Mathematics - 2011-12-13 (page 1 of 3)

Gigili

00:10

@Templar I guess it should be $2^n-2$

Templar

what?

Gigili

I thought you asked a question.

Templar

so what do you guess?

that I need to sum 2^n-2?

Gigili

The probability of getting two heads is (1/2)^2 out of (1/2)^n

But I'm not sure

Templar

..

probability is 0.5^n*(n-1) because you throw it until i get 2 heads and then I stop

and i am asking not about probability but about expected value

Gigili

00:25

E(x), didn't read carefully. Anyway, your probability doesn't seem true to me.

Templar

it's true

have you understood question correctly?

if n is 2 then you can get HH

so probability is 0.5^2

if n is 3 you can get THH or HTH

so probability 0.5^3*2

if n is 4 you can get:
TTHH
THTH
HTTH

so probability 0.5^4*3

it's always same format 0.5^n*(n-1)

and you can see that last throw you always get head

Gigili

00:39

Got it, right. it's (n-1)/2^n ..

The expected value, would be sum of [n* (that)] from 2 to infinity

What's the problem?

Templar

we didn't learn how to do such sum

so i am asking is there any way to get it, maybe with Poisson distribution?

Gigili

You can separate it

Sum of n(n-1)

And sum of 2^n

Can you do it now?

Templar

no, i know how to do sum only if it's infinity geometric progression or if it's arithmetic progression but not to infinity

Gigili

The second one is geometric progression

The point is its first sentence is 2^2

See these: 1, 2 and 3

Wait, shouldn't it be from 2 to n?

Templar

00:54

yeah from 2 to n

yeah i know one of them is geometric, but i mean

i can solve geometric progression only if it goes down

for example 1+1/2+1/4+1/8+...

The Chaz

That's what you have, more or less, Templar...

Isn't the "2^n" in the denominator ??!?!

Templar

yeah it is

then i can do it

but what about n(n-1)

The Chaz

well 1 + 2 + 3 + ... + n = n(n + 1)/2

Maybe there's a relation... I just stumbled upon your discussion so I'm not sure of the context.

Templar

I have coin, and want to get 2 heads exactly. I will throw it until this condition is met. What is expected number of tries for this condition?
I know that it would be sum from 2 to infinity P(X=n)*n=0.5^n*n(-1)*n
however I don't have an idea how to solve that sum
is it possible to get expected value using Poisson distribution?

Gigili

@TheChaz You cannot do it

The Chaz

01:00

@Gigili Maybe not, but I was just notified by a snowman!

Templar

n(n-1) from 2 to infinity will be infinity isn't it?

Gigili

@TheChaz I mean, we got that by separating the sum to denominator and numerator ..

The Chaz

Are we talking about the sum (n ≥ 2) of (n(n -1))/2^n ?

Gigili

Yes

Templar

yes

The Chaz

01:05

Then we need more advanced techniques!

Templar

but I haven't learnt advanced techniques

The Chaz

W|A says it's 4. I'll try to refresh myself on said techniques.

Templar

yeah i know answer myself

but don't know how to do

t.b.

@Gigili Did you see this? That was due to a question of yours.

@JonasTeuwen you got to be open minded :)

Gigili

@tb Nope, I didn't. Someone got offended by me =\

Hello, by the way.

Luckily, I asked before deletion.

t.b.

01:18

@Gigili I don't think he got offended. But it is somewhat annoying if a thread is closed or deleted for which you were about to write an answer.

Happened a few times to me, too.

And hello, you too, by the way :)

Gigili

@tb It is indeed annoying, in that situation I'd kill the OP. =)

Zhen Lin

@tb Goedecke and Van der Linden (2007) "A comparison theorem for simplicial resolutions" gives a definition of exact (2.13) for augmented simplicial objects. Unfortunately it's not as simple as I had hoped...

t.b.

@ZhenLin But it does look like something natural, no?

Gigili

@Templar Why don't you ask a question?

Zhen Lin

@tb Yes. It looks like what I had in mind (forks are exact), except it's demanded for every fork at once at each level.

t.b.

01:30

@Gigili Well, I might just slap the OP on the fingers. Which for you amounts to a smack in the face(s) :)

J. M.

Plural? :D

Gigili

@tb Oops sorry, I meant @Templar

Templar

because they will show me how to solve it that way which i haven't learned

t.b.

@JM sure, judging from the Gravatar Gigili has at least three of 'em

J. M.

Ah. That was slow of me... :D

t.b.

01:35

@ZhenLin Did you look at Tierney-Vogel?

Gigili

@Templar You can ask them to use what you have learned, if not possible so you can stop trying to solve it. Actually I didn't come up with a solution myself, I'm curious to know the solution.

Zhen Lin

I am about to.

Gigili

@tb What you see less than half of it ..

Zhen Lin

From the MathReview: "satisfy an obvious acyclicity property"... what is acyclicity, really, and why is it called that?

t.b.

@ZhenLin exactness. All cycles are boundaries $\approx$ no true cycles.

Zhen Lin

01:43

Ah. I suspected as much. But I also see the word "contractible" floating around, and I would have guessed that means $H_i = 0$ for $i > 0$.

t.b.

Well, contractible is stronger: you have a contracting homotopy.

