Mathematics - 2014-12-23 (page 8 of 10)

skullpatrol

9:00 PM

:)

Studentmath

@Lord the latter two turn to $q/p$

I think that's right, that's was where I was heading to at least with my hints

Lord_Farin

@Studentmath Ok, I was expecting worse :).

Ted Shifrin

Something bothers me. The exponent $n$ doesn't seem right.

Oh right, I keep forgetting the problem.

Lord_Farin

@TedShifrin $n$ from "$n$ subsets". We first pick $i$ elements as an overlap, then $n$ times $q - i$ other elements, and lastly divide over selecting $n$ times a $q$-subset from $p$.

Studentmath

The probability that each subset has specific $i$ items in it is idendepent. There are $n$ such subsets at any case

@Ted lol

Ted Shifrin

9:04 PM

@Studentmath: I hereby abdicate my probability knowledge. It's all for you in the future. I'm sticking with differential geometry :P

Studentmath

@Lord post it as answer mayhaps? He was really there, just simplified the latter two to $q/p$ in the case of $i=1$ which got him a bit further. Sometimes simplifying isn't good, especially in inclusion-exclusion :P

@Ted too much responsibility..

Lord_Farin

@TedShifrin Oh, DiffGeo. That's one subject I have admitted not to be gifted for :P.

Ted Shifrin

Well, @Lord, I won't take it personally.

Studentmath

@Ted oh, I started to read your class-notes (just the first couple of pages)

Lord_Farin

@TedShifrin A big relief :P.

Ted Shifrin

9:07 PM

That doesn't mean I won't take other things personally, @Lord :P

Is that good or bad, @Studentmath? :D

Lord_Farin

@TedShifrin Haha, I won't get too comfy then.

Studentmath

@Ted You tell me the moment I start asking questions in that :P

Ted Shifrin

I hope I'm slightly more competent on that, @Studentmath :D

Or else I really should have retired already ...

I guess our new mods have disappeared to learn their authorities.

Studentmath

Just like after real elections, we won't see them ever again

Jasper Loy

I am thinking that the site can still function without any mods.

Ted Shifrin

9:10 PM

Well, I doubt that.

Do you recall the personal attacks on me by a certain person, @Jasper? I think not.

Jasper Loy

@TedShifrin Oh, Mr Rene, lol.

Studentmath

@Jasper I know many people would leave this website had it not been for some mod's work

Ted Shifrin

Yes, I suspect his mental state is far worse than yours, @Jasper.

@Studentmath: Some claim they have left because of mods. It's not a perfect world.

hi @Henning

Jasper Loy

@TedShifrin Well, I don't know about that. He could just be evil and not sick. There is a difference.

Studentmath

@Ted that's also true..

Henning Makholm

9:11 PM

@TedShifrin Um, hi.

Jasper Loy

@HenningMakholm I am still here, after deleting so many accounts, lol.

Ted Shifrin

Didn't mean to broadside you :)

Henning Makholm

@JasperLoy I see. Still blue.

Ted Shifrin

He was bronzed last week, @Henning.

Mike Miller

hi

Henning Makholm

9:13 PM

Smurfs > statues.

Ted Shifrin

You back from cavorting with the new mods, @Mike?

Daniel Rust

I've received so many congratulatory pings :P

Mike Miller

No, there's a dog on my lap I was paying attention to.

Someone just got home so he left.

He probably thinks he's gonna get fed.

Ted Shifrin

LOL

It's awfully early for dinnertime ... even for a dog.

Jasper Loy

@DanielRust Do you get pinged if they use DanielFischer?

Ted Shifrin

9:13 PM

@DanielRust: Now you appreciate my trying to take so much care not to misping you :P

Daniel Rust

haha yes @Ted

Mike Miller

Yes, @Ted, he's not really going to get fed. I wish I still had his naïveté.

Don Larynx

Hey all. I am making progress on Problem Euler 494 :O

skullpatrol

Welcome @HenningMakholm

Back :-)

Lord_Farin

@DonLarynx 494...! It's been so long since I've been there; I've witnessed the release of 200 :P.

