It is important.

Then I will return to Galois fields

@Alizter It went terrible. Somehow, someone kept pinging him about how his dinner went :P

@Semiclassical Hello!

hi

@alitzer meant to ask, did you get that generating function problem to work?

the one with decomposing $n$ into $k$ distinct parts

@Semiclassical no:(

dang

I have answers but nothing like I would like

I thought you got it, @Alizter

I had a good think about it with Mike and I think I got close however I could not find the mistake

Mike I get mistakes :(

So lets run through this again. I have notes with me.

Gtg, it's 1 AM here and I need to get up for school at 6.30 tomorrow today, see ya

WLOG assume $k_{j}<k_{j+1}$

we will address this log later

If we add n to both sides we get $k'_1+\cdots+ k'_n=k+n$

@Hippalectryon I accept I might be wrong, but not that wrong. The drama, if I can say that, is that even if we know people in reality it's hard to entirely know them especially when some want to be fully understood. :-)

where $k_j'=k_j+1$

so this is partitions of $k+n$ into $n$ distinct parts

Yup. so this is $p_n(n+k)$

now I am slightly skeptical about this part

I just had breakfast. Noodles and egg.

the part we just did, or the next part?

can we just times by $n!$ to consider the other things

Yes.

@Alizter In the UK, it is sceptical.

They're distinct so you're not double counting.

OK so we have $a_n=n! p_n(n+k)$

mhm

now to find the generating function

well

you should really say

$q_n(k) = n!p_n(n+k)$

we can write this as $$k!\sum_{n\ge 0} p_k(n+k)x^n$$

also I deeply object to having the index be $n$ so I'm going to rewrite this as

$q_k(n) = k!p_k(n+k)$

OK.

Here's where you ran into trouble, @Alizter

$n!p_n(n+k)$ are not the coefficients of the generating function for $q_n$.

hmm

What you need to do is find the generating function for $k!p_k(n+k)$ and that is the generating function for $q_n$.

@MikeMiller I have confused terminology then. $q_k(n)$ I want to be coeffs

Ooh, no, I'm sorry.

I got confused. You're right.

According to the Wikipedia article on integer pratition

$p_k(n)=p_k(n-k)+p_{k-1}(n-1)$

And it was 3am when I wrote the next part so here goes

I deduce that $p_k(n)=\sum_{k \ge j \ge 1}p_j(n-k)$

nooo

The next step is that there is a very explicit generating function for $p_k(n)$.

Hello @justabrickinthewall

@MikeMiller This was to eliminate the $n+k$

I write it as a series and a partial sum and swap

If $f(x)$ is a generating function for $p_k(n)$, then $x^kf(x)$ is a generating function for $p_k(n+k)$.

What happens to the other terms?

Because now the sum starts at k no?

$x^k \sum_{n=0}^\infty a_n x^n = \sum_{n=k}^\infty a_{n-k}x^k$

You've just shifted everything up by $k$. You didn't kill off everything below $k$.

so $x^k f(x)$ would be the gen. function for $p_k(n-k)$ not $p_k(n+k)$

hmm

now that semiclassical has stopped me from saying wrong things, it should be clear what the right approach is

I am confused.

Not bad @Alizter

Hi, confused all.

@Alizter, what's wrong with bof's answer?

I've been confused, myself, by two perfectly reasonable interpretations of a probability problem that give different answers. Ah, analysis is so much saner.

@KajHansen I have no idea. Where is he getting 3 from?

@Semiclassical So I get $$x^{-k} \sum_{n=k}^\infty p_k(n)x^n$$

So have can this be evaluated if the generating function for $p_k(n)$ is known?

@TedShifrin, have you forsworn your intuition yet? That sounds like me whenever I take a macroecon course.

@Alizter What do you mean by evaluated?

I thought my solution was perfectly reasonable, @Kaj, but I think Rachel convinced me otherwise. One answer gives 7/36, the other 3/16. Not far off, but quite different.

@Alizter The generating function for $p_k(n)$ is $$\frac{x^k}{\prod_{i=1}^n 1-x^i}.$$ bof's answer is right in this respect.

Or the function found

I don't understand anything he's doing other than that.

But, luckily, I get to go to a piano recital and not think about math tonight :)

LOL

I guess @Mike is still breathing.

@MikeMiller I am confused because that sum up there starts from $n=k$

@TedShifrin Is it classical?

