12:00 AM
Yeah I just caught that :-/ Sometimes when you find one error you don't see the others...

@PeterTamaroff I know the answer. I was playing around with an identity and mangled it into this.

Like, you want it to be a manifold. And every open subset of Euclidean space has an obvious (global!) chart.

@robjohn Is it $\pi$ish?

@PeterTamaroff No it's smaller than that

@DylanMoreland Hmmm not obvious to me how to declare if a set of invertible matrices is open.....
or closed.....
LIke we would need some kind of norm on the matrices yes?

12:07 AM
The determinant is a continuous map from $n \times n$ matrices to $\mathbb R$.

Use the determinant map
$GL(n) = \det^{-1}(\Bbb R \setminus \{0\})$

@HenryT.Horton but the determinant map is not injective

"Inverse image"

yes

Inverse image of an open set under a continuous map is open

12:09 AM
@HenryT.Horton wait why are we talking about the determinant?

Doesn't matter if the map is injective

Because it makes it easy to see that this set is open.

@DylanMoreland Ah I get it: $\Bbb{R} - \{0\}$ is open in $\Bbb{R}$ under the usual topology :D

Yep.

Yesterday he was saying about how I think as group you can identify $SO(2)$ with $S^1$
but then suddenly jumped to saying that because $S^1$ is compact so is $SO(2)$
So I assume that $SO(2)$ has some topology on it and he's now talking about the two objects being homeomorphic

12:12 AM
Even more, they are isomorphic as groups.

yes
but I don't see the homeomorphism....

But yeah. $SO(2)$ is a closed submanifold of $GL_2$.

right.

I think. I'm really rusty with this stuff. That needs to change.

@DylanMoreland It is a closed subspace of $GL_2$ I think otherwise it would not be a matrix lie group

12:13 AM
Well, let's see.

Take any sequence of matrices $A_n$ in $SO(2)$
Oh I think this comes from the fact that for each $n$, $\det (A_n) = 1$

$S^1$ consists of stuff like $e^{i\theta} = \cos \theta + i\sin\theta$ for $\theta \in \mathbb R$. $SO(2)$ has stuff like $\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$.
And, well, there's a reason I used $\theta$ in both places.

@HenryT.Horton Now that I call a smart arse.

No, do it your way :)

@HenryT.Horton Why did you remove that?
the determinant map is surjective

12:17 AM
Both of the conditions ${}^tAA = I$ and $\det A = 1$ are closed, yeah.

Yeah I left out the orthogonal condition

@DylanMoreland wait I don't get the bit on the orthogonal condition

But $F(A) = A^T A - I$ can be shown to be continuous, so $\mathrm{O}(n) = F^{-1}(\{\text{zero matrix}\})$ is closed
And the set $S = \det^{-1}(\{1\})$ is closed

yes

So $\mathrm{SO}(n) = \mathrm{O}(n) \cap S$ is closed

12:21 AM
intersection of two closed sets is closed
Hmmm I need to get used to thinking about matrices this way

That is a true fact!

I should go and prepare for class at 11 now
Thanks @DylanMoreland @HenryT.Horton

Implicit function theorem is a good thing to learn, by the way.

You should be able to deduce that a element of $\mathrm{O}(2)$ has the form $\pmatrix{a & b \\ -b & a}$. Then the determinant $1$ condition for it to be in $\mathrm{SO}(2)$ says $\mathrm{SO}(2)$ is the set of all $a, b \in \Bbb R$ such that $a^2 + b^2 = 1$, i.e. it's a circle

12:26 AM
takes notes

Hey Jiggly.

Hullo.

It doesn't seem that you are registered for this class. I'm going to have to ask you to leave before I get campus security involved.

Go call the police, I took notes long ago and am ready to leave. Not that your lines are noteworthy, ish.

Please don't leave, everyone else walked out.
I won't get tenure if everyone drops my class.

12:31 AM
whistles
If you insist ...

I'm not insisting anything. I don't want to pressure you.

Well, there's this transcript thingy, I let the readers be the judge.

@PeterTamaroff somewhat... did you have a guess?

Is there an exact answer, Roberto?