Don Larynx

9:16 PM

Wow. What made you stop? @Lord

I think PE will help me in my cognitive abilities

Lord_Farin

@DonLarynx Lack of time, mostly. Other interests were assigned higher priorities.

@DonLarynx Definitely. It aided mine.

Mike Miller

Well, @Ted, Lawson's going into extreme detail about the representation theory of Clifford algebras now. I have not yet been convinced to care, even though I know I'm supposed to.

Don Larynx

Interests? @Lord

Jasper Loy

@MikeMiller What course is this?

Mike Miller

It is Winter vacation; I am not in any courses.

Lord_Farin

9:18 PM

@DonLarynx My studies, MSE, games. It just waned, I guess.

Mike Miller

The book I'm reading is Lawson's "Spin geometry".

evinda

Hello!!! Could you explain me why $n$ is the nimimum element of a set $X \subset \omega$ iff $n \in X \wedge (n \cap X=\varnothing)$ ?

Don Larynx

@Lord: I embarked on PE because my interests shifted to computer programming, so I understand. Games?

Jasper Loy

@MikeMiller I can never make the table tennis ball spin.

Studentmath

@Jasper it's in the swing

Huy

9:19 PM

@JasperLoy: I can teach you. I used to play professionally.

skullpatrol

:O really @Huy?

Lord_Farin

@DonLarynx Yes?

Don Larynx

@Lord: What games do you speak of?

Lord_Farin

@DonLarynx Computer games :).

Don Larynx

Oh

been there done that :p

Mike Miller

9:20 PM

Yes, @skullpatrol. Huy's real name is Pete Sampras.

Ted Shifrin

Wrong sport, @Mike.

Mike Miller

You're my hero, @Huy.

skullpatrol

Lol

Huy

@MikeMiller: I wish I could tell you the same.

Ted Shifrin

Well, I just crossed another K threshold, @Mike.

Mike Miller

9:21 PM

I'm still down in the 7s... and I'm going to lose 100 on a bounty I give out later.

Ted Shifrin

Well, I made someone very happy with today's answer :)

Mike Miller

I need to go answer a hundred questions and make it past the big 10.

Lord_Farin

@DonLarynx :)

Ted Shifrin

No, I think you're better off being a graduate student, @Mike.

Studentmath

@Mike do you have the Hannukah hat? Since it ends in... well, supposedly today

Mike Miller

9:22 PM

@Ted Convince me to care about the representation theory of Clifford algebras.

evinda

Could I ask someone something about set theory?

Mike Miller

I've everything I could have by now, @Studentmath

Ted Shifrin

@Mike: Not I. I've never learned enough representation theory.

Studentmath

And I dare asking

Mike Miller

OK, then I won't.

Henning Makholm

9:23 PM

@evinda Do you remember each natural number is the set of all smaller naturals?

Mike Miller

I have to admit it's one of those taste barriers I never got over.

Ted Shifrin

Although I did ask someone else the question I gave you months ago ... about giving $T(G/H)$ as an associated bundle.

Mike Miller

That was nice.

evinda

@HenningMakholm Yes...

@HenningMakholm What do we get from that?

Ted Shifrin

I wonder if @Pedro and @DanielF will be too good for us, now :)

Henning Makholm

9:24 PM

@evinda So $n\in X$ means that, um, $n$ is in $X$, and $n\cap X=\emptyset$ means that $X$ doesn't contain anything smaller than $n$.

Lord_Farin

@Studentmath @TedShifrin @A.E My attempt. 't Feels good to write MSE answers once more :).

Mike Miller

Pedro's already too good for me. I said hi on facebook post-election and he told me he was leaving.

Lord_Farin

I will have to go now, see you all around!

Ted Shifrin

We're glad to have you back, @Lord. Take care.

skullpatrol

Later pal @Lord_Farin

Lord_Farin

9:25 PM

@TedShifrin Thanks for the warm welcome, bye :).