Yes, @Alizter.

@TedShifrin What are they going to play?

@DanielFischer Can I ask you a question?

I don't know, @Alizter. I can tell you after the fact :)

@Alizter That formula there is wrong, sorry, because we want distinct parts, not just parts. But how many ways can you write $n<k$ as a sum of $k$ (positive) parts?

@TedShifrin OK. Let me know :)

@MikeMiller OK. So we shall go back to the beginning?

god no

LOL ... bubye, all.

@Ted Unfortunately I'm still not doing so well... I actually need someone to sub for me tomorrow.

@MikeMiller Eekk.

@Alizter We just need a generating function for $p_k(n)$, the number of ways to write $n$ as a sum of $k$ distinct parts.

Yikes, @Mike, that's not sounding good. I hope it isn't the 'flu. It's going around.

@TedShifrin Bye Ted.

No need to go back to the beginning.

It is, @Ted.

I'm getting over it but not as quickly as I'd like.

Did you go to the health service and get Tamiflu?

Well...

smack

I have not been sick for a while luckily.

And you didn't get a flu shot, either.

I get those

Perhaps the two are related

I thought Tamiflu was the flu shot.

@MikeMiller I will leave this for another time. I need to do some homework.

at 00:36

No, @Mike, it's a drug that helps you get over 'flu if you get it within the first 24 hours or so.

Night, @Alizter. Bye all

Oh.

Well, I didn't realize it was the flu in the first 24 hours.

As usual. Headache medicine is very effective if you take it 2-3 hours before headache begins.

lol

Should I get a free flu shot

yes

Is it really worth it

There's no negative side effects or risks in getting a flu shot?

I had 2 colds in the past 2 months so far :(

I was in agony on Saturday and am missing classes and TAing because of the flu

Get the damn shot

Hi I would like to prove that if $f:\Bbb{R}\rightarrow\Bbb{R}$ is continuous then the graph is closed (without sequence because I am terrible without it).

what's your definition of continuous

You know, I am not in the frequently in room list

hmm

i need to either find an interesting problem to work on, or something interesting to read

@MikeMiller Let $a\in A$ and $f:A\rightarrow F$ such that $f$ admits a limit $b$ in $a$.

(real analysis doesn't count for me, sry)

@Semiclassical Like linear algebra ?

eh, could work depending on the problem

what does $A\geq B$ mean here?

element-wise?

$A\ge B$ if $A-B$ is positive semidefinite.

ah. i don't deal with definiteness of matrix stuff a lot, tbh

(in physics that's mostly questions about stability of systems, and that was never much within my purview)

hehe okay, it's called Löwner-Heinz inequality :-)

kk

also, how are fractional matrix powers being defined here?

@Semiclassical Interesting to read, have you read 'Harry potter and the methods of rationality'?

@Semiclassical For any positive operator $A$ and positive integer $n$ there is a unique operator $B \in k[A]$ (i.e., $B$ is a power series in $A$) such that $B^n = A$.

Harry is a rationalist in this story, and it has millions of followers, he breaks down the logic of magic and abused it to his advantage

Also he gets rid of ron as useless when he first meets him

Probably I need to be more careful and say positive bounded operator on a Hilbert space or something.

@MikeMiller: does that help here, though? MarcGato's problem above had $0\le r \le 1$ for the exponent

@Semiclassical Yeah, $B$ is the $n$th root of $A$, so you can define fractional powers for all $0 \leq q \leq 1$. You could define them for any $r$ in that range if you wanted, but you asked for fractional ones.

ahh, point

Also above should have said $k[[A]]$

@Committingtoachallenge: can't say i read a lot of fanfiction, sry.

@Semiclassical I dislike fanfiction, this is the only I have bothered to read, and I read it all the way through in a month(1866 pages worth)

...that's a lot

@Semiclassical Honestly it is worth it, I would never recommend essentially anything to anyone

@MarcGato: all cool with unions?

Guess I missed him.

yep

@user130018 what's free flu?

Freeeee

`Quick question: Does Lusin's Theorem imply that for a real valued measurable function $f$ on $E$. That there is a continuous function $g$ such that $f=g$ almost everywhere on $E$?

It usually presented in the context of "There is a closed set $F$ in $E$ for which $f = g$ on $F$ and $m(E-F) < \varepsilon$

It is almost 3am and this physics homework will not end

But this seems like since this holds for all $0 < \varepsilon$ it should imply that $m(E-F) = 0$

ooo physics

@Semiclassical Stress-strain

ehhh

that could annouying

depending on the level. i'm guessing it's not just hooke's law stuff?

