo

3:49 PM

@robjohn yes, then it is true. But I'm trying to prove that this is true for every compact $K$

Frank Science

@RajeshD I'm editing mine.

robjohn

@leo no it is false

leo

@robjohn I just realized that

Rajesh D

@FrankScience post a link when done

Frank Science

@RajeshD Okay.

leo

3:50 PM

circle insted of ball

@robjohn thanks

Chris's sister

Could any help me with the problem above? I try to figure out where I'm wrong.

Gustavo Bandeira

This is amazing:

http://youtu.be/V3B7uQ5K0IU

Jayesh Badwaik

@FrankScience if $\log n < n$ as $n \rightarrow \infty$ then wouldn't be an algorithm of $O(\log n)$ faster than $O(n)$. If that is what @RajeshD meant?

Rajesh D

@JayeshBadwaik by faster I mean grows faster with n.

didn't think in terms of algorithm execution

Jayesh Badwaik

4:05 PM

@RajeshD ohh! :-)

Frank Science

Is it articulate?

Rajesh D

@FrankScience which question

Frank Science

@RajeshD Um, is it a question?

Rajesh D

post the direct link

Frank Science

@RajeshD Click that "it". It's a hyperlink.

Rajesh D

4:09 PM

@FrankScience sorry it took me to a wrong page last time

its fine now

lol

"look deeply into the night sky"

Frank Science

@RajeshD Maybe because of my bad English, it's hard to understand my question.

Rajesh D

@FrankScience I am a bit sleepy now so can't say

also I am not much familiar with the content though

Frank Science

@robjohn Long time to see.

user19161

4:40 PM

So I went out and came back and got one upvote with accept for one answer and no votes for the other, sad panda.

skullpatrol

hi @Charlie

wassup?

user19161

@skullpatrol You did not say hi to me? Sad panda.

Charlie

Because I'm TNT

@skullpatrol hi skull!!o am good!you?

skullpatrol

I'm dynamite!!!

@JasperLoy hi panda boy :-D

user19161

@skullpatrol Every time you don't say hi, a panda in the world dies. =)

Charlie

4:47 PM

@JayeshBadwaik Hiiiiiiiiiii!!!!!!!

Jayesh Badwaik

@Charlie Hellloooooooo!!!!!!

user19161

@JayeshBadwaik So many exclamation marks! Something is going on...

robjohn

@FrankScience I am like the wind...

user19161

@robjohn ...that comes out from behind. =)

skullpatrol

I thought she's supposed to be like the wind beneath my wings...

robjohn

4:51 PM

@JasperLoy No one likes Sad Panda

user19161

@robjohn What a lovely picture of me!

robjohn

@JasperLoy Glad to help :-)

skullpatrol

Is that a self-portrait above him?

Jayesh Badwaik

@JasperLoy heh. I answered your question. You got your schedule on proper time yet?

user19161

@JayeshBadwaik Nope, but never mind. I realised that the answers I got no votes for are usually the ones which are too good for the OP to appreciate. =)

Jayesh Badwaik

4:59 PM

@JasperLoy =)

skullpatrol

hi @NicolasEssis-Breton

Nicolas Essis-Breton

Hi @skullpatrol, thanks for the greeting, I'm trying to familiriaze with the chat rooms

Charlie

@JayeshBadwaik how are you???

skullpatrol

@NicolasEssis-Breton Your very welcome :-D

Charlie

@JasperLoy what is going on?

user19161

5:06 PM

@Charlie I smell romance in the air...

Charlie

@JasperLoy i do not...

Jayesh Badwaik

@Charlie I'm good. You?

@JasperLoy you are four months early.

Charlie

@JayeshBadwaik good.octupus in the shade!

Jayesh Badwaik

@Charlie OH YEAH! currently listening to Torn - Natalie Imbruglia.

user19161

@JayeshBadwaik Haha! What's happening in February?

Jayesh Badwaik

5:09 PM

@JasperLoy Romance will be so thick in the air that there will be condensation in form of cupid couples on the street.

user19161

@JayeshBadwaik Oh I see it is V day. Also perhaps Benja might come and visit me that month.

Jayesh Badwaik

@JasperLoy Hmm. Nice. :-) He is coming for holidays?

user19161

@JayeshBadwaik I don't know, he mentioned it and I guess it's not really secret so I said it just now.

user19161

I might be dead by then. =)

Jayesh Badwaik

@JasperLoy yes, I saw him say that on chat. And you better be alive, else Benja will raise you from the dead into an eternal zombie torture.

Charlie

5:13 PM

@JayeshBadwaik sssssssssss.....

user19161

@JayeshBadwaik I would like to be born in Germany in my next life I think. They ask me why and I say I think that Germany has the best mathematicians and the most beautiful women.