Daniel Fischer

@TedShifrin At the moment, we're busy learning new stuff. So I think we'll be around here a little less in the next days.

Ted Shifrin

Should I cook you dinner and send it over, @DanielF? :D

Jasper Loy

@DanielFischer I see that your name is in blue.

Ted Shifrin

Yup, he's now officially officious :)

evinda

@HenningMakholm Could you explain me why this is the condition for a natural number to be minimum?

Jasper Loy

9:26 PM

To the new mods, it's not too late to retire, lol.

Mike Miller

Did you learn any nasty secrets about me, @DanielF?

Daniel Fischer

@TedShifrin With the speed of mail, I think that's a bit much of a gamble.

Henning Makholm

@evinda What else would you want "minimum" to mean? It is a member of X, and nothing smaller is a member of X?

Ted Shifrin

Especially around Christmastime, @DanielF :)

skullpatrol

Did they take away your picture @DanielFischer?

Daniel Fischer

9:27 PM

@MikeMiller Yes. You can't cook. Ted told me.

N3buchadnezzar

@DanielFischer I guess he can cook, it's just the result which could be better

Daniel Fischer

@skullpatrol I had to change my chat-parent from SO to Mathematics to get into the mod chat room.

skullpatrol

Icic

Jasper Loy

I have never cooked anything in my life.

Don Larynx

@Jasper: The first time I tasted bacon was at 18, in college

N3buchadnezzar

9:28 PM

@DanielFischer Is there a mod chatroom? Do tell dirty jokes and plan trips to las vegas?

evinda

@HenningMakholm Could you maybe explain it further to me? I am confused now... :/

Jasper Loy

@DonLarynx The first time I ate an oyster, I spit it out.

Mike Miller

That's not true, @DanielF. I know how to make chicken, brussel sprouts, and pasta sauce.

Don Larynx

@Jasper: I had to hold (the oyster) in to not look doofy in front of the chess team.

Huy

Which pasta sauce, @MikeMiller?

Daniel Fischer

9:29 PM

@N3buchadnezzar I haven't been there much yet, so far, it was all "hello" and tips for what moderators should know. The dirty jokes will surely come later.

N3buchadnezzar

$$ \int_0^\infty 1 - \left( 1 - e^{-a w} \right)^n \,\mathrm{d}w = \frac{1}{a} \sum_{i = 1}^n \frac{1}{i } = \frac{H_n}{a} $$

Mike Miller

Meat sauce. I can't make pesto.

Ted Shifrin

Brussels sprouts are easy to ruin ...

Khallil Benyattou

I'm liking the blue name, @DanielFischer! ^_^

Mike Miller

I had the best pesto of my life two days ago.

skullpatrol

9:29 PM

How many members are in the smallest set @evinda?

Ted Shifrin

Pesto is easy, @Mike. I see we'll need lessons.

Jasper Loy

@DonLarynx I used to play chess too, but stopped after elementary school, lol.

Mike Miller

But is the best pesto of my life easy, @Ted?

I won't settle for less.

Ted Shifrin

It's good to set the bar high.

Mike Miller

I see grades are still not submitted yet. Hmph.

Jasper Loy

9:30 PM

I think the easiest dish to cook is salmon. Just add hot water and boil it and you are done.

evinda

@skullpatrol In the smallest set generally?

Huy

Good pesto is rather difficult to make, imo.

N3buchadnezzar

@TedShifrin That excludes the dwarves.

skullpatrol

@evinda yes

evinda

@skullpatrol $\varnothing$ contains the less elements...

Don Larynx

9:31 PM

@Jasp: I played chess professionally.

Jasper Loy

@Huy I think imo is not needed, as everything one says is his opinion.

Ted Shifrin

No, @Huy, like most good Italian food, it's about the quality of the ingredients. Very few needed, but one needs good fresh basil and top quality olive oil. Garlic, parmeggiano reggiano, pignoli nuts. Done.

Salt and pepper, of course.

I might put in a bit of cilantro or arugula to be different sometimes.