It is hooke's law and youngs modulus and stuff

gotcha. stuff that's more exploited in engineering than in physics

Every last sentence of a paragraph in this book refers to a use in engineering.

It is getting annoying.

would agree with that

The youungs modulus is user by engineers to make sure their materials can withstand sufficient forces.

that's a quote?

shocking revelation

yup

lol

Two more reopen votes on an improved question, if you don't mind.

when i was ta-ing for a pre-med/biology-geared intro physics course, our text was really into applications in that directions

@justabrickinthewall Context added. What a good OP.

so the chapter on stress-strain stuff had a lot of stuff about bones fracturing and the like

It is difficult to find a non-applications directed probability text.

I ended up learning from one of those mathematical methods for engineering books.

if i were looking for non-applications based probability material, i'd probably look for a big book on discrete mathematics

perhaps 'discrete mathematics and ducks'

Hmm. I always wanted to try the measure theory approach

if you know measure theory it's not a bad way to go

What is measure theory about?

@MikeMiller I started learning it over the summer however I had to give the universities books back

i think the most measure theory i'd know is the stuff i've picked up randomly in physics, aka distributions and the like. (which is to say, not really measure theory)

with analysis

distributions are functional analysis, buddy

hence why i say not really :P

@MikeMiller Is there a category of categories?

Nope

@MikeMiller Is it paradoxical?

yes

wait, do you mean 'category of all categories'

Yes.

then everything above holds

there're certainly categories whose objects are categories, just not all of 'em

Yes, of course.

Is this the same argument as the universal set?

Sort of, yes

Can we have a set of all categories?

lord no

Which would include a category of all sets ... ew

here's a reference for the category of all categories thing; ctrl+f "just for reference"

that makes me wonder (which is dangerous, given how piecemeal my knowledge about the paradoxes of set theory is)

Can you map a category to a graph of sorts?

when talking about resolving those paradoxes, i know people invoke the concept of classes rather than sets

it's dangerous to wonder

@Alizter you'd be better off grabbing a book than asking all these questions :P

simple question: for a function, does the preimage of a set require axiom of choice?

what i'm wondering is whether there's some concept which relates to categories in roughly the same way as classes relate to sets

don't believe so

that was to @simonzack

i'll admit, i don't even know if that's a question that's well-formed

@Semiclassical probably but I don't know if anybody thinks about it. NBG (the conservative extension of ZFC that introduces classes) seems to be all people need for category theory

hmm, okay. you see why i'd wonder about that, though, in the context of "a category of categories"

though I'm sure it perturbs people sometimes that they can't have a category of all categories and they often have to work with, say, the category of all small categories

i'm tempted to put that up as a question, albeit a community wiki one since i don't know how to formulate it properly

I guess something like the category of all "countable" categories?

a small category is one whose class of objects is a set and whose class of morphisms is a set

I should get back to physics :P

Can someone actually explain the Axiom of choice to me?

there's lots of good sources if you google

I have read the wiki page before, I didn't really understand it

Is there a thought experiment that makes you 'get' it?

dunno

It seems to say that without AC, you can't take a specific object from an infinite number of sets if they are indistinguishable

But if they are distinguishable you don't need AC

"The Axiom of Choice is this: If you are in an old fashioned sweetie shop, with some infinitely big sweetie jars, then you can take one sweetie from each jar and put them all in another jar. Just take the top sweetie, you say? Well, that requires that the sweeties in each jar are well-ordered..."

i am annouyed that i can't find another way to compute this integral than by the residue theorem. (which of course works, but i'm trying to find another route)

@Semiclassical What is the integral?

$$\frac{2}{\pi}\int_0^{q_0}dq\,\sqrt{\tan^2{q_0}-\tan^2q}=\sqrt{1+\tan^2{q_0}}-1‌​,$$

damn physics notation :P

lol

blah

I guess you could say $\sec q_0-1$

yeah. and moreover you can rearrange it to $\tan(q_0/2)$

by double-angle formulae

hmm

eh residues

it's not terrible by residues, to be fair. you can treat the integral as one-fourth of an integral that winds around the branch cut along [-1,1]

then you just note that's the only branch cut, so you can equate the integral to the sum of the residues (including one at infinity)

keep track of the signs correctly, and it comes out right

@Committingtoachallenge I like this puzzle: en.wikipedia.org/wiki/…

but, meh, i'd like to be able to provide a real-analysis answer

Really tricky and unintuitive solution that requires the axiom of choice.