@HenryT.Horton There is an exact, closed form answer in terms of common constants.
Hints?

12:47 AM
@HenryT.Horton Which problem are you talking about?

55 mins ago, by robjohn
Here is a cute problem: $\displaystyle\sum_{k=2}^\infty2^{-k}\tan(2^{-k}\pi)$

@robjohn OH! Cauchy condensation?

Do you think it expressible?

@PeterTamaroff No, that only tests for convergence, it doesn't tell you to what it converges.
10 mins ago, by robjohn
@HenryT.Horton There is an exact, closed form answer in terms of common constants.

@robjohn Well, but it certainly does, and it gives bounds in terms of the uncondensed series.
I'm just going with $\pi$. Whatever.
=)

12:57 AM
@PeterTamaroff True, it does give you bounds.
@PeterTamaroff Have you tried summing it? It converges very quickly.

@robjohn I haven't

0.31831

@PeterTamaroff Think about $\cot(x)-\tan(x)$ and how it might be simplified.
@FrankScience Yes, so the answer is not $\pi$.

@robjohn OK. $2 \cot(2x)$
I see.

@PeterTamaroff close, but not quite

1:06 AM
1 hour ago, by Peter Tamaroff
@robjohn Have you tried a duplication formula?
I didn't pay many attention.
Let me retry.

@PeterTamaroff look at the numerator and denominator

$\tan\alpha=\cot\alpha-2\cot2\alpha$ and telescoping.

@PeterTamaroff That's correct!
@FrankScience telescoping to what? :-)

$1/\pi$

@HenryT.Horton and how?

1:12 AM
$\lim_{n\to\infty}2^{-n}\cot 2^{-n}\pi$.

Hello boyos! All bugs which I have found are... fixed!

@anon: wekcome back

@FrankScience You ruined it.

@FrankScience indeed !

1:12 AM
@PeterTamaroff Looks like my hair this morning.
But my beard is bigger.

$$\sum_{n=2}^\infty 2^{-n} \tan( 2^{-n} \pi) = -\frac{1}{2} \cot \frac{\pi}{2} + \lim_{n \to \infty} 2^{-n} \cot(2^{-n} \pi) = 0 + \frac{1}{\pi}$$

Another approach?

@HenryT.Horton That's it

It is clear the $^{-1}$ in my previous answers wasn't rendering.

@FrankScience do you have another approach?
@PeterTamaroff Ah, yes ;-)

1:16 AM
@robjohn Where did this arise, then?

This problem came about when I was working out a way to do $\sum\limits_{k=1}^\infty\frac1{k^2}$
The function $\pi\cot(\pi z)-2\pi\cot(2\pi z)$ has poles at the odd integers with residue $1$

@robjohn Sick! Why would you want to do series?
@robjohn OK.

and I worked out that that was $\pi\tan(\pi z)$

So?
Does this integral even make sense? $$\int\limits_{ - \infty }^\infty {\frac{{dx}}{{\sqrt {{k^2} - {x^2}} }}}$$

@PeterTamaroff Then you can use residues and the function $\frac1{z^2}$ to get the sum of the series.

1:19 AM
@robjohn Cool.

not outside of $[-k,k]$

@PeterTamaroff it is wonderfulness

Too little residues for my distributions.

@PeterTamaroff actually you could compute the outer parts as a complex number
@PeterTamaroff but it doesn't converge unless you consider it a Principal Value and take the sign of the radical opposite on both sides.

1:22 AM
@robjohn I see,

That is a forward reference :-)
@PeterTamaroff The integral from $-k$ to $k$ is $\pi$ I believe

@robjohn Yes, just some $\arcsin$ery does it.

@PeterTamaroff just seeing if I could do it in my head.
only two substitutions
Off to run some errands and take the dog to the park. We have company over for dinner, so I won't be back for about4 hours

1:39 AM
@robjohn What about $\tan z=\sum_{n\ge0}T_nz^n$?
Incidentally, like Knuth, I would prefer to use pencil rather than pen.

2:01 AM
The primary reason why I prefer computer to pencil is that I don't accidentally erase half of the line above (I write small).

You cannot always use computer.

Yes, but that does not change my preferences.