Huy

@TedShifrin: Then, my ingredients were mostly of bad quality, because my pesto is never as good as I imagine it to be like.

Jasper Loy

My favourite Italian food is spaghetti carbonara.

Mike Miller

That and being willing to start from scratch... the way most people ruin their meat sauce is by starting from pre-made stuff.

Henning Makholm

9:32 PM

@evinda Okay, what is your definition of "mimimum element"?

Ted Shifrin

@Jasper: It's not good for me, but I love it. I make it occasionally.

@Huy. Really good olive oil is expensive.

Don Larynx

Mine is pizza

Jasper Loy

@TedShifrin Why is it not good for you, too fatty?

Ted Shifrin

We're about to be flooded and have tornados here. So if I disappear, send a boat for me.

Huy

@TedShifrin: What does expensive mean, in terms of olive oil?

Mike Miller

9:33 PM

Jasper called you a fatty, @Ted

N3buchadnezzar

@JasperLoy that sentence becomes much funnier if you remove too-

Ted Shifrin

Yes, @Jasper ... bacon and butter and eggs.

evinda

@HenningMakholm Let $(A, \leq)$ be an ordered set.
We say that $a \in A$ is:
mimimum when $(\forall x \in A) a \leq x$

Ted Shifrin

smacks @N3B and puts him on ignore

Mike Miller

I said it first. Shouldn't it be me?

skullpatrol

9:34 PM

:O

Henning Makholm

@evinda Right, and if you say $n$ and $X$ instead of $a$ and $A$, you get ...

You, too, @Mike.

Finally!

:(

@evinda ... $n\in X$ which you already have, and $\forall y\in X: n\leq y$.

Don Larynx

9:36 PM

Who here is a fan of Tycho?

Henning Makholm

But the ordinals are totally ordered so $n\leq y$ is the same as $y\not < n$. And $<$ on ordinals is the same as $\in$, so $n\leq y$ is the same as $y\notin n$.

So for the second condition we have $\forall y\in X: y\notin n$, which says exactly that $X\cap n=\varnothing$.

skullpatrol

Got it^ @evinda?

Take your time and think about it :-)

Don Larynx

@evinda $a$ is a minimum of the set $A$

induction proofs are so awesome

and so are RAA's

Khallil Benyattou

What's an RAA, @DonLarynx?

Don Larynx

Reductio ad absorptum, also known as contradiction

Chris's sis

9:47 PM

Let $(a_n)_{n\ge 1}$ a strictly increasing sequence in natural numbers. Prove that

$$\lim_{n\to\infty} \left(\frac{1}{a_1} +\frac{1}{a_1\cdot a_2}+\cdots + \frac{1}{a_1\cdot a_2\cdots a_n}\right)\in \mathbb R \setminus \mathbb Q$$

Don Larynx

@Chris'ssis HINT: $\sum_1^{\infty}\frac{1}{n!}$

Khallil Benyattou

I thought it was reductio ad absurdum, @DonLarynx.

Don Larynx

@Chris'ssis the sum is $e-1$, which is in $\Bbb{R}$ and not in $\Bbb{Q}$

I think you should stick to your integrals from level 9 in Dante :)

Chris's sis

@DonLarynx I don't think that is relevant for proving this fact, maybe for proving the fact the limit exists. And yes, as @HenningMakholm said, you only proved a particular case.

Henning Makholm

@DonLarynx Only if $a_n=n+1$.

@Chris'ssis Suppose the sum is p/q. What happens the first time a_n is larger than q?

Don Larynx

9:52 PM

Let $q \in \Bbb{Q}$. then the sequence can be written in the form $\sum_1^{\infty}\frac{1}{{q_n}{a_1}}$

Chris's sis

@HenningMakholm I know how to do it. I posted it here for you (it's just for fun).

Don Larynx

Then we get $\frac{1}{a_1}\sum_1^{\infty}\frac{1}{q_n}$

Henning Makholm

@Chris'ssis Oh, sorry.