@Semiclassical Just treat $\Bbb C$ as a special $\Bbb R^2$ the whole time

real enough? ;)

pfft

read the wiki page on ZFC but it's still not that clear to me :( does testing whether an element exists in a set require axiom of choice?

@simonzack Axiom of choice kind of says that everything is a set so we can take things from it

huh

how does it say that at all, @Alizter? :/

@MikeMiller I am blabbering again

the way i read it you can take elements from each set in a set of sets, but i'm not sure when it's applied or not

It is 3:23

I am going to pause my homework for today

slight error earlier, should've said the RH-side can be rearranged to $\tan(q_0)\tan(q_0/2)$. but that just requires pulling $tan(q_0)$ from the LHS

which is a cute statement, but not actually helpful

I got 96% in a maths test today

There was a question on solving trig equations and I was adding 180 to my tan arguments for the range.

180 + 30 = 220 in my head :P

i'm just going to leave this here

phdcomics.com/comics/archive.php?comicid=1356

I am no where near grad school at the moment so is this good or worrying?

@MikeMiller What do you do in Grad school?

whine that there aren't any couches in the grad lounge

@MikeMiller Are you going to do a PhD?

I'm in the PhD program, yeah

Ah

@MikeMiller Can we see your undergrad transcript

@Semiclassical There's a second variable; whether or not you're in front of a board, in which case both drop dramatically

What are you researching on?

No, @user130018

Ok @MikeMiller

@MikeMiller yes, this is quite true

am i right in saying that you can take the union of sets without axiom of choice, but not the cartesian product?

@simonzack You can take the cartesian product; in fact the cartesian product of finitely many sets is always nonempty. But if you take the cartesian product of an infinite number of sets (i.e., a product of sets indexed by some infinite set $S$) one needs some form of the axiom of choice to prove that it will always be nonempty.

We don't have decided topics at the start of the PhD, @Alizter. At present I'd say I want to work on low-dimensional topology.

Ok :)

@MikeMiller That such products are nonempty is precisely the statement of the axiom of choice

@leo I know, it was poorly phrased. Thanks for pointing it out.

@MikeMiller so axiom of choice is not really about constructing sets but about the properties of sets? as you said the product exists regardless, but it's the properties it talks about

They also seem to be talking about definitions on the left, @Semiclassical, and on the right they should just be guessing what 15% is and rounding to the nearest dollar

@MikeMiller no problem :-)

true. i tend to divide by 6 instead

which is slightly more generous but easier for me to do in my head, for whatever reason

wish i could use a theorem prover to learn things like ZFC, would be a lot easier for me to know when to use what

2 hours of sleep. Let us make the most of it.

@ypercube Thanks for that. I hadn't heard of this hat problem, I have read through to the three hat variant so far, and I will read on after time to absorb. Very nice so far

@Alizter I got the most sleep in ages last night, a good 8 hour rest(alarms failed me)

Please help to close this

I has two questions.

Lay 'em on

(1) Let $\Sigma_p$ be the subspace of $\Bbb Q^{\Bbb N}$ comprised of $p$-adically convergent sequences, with $p\le\infty$ (and $\infty$-adic meaning the Euclidean metric). What is $\sum_{p\le\infty}\Sigma_p\subseteq \Bbb Q^{\Bbb N}$? If it's not the whole space, what is an example of a sequence not in it? I feel that partial fraction decomposition (for rationals) could be helpful here.

yikes notation

(2) Suppose $x_n\to x$ in $\Bbb Q$ wrt the $p$-adic metric, $p<\infty$. Define $f(u)=\sum_{n=0}^\infty x_nu^n$ and analytically continue it (in $\Bbb C$); does $f(1)=x$? This is true for $p$-adically convergent geometric series at least, but I doubt it's true in general.

@leo in the SO chatrooms we use [tag:cv-pls], which renders cv-pls

@ZachSaucier what for?

to link to tags?

to request for a question to be closed like you did :P just makes it more noticable/consistent if it happens fairly often

I see

ahh 2 hours of sleep and I end up dreaming about the axiom of choice >:(

how do you deal with PSQs that were still fun to solve?