I just like the PC because of the templates and Ctrl+V Ctrl+C

PC is scratch paper?

@FrankScience Personal Computer!

2:08 AM
@PeterTamaroff I meant that you use PC as scratch paper?

@FrankScience Oh. Well, sometimes.

@PeterTamaroff I only use substantial scratch paper or my brain.

@FrankScience I don't like to waste paper. I don't have much and I need it.

@PeterTamaroff But there are some calculations I cannot work on computer.

@FrankScience Don't you have MathType of something of the sort?

2:13 AM
@PeterTamaroff No longer.

@FrankScience Why no longer?

@PeterTamaroff No longer do I use Windows.

@FrankScience What are you using now?

@PeterTamaroff There are many recyclable thing, such as advertisement with only one page printed.
@PeterTamaroff Newspaper is sometimes available.

whats up nerdizzles
from the big bang theory (not offending)

2:26 AM
up is the local outward normal of a surface

@ChuckFernández Say fo shizzle to my nerdizzles!

@anon I tried that and fell down when I got to a wall.

WHAT IS THE BIG BANG THEORY?

@FrankScience A very funny TV show
off to pick up dinner. bbl

@robjohn OH, I DO NOT LIKE THE TV SHOW.

2:43 AM
Do you have caps lock on because you do not have to use the Shift key to start a sentence and type proper nouns?

<Khassaki> HI EVERYBODY!!!!!!!!!!
<Judge-Mental> try pressing the the Caps Lock key
<Khassaki> O THANKS!!! ITS SO MUCH EASIER TO WRITE NOW!!!!!!!
<Judge-Mental> fuck me

your backquotes don't work, for some reason...

'parrently they don't combine with linebreaks

CAPS LOCK IS A GOOD WAY TO GET PEOPLE TO PAY MORE ATTENTION TO YOUR MESSAGES.

what's an easy way to see that $\pi_1\otimes\pi_2$ is irreducible iff $\pi_1,\pi_2$ are irreducible? the $\implies$ direction I see by contraposition and distributivity of tensor products through direct sums, but I don't see the "only if" direction. (I'm think in $\Bbb C$.)

2:52 AM
ALL CAPS SUGGEST URGENCY AND IMPORTANCE.

CRUISE CONTROL FOR COOL

actually my computer does not even have a caps lock. I had to type that with my finger on the shift key.

Topologists! I need your help in proveing $\operatorname{int}(A\cap B)=\operatorname{int}(A)\cap \operatorname{int}(B)$
I have the proof that $\overline{A\cup B}=\overline A \cup \overline B$
Maybe I can mimic some of that.

The first thing that comes to mind is that the interior of $X$ is the union of all open sets contained in $X$.
And then use De Morgan's law or something.
I hope that wasn't too much of a spoiler.

I'm writing something by myself. THen I'll see that

3:05 AM
I tend not to be cautious about answering questions addressed to "topologists" since I am not a topologist.

if you're working with a metric definition of interior, proving inclusion in both directions should be straightforward

@anon I'm on da topological spaceshizzles now.
I have proved that if $x\in \operatorname{int}A\cap \operatorname{int}B$ then $x\in A\cap B$
That is a start.

Are you familiar with $\mathrm{int}(S)=X\setminus(\overline{X\setminus S})$ and $X\setminus (A\cap B)=(X\setminus A)\cup(X\setminus B)$?
(when $A,B\subseteq X$)

Yes!

then $$\mathrm{int}(A\cap B)=X\setminus(\overline{X\setminus (A\cap B)})=X\setminus(\overline{X\setminus A}\cup\overline{X\setminus B})=X\setminus(\overline{X\setminus A})\cap X\setminus(\overline{X\setminus B})$$

3:11 AM
Good night, gentlemen.
And ladies, if any are present.

@anon Sweet

If only I knew some girls... a girl...

@CLarue I did not have caps lock on. I just keep pressing Shift.

@anon That is the set theoric stuff I want to get more used to using.

@anon Reference?

3:17 AM
@PeterTamaroff distributivity is your everlasting friend

googlability?