Jayesh Badwaik

@Charlie ssssssss.

@JasperLoy you might accidentally be born a jew in 1920's.

user19161

@JayeshBadwaik And the two shall become one flesh!

Jayesh Badwaik

@JasperLoy gross

user19161

@JayeshBadwaik Or be knifed by the Khmer Rouge.

Jayesh Badwaik

5:16 PM

@JasperLoy yeah, so enjoy the life which you have. make the best of it.

user19161

@JayeshBadwaik QED.

user19161

@JayeshBadwaik Interestingly that has got to do with Germany as well.

Jayesh Badwaik

@JasperLoy it was intended to be ;-)

robjohn

5:34 PM

@Charlie losing air?

@JayeshBadwaik oh, no! it's catching!

Jayesh Badwaik

@robjohn hahaha......

robjohn

@JayeshBadwaik No, this is serious, all the hot air is being released into the atmosphere. Global Warming will ensue!

skullpatrol

@robjohn would you mind re-installing the drop down new question notifier?

robjohn

@skullpatrol That was annoying.

user19161

The Canadian girl I liked told me that there were so many factors in sea levels that nobody really knows for sure whether sea levels will rise or fall.

robjohn

5:38 PM

It would be nice if it could be a personal setting

skullpatrol

@robjohn Ok np :)

Jayesh Badwaik

@robjohn ohh :-)

skullpatrol

hi @MichaelGreinecker

Charlie

@robjohn Jay knows what it is.

Michael Greinecker

hi

skullpatrol

5:42 PM

@JasperLoy Did she like you back?

user19161

@skullpatrol I don't know. That was many years ago. She is married now.

skullpatrol

@JasperLoy If you asked her maybe you would be the one she's married to now.

user19161

@skullpatrol Hehe. How about you skullie? Have you met the person of your dreams yet?

skullpatrol

@JasperLoy skullpatrol does not dream...

user19161

@skullpatrol Oh, but you know what I mean.

skullpatrol

5:49 PM

hi @NominSim

Ed Gorcenski

afternoon

skullpatrol

hi

user19161

6:04 PM

Hey @anon are you going for the costume party after all?

anon

no it was yesterday

skullpatrol

We can have a costume party right here :-D

or not.

user19161

Skullie's costume is made up of six cubes.

Jonas Teuwen

@leo Why do you ping almost everybody you think can answer your question? Does not really sound like you're asking a friendly question 8-(.

Also, it makes me uninclined to answer it.

@JasperLoy Cubes.

skullpatrol

Boo!

user19161

6:18 PM

Ah!

Charlie

6:34 PM

THUNDERSTRUCK!

@JasperLoy nice pic

user19161

@Charlie Refresh, you should see the default one now, hehe.

Charlie

@JasperLoy not yet

skullpatrol

You've been THUNDER STRUCK!

you got me shakin at the knees

Charlie

@skullpatrol THUNDER

skullpatrol

beggin you please

Charlie

6:39 PM

@skullpatrol to sing?

skullpatrol

it's alright

Charlie

I was caught
In the middle of a railroad track (Thunder)

skullpatrol

THUNDER

Charlie

I looked 'round,
And I knew there was no turning back

skullpatrol

I'm on the HIGHWAY to hell....

Charlie

6:44 PM

@skullpatrol HIGHWAY TO HELL

HMMM

DON'T STOP ME

skullpatrol

"No stop signs... speed limits... nobody gonna slow me down."

"Ain't nothing I would rather do. Going down, party time, my friends are gonna be there too."

Charlie

Aoww

I'm on the highway to hell

Hey Satan! Paid my dues
Playin' in a rockin' band
Hey mama! Look at me
I'm on my way to the promise land

skullpatrol

Thunder only happens when it's raining...

...players only love you when they're playing

Charlie

Back in black, I hit the sack
I've been too long, I'm glad to be back

skullpatrol

They say, women, they will come and they will go
When the rain washes you clean you'll know
You'll know

You'll know
You will know
Oh oh oh you'll know

Charlie

6:54 PM

@skullpatrol i don't know this song :(

skullpatrol

@Charlie Sorry, I switched gears on you. here

user19161

Hey guys it's Friday, time for a song...

Charlie

@JasperLoy YEAAAHHHH!

not that

that is in your mind

user19161

@Charlie You know what I am thinking of?

Charlie

@skullpatrol OOhH

that i know

@JasperLoy yes

user19161

6:57 PM

@Charlie Hehe, I love RB.

Charlie

Arm yourself because no one else here will save you

2

the odds will betray you and i will replace you

2

Transcript for

$Mathematics$ Mathematics