I don't want to put an answer since it was a PSQ, but it's also not completely trivial

I guess I could just be satisfied in solving it and move on

You could put a comment with a hint, based on your solution.

I suppose I'll do that, but leave it quite general.

I closevoted this as too broad, but it's easier to close as a duplicate: see the comment.

How do you guys pen $\mathbb Z$ ?

two untouching angles?

or do you connect somewhere

@IceBoy The 2nd paper went bad :'( lost 12 marks. That means Il only get 182/200 :'( I wanted atleast 188 :'(

@GBeau a normal Z with a slash.

@anon that's a very interesting question

but i don't think that is true, no.

@Balarka school chilo na ajke ?

@r9m exam in 3 days.

@BalarkaSen aha !! best of luck !! :D

i know i am going to get a 90 at math. but i also know that i am going to get zeroes in everything except math

that's the spirit !! :D

:P

hey @r9m does higher secondary/secondary exam really really matters for getting into good uni?

the thought that i am going to fail in one of those has been haunting me for a while

@BalarkaSen well its always a safe play to have good scores in exams .. these two (secondary and HS) matter a lot in future (ie applying for scholarships, interview, and so on) .. as for uni .. yea I guess it does if you are applying in an uni that does not have a separate screening exam and selections are solely based on final marks

but there are unis with separate screening exams right?

right .. like isi, iisc (ug), ima and the one I'm in :P

right

so those do not bother anything whatsoever about the results in HS/S?

@BalarkaSen nah .. just number jiggas kore chere dey .. ogulo khub ekta matter kore na .. tai bole fail thakle cholbe na .. ar isi er application e bodhoy rule ache 60% minimum thaka dorkar

actually the main problem with me is S. i feel the content of the subjects in HS would me far more logical (i am not gonna take biology)

failing S won't let me get into math on HS =(

@BalarkaSen ami S er 6 mas aage theke seriously gant dite start korechilam :P .. utre gechi :P

haha

bbl (classes) :-)

just before leaving @r9m. how much did you get at bengali on S?

:P

heya @Kaj!

Hey there

I see you've woken up

yep

It's 3:20 AM here, lol

What is S?

well just one of the tiny drawbacks of being in the opposite side of the world, i guess

@KajHansen just a stupid examination you need to sit on at grade 10 which will enable you to pick up your favorite subjects at grade 11

HS is taken at grade 12, usually matters a bit if you are getting into colleges.

Don't complain too much @BalarkaSen. At least you get to pick your favorite subjects. We have to do all subjects throughout high school and even well into undergrad at college.

i can't talk about them any more, i'll get sick

Which essentially results in a very broad, yet unfathomably shallow knowledge of "everything" that people like passing off as being "well-rounded".

right

i call that "name throwing"

Yeah, now you got me going.

"I know the Huygen's principle says X but the only problem is that quantum field theory doesn't support the experiment Y which is a bit interesting" <--- overenthusiastic person name throwing after watching a talk on string theory at discovery channel.

@KajHansen Did you get a peek at p-adics?

Not yet. Bogged down with work of my own for the next few days.

Pfeh PDEs

@BalarkaSen 67 on paper 1 and 86 on paper 2 :P lol

back .. no classes :(

@r9m well the first one sure sucked

67/100 you mean?

@BalarkaSen yes

wait @r9m did you guys use to take grade 9 and grade 10 together in your time?

so what's this paper 1 and paper 2 stuff?

@BalarkaSen no .. we were the first ones to experience the delight of having 9th and 10th grade separated :-) (paper 1 was the prose-poetry stuff and paper 2 was grammar and bangla bhasar itihas :P)

oh thank god

@BalarkaSen if I remember correctly .. there were 40 prose and poetry (20+20) for us .. is it still the same ?

no idea. i am in 9th at the moment.

@BalarkaSen 9th e kota thakbe sob miliye ?

5 + 5 = 10, methinks

i'll suck at bengali anyways

@r9m Had to play the theme song

my priority at the moment are the two sciences and geo, @r9m. i have studied them hard throughout the two months

@BalarkaSen uff !! jaata .. tahole e to pray kono jhamelai nei :P

@BalarkaSen good good !!

@robjohn ;) :D .. to do .. to do ..to do do do doodoo doodoo :P

How is everyone today/tonight?

theme songs !! my favourite :D