@anon Well, and mostly using all together. Getting the big picture.

google-fu refers to one's ability to research and find sources on the internet, typically via search engine functionality specifically. just google the convo I posted and you'd find bash.org.
@PeterTamaroff every single step in the derivation was distributivity here. :) how complements distribute through unions and intersections, how closures distribute through unions (in order to show how interiors distribute through intersections)

I googled and entered a website not appropriate for adolescent/children, therefore I closed that page immediately.

im 12 and what is this

3:21 AM

not bash.org

oh
ie porn
(not to conflate with "porn involving internet explorer," ye avid 34ers)

@FrankScience How old are you?
@anon Well, yes, but as you note you did use other stuff. Whatever. Let me hate myself for a while.

@PeterTamaroff This sentence suddenly reminds me of caution on avoiding discussing age in western country.

What?
What do you mean by "caution on avoiding discussing age in western country."?

3:28 AM
@PeterTamaroff Our teacher told us that we should avoid discussing the age with westerners.

@FrankScience I'm a little baffled by the constant distinction of "east" and "west".

I'm Japanese, can you tell me?

I'm in Argentina. I don't give a rat's ass about the "east" or the "west".

discussing age in a chatroom shouldn't be a problem unless: (1) you are in an unfriendly environment, (2) you have a stalker, (3) you are underage, (4) you simply don't want people to know how old you are. you have no social obligation to discuss it, but conversely there should be no problem in this situation if you want to. also, "westerners" is a rather broad category of people.
I watch some anime, can I be Japanese?

Gaijin

3:33 AM
:(

I remember watching Ranma as a kid.

It's famous for Atman?

@FrankScience The boy that turned into a redhead when put into cold water.

@PeterTamaroff I cannot get a clear picture.

3:37 AM
I liked: ghost in the shell, samurai champloo, death note, monster. Many others I like purely out of nostalgia (eg dragonball z).

DBZ was pure awesome.
Specially DBZ GT

DBZ nostalgia is the best
Wtf DBGT? Get out

GT = nope

@anon Whaaa....?
I'm leaving now....
Bye byes.

later

3:39 AM
@HenryT.Horton Expect toplogy questions tomorrow!

Good, I'll take one of my vacation days tomorrow.

I hope @PeterTamaroff knows "get out" is a figure of speech.

GET OUT is a way of life

do you know rep thry Henry?

No I'm just a middle schooler

3:43 AM
@HenryT.Horton And you participate in Mathematics Olympiad?

No I'm not smart enough...
@anon What representation theory are you looking at currently

wait, do you know about braid groups? I think you know about algebraic topology

@HenryT.Horton just some basic stuff. in some chap 1 notes there was a proof that the tensor products of two irreducible reps is irreducible (everything in $\Bbb C$), but I couldn't understand it

@anon I saw the Burau representation at a conference talk...

3:47 AM
eh. I ask because I had an error in a braid group question I answered, and while I could just alter the example I gave I wanted to know more general theory behind ameliorating the sort of error I had

Your middle school is teaching representation theory?

wooooooooosh

@HenryT.Horton ?

Last year we did
This year is Langlands correspondence

Oh, astonishing.

3:50 AM
Reminds me of Ilya and Rajesh discussing Galois.

In our school, no further mathematics was taught.

4:01 AM
Hey guys...I'm reading through "Baby Rudin" and I'm currently on the chapter on the Riemann-Stieltjes Integral. I'm having some trouble finding the motivation for the generalization from the Riemann integral to the more general Stieltjes case. Any pointers?

it allows you to express many number-theoretic sums in calculus terms and thereby allow useful calculus-like manipulations (like by-parts and substitution). personal example
also, your name sounds familiar, can't pin down how though

Haha do you go to Stevens?

nope

Where are you from, if you don't mind me asking?

4:08 AM
Haha doubtful then. I've lived in New Jersey all of my life.

4:46 AM
well let's see here. every subspace of $V\otimes W$ should be a direct sum of tensor products of subspaces of $V$ and $W$. Say $U\subset V\otimes W, U=\bigoplus_{i}V_i\otimes W_i$. If $\pi_1,\pi_2$ are irreducible and $U$ is an invariant subspace, then each $V_i$ and $W_i$ is a subrep of $V$ and $W$ resp. and so is $V$ or $0$ or $W$ or $0$ resp.; tensoring with $0$ is annihilation so the only invariant subspaces are $0$ and all of $V\otimes W$.
no, that's not right, we can't say each $V_i,W_i$ is a subrep. drats.
picking $v\in V,w\in W$, $\langle \pi_1(G)v\rangle=V$ and $\langle\pi_2(G)w\rangle=W$ for any nonzero $v,w$, so clearly any nonzero pure tensor in $V\otimes W$ will span every pure tensor under the linear action.

2 hours later…
6:24 AM
Hmm, we can write an invariant $U\subseteq V\otimes W$ as a direct sum of complementary invariant subspaces $$U=\bigoplus_i \left\langle(\pi_1\otimes\pi_2)(G_1\times G_2) \sum_j v_{ij}\otimes w_{ij}\right\rangle.$$

6:41 AM
4

user19161
@robjohn Hey! Do you pin this every week?

more like every month

@JasperLoy whenever it falls off the board

or biweekly, I can't remember

user19161
@anon Yeah, I can't figure out which n for 7n days where n goes from 1 to 4.

6:45 AM
:)

@JasperLoy most likely not $3$

user19161
@robjohn Yeah, the real world prefers even numbers to odd numbers somehow.

@JasperLoy at least things that go into evenly ($1,2,4\mid4$).

odd numbers are...well, odd

6:50 AM
2 is the oddest prime

user19161
Yeah, 1 is the only positive integer that is neither prime nor composite.

user19161
So what happens when one tries very hard to solve a research problem but fails, or finds out that someone else just solved it a month ago?

@JasperLoy 5 is both prime and composite (depending on whether Gauss or Rational)

user19161
So it seems it can be hard to make a living out of mathematical research.

@JasperLoy I think one generally drinks heavily.

user19161
6:54 AM
@robjohn Haha, I am having coffee now.

@JasperLoy I am having tea

user19161
@robjohn Maybe Jonas is having whisky in his sleep now, hehe.

(1+2i)(1-2i)?

@JasperLoy sleep drinking?
@DavidWheeler $1-2!^2$

(that was supposed to be read: "five"?)
trying to parse your reply: one minus two! (emphatic, see reference number two)

7:02 AM
There are FOUR lights!

that is not your father's oldsmobile

I have no idea what's going on.

Zhen Lin made a reference to Star Trek (Next Generation, i think)
to which i countered with a twist on a general motors advertising campaign
it's like a geometric series....the first reply kinda sorta makes sense, the second reply is a lot more goof-ball, and the third reply is totally out of the ballpark
i'm pretty sure you understand such things

par for the course

eventually the conversation will become unbounded, and we can start over

7:08 AM
Good morning

@OldJohn technically, yes, it is morning, here. and it's good. so...yes. hi there!

7:39 AM
Aha! Character theory! $\langle\chi_\pi,\chi_\pi\rangle_G=1$ iff $\pi$ is irreducible, and so $\pi_{1,2}$ irreducible implies $$\langle\chi_{\pi_1\otimes\pi_2},\chi_{\pi_1\otimes\pi_2}\rangle_{G_1\times G_2}=\langle\chi_{\pi_1}\chi_{\pi_2},\chi_{\pi_1}\chi_{\pi_2}\rangle_{G_1\times G_2}=\langle\chi_{\pi_1},\chi_{\pi_1}\rangle_{G_1}\langle\chi_{\pi_2},\chi_{\pi_‌​2}\rangle_{G_2}=1.$$ Glad that's over with.

8:10 AM
i m in urgeny to solve to questions on probability. any one interested to help?

user19161
@narayanpatra You should post on the main site for more people to see. That way you get the most help.

just state the question

0

Identify the random variable in each of the experiments and the probability distribution followed by the variable. Also find the expected value of the random variable. a)Number of print mistakes in a text book are being estimated. b)200 cars are expected to arrive at inspection station during c...

i m pretty weak in probability

user19161
...

and tomorrow my exams are scheduled
will be a gr8 favor to me if any one solves these two

user19161
8:14 AM
@narayanpatra If your exams are tomorrow and you are so weak, you should study the theory first instead of working on examples. And I forgot all my probability, sorry.

@Jasper: Nice suggestion bro, but little time is there for me to study. solution for these two questions will help me a lot.

user19161
@narayanpatra The time you spent posting here you could have spent studying, and you might have finished reading by now.

user19161
Doing these examples may not help you solve the problems you will meet tomorrow on the exam.

Chances are very high that I will get this question in my exam. and my further study do not involves statistics.

user19161
Good luck for tomorrow, but note that this is not the right way to learn mathematics.

user19161
8:21 AM
@narayanpatra Chances are very high that you can finish studying these topics before the exam...

user19161
Morning @jonas! Have a nice day!

Hi.

haha. I also think so. I have to study it myself.

@JonasTeuwen Hi there

user19161
@narayanpatra Yes, you need an attitude change to make progress.

8:26 AM
I need to call the doctor, stressy 8-).

user19161
@JonasTeuwen OK, do whatever you have to. There can be miracles when you believe.

Sure 8-).

user19161
@OldJohn Hi there. Now we have a chain of hi's.

@JonasTeuwen What is stressing you out today?

Bloody monkey, why does the mofo go on a holiday! 8-)!!!

8:31 AM
@JasperLoy na, I don't think. I have to change my attitude. many times we require a stop gap arrangement, and its same here. I m pretty good in many things which I love to do. stats is not what I am interested in. Its a compulsion from my college and I am just completing formalities.

user19161
@narayanpatra OK, if you need this kind of help you should be asking the people around you. SE is not really a site for solving your last minute problems.

user19161
@JonasTeuwen It's OK, I am here for you bro...

user19161
@jonas I did not see that.

user19161
@jonas OK I saw that.

Oh well.

8:37 AM
Planning to go and do a bit of hill walking while the weather is good - back later, folks

user19161
@JonasTeuwen I sent you a short email.

user19161
@OldJohn Enjoy the walk.

@JasperLoy Thanks - take care!

@JasperLoy Thanks :-).

9:05 AM
@Jonas: hi

@Ilya Hi!

@Jonas: so have you read it?

@Ilya No, I was too ill :-(. I will try to do today!

oh no! ill? what had happened?

@Ilya I don't know.

2 hours later…
10:47 AM
How can I solve the third order differential equation with non-constant coefficient.
$$\frac{1}{30} \frac{\partial ^3}{\partial x^3} S(x,y) - 2S(x,y)\left \{\frac{y^3}{(1-xy)^3} + \frac{1}{(1-x)^3}\right \} \\ ~=\frac{24y}{1-y}\left \{\frac{1}{(1-x)^7} - \frac{y^5}{(1-xy)^7}\right \}$$

Whoah. Try with a computer.

@JonasTeuwen with Mathematica?

@ami Yeah, a numerical solution looks plausible.

For example...

I tried with wolframalpha.com. It gave timeout :P
@robjohn thanks. how can I approach in case I want to solve it by myself.

10:52 AM
Learn more mathematics, I suppose. If you don't know how to begin you will fail I am afraid.

@JonasTeuwen At university, I read only 2nd order and 1st order DEs.
@JonasTeuwen never come across third order DEs

If the $3$ would be a $2$, would you know how to solve it?

Is it possible to reduce the order of 3 to 2 ?
@JonasTeuwen $y = c_1y_1 + c_2y_2$ -2nd order homogeneous eqn

I know. But you don't seem to know much about them then I think you should learn more first before you attempt this equation because it is quite hard (at a first glance).
And if you got them from a model you are required to solve you probably have done something wrong.

@JonasTeuwen I am trying to find the average cost of median of 5 - Quick select using Generating function. This is the closed form equation which I want to solve.
@JonasTeuwen If you are interested, I can email you the paper which I am writing.

11:03 AM
I doubt you will get a closed form. Need to go now. See you later. Good luck, perhaps I can look at it later on.

Sure.
Do you want the paper?
I am more interested in finding the coefficient of $[x^ny^i] in S(x,y)$
@JonasTeuwen I don't have PHD degree. I work for a software company.
@JonasTeuwen Take care.

11:52 AM
@JonasTeuwen hey