Zhen Lin

Ah, of course.

t.b.

See also this thread for acyclic vs. exact.

Gigili

Good night.

t.b.

Good night, Gigili

Zhen Lin

01:46

Goodnight.

@tb Thanks. I think of them being the same, since truncating the complex after the 0-th term means that $H_0$ no longer makes sense.

Templar

@Gigili math.stackexchange.com/questions/90987/… posted it

but u are off

t.b.

and I was just about to ask: where's Srivatsan? Think of the devil...

Srivatsan

"@all: You guys are awesome. I had only one simple question and I got so much worthful and interesting answers. Thanks! I have never seen this phenomenon more often as I have seen it here on math.stackexchange. Maybe it is a mathematician-related thing... :-)" - Aufwind

t.b.

Hi Srivatsan

Srivatsan

Good to hear such wonderful comments :=)

hi tb

J. M.

01:50

@Srivatsan Where is this? :)

t.b.

@JM here

Srivatsan

@JM This question: math.stackexchange.com/questions/90629/…

J. M.

(It's fun to do math; knowing that it helps people is a nice cherry on top.)

Srivatsan

Um, my bounty expires in 5 hours. [I think I can add a 1 day grace period.] Do I get to lose my points even if there are no answers? =/

t.b.

@Srivatsan You know what "Aufwind" means, literally? Upwind/updraft. "Aufwind haben" means "to be on a tear"...

@Srivatsan Yes, that's the rules of the game it seems.

Srivatsan

01:54

@tb I didn't know. But why did you say it now? :)

t.b.

@Srivatsan I never intended it to be relevant...

Just an aside.

Srivatsan

@tb Ok =) .

Gigili

@Templar I've got an idea, why it's sum and not integral? For n>=2 it should be integral, no?

Srivatsan

What's up with this comment: chat.stackexchange.com/transcript/message/2685494#2685494. Why is it pinned to the wall?

Gigili

Umm, doesn't make a big difference .. I'm sure I'll dream of your question tonight!

t.b.

01:59

@Srivatsan I guess the train of thought was: Aufwind's comment gave you Aufwind, or something to that extent.

As for the "No". No clue. But I gave up a long time ago to make sense of much of what Asaf says, especially what's pinned there.

5

J. M.

Now *that*, I'm starring...

Srivatsan

@tb Well, I am just happy for the community -- I am sure JM's and Mike's comments deserve most of the compliments. That it's under my answer is just the cherry on top -- to reuse JM's words.

J. M.

@Srivatsan Well, you did much of the grunt work; all I did was to say that those entities had names. ;)

Srivatsan

Oh yes, now I must go and fix that answer.

J. M.

...and on a more selfish note: at least now it will be part of the search results when I type site:math.stackexchange.com bell dobinski in Google. Very convenient for me! ;)

Templar

02:05

@Gigili it's sum because it's the definition of expected value

Srivatsan

@JM To be honest, it will be convenient.

t.b.

What does "WTS" mean?

Templar

and it's discrete probability problem

Srivatsan

@tb want to show?

Lame, but fits.

t.b.

@Srivatsan Oh, yes. I should have remembered this

I hate acronyms.

(But I guess I already mentioned that)

Srivatsan

02:20

tb: I forgot about tagging math.stackexchange.com/questions/22265/… [and didn't appear online for almost all of today].

t.b.

@Srivatsan Well, I took care of it, following suggestions of Henning's. (My 10 pts of yesterday were due to that :))

Srivatsan

I thought [matrices] and also [ring-theory] tags are somewhat relevant. (The algorithm works for matrices over an arbitrary commutative ring.)

Although ring-theory is a somewhat iffy suggestion.

I just learned about the complexity tag. =)

t.b.

Well, [ring-theory] is more for *non*-commutative rings, maybe matrices would be okay. I was more concerned with replacing [soft-question] and [analysis] by more relevant tags. If you want to add [matrices] that's fine with me.

Srivatsan

@tb Done, thanks.

t.b.

@Srivatsan I'm in good company :)

Srivatsan

02:45

I don't understand these edits: math.stackexchange.com/posts/21367/revisions. Needed?

t.b.

@Srivatsan I don't think so, on the contrary, in fact. I think it has to do with this.

Srivatsan

In conjunction with this edit, it looks quite funny. After many months, U becomes $U$, which after several more months becomes $\operatorname{U}$ .

Zhen Lin

Hmmm, should I mention that Grothendieck worked in functional analysis first? :p

leo

hi everybody

Srivatsan

@ZhenLin Well, what does that say? =)

Hi Leo

leo

02:50

can you help me. I don't understand how the general case can be deduced at the end of the answer here: math.stackexchange.com/q/90303/8271

t.b.

@ZhenLin well, if you want. It's one of those completely unanswerable questions anyway...

leo

how can I put links here?

Srivatsan

[google](http://google.com) gives you google

leo

... there exist a FAQ.

ha thanks

t.b.

leo use [text](http://...). I had quite a chuckle at your comment to Didier's answer :D

leo

02:54

(: does the series do that?

Srivatsan

@leo Is that a genuine question or a joke?

t.b.

@leo The formatting is summarized here. By the way, do you have the MathJaX bookmark installed

leo

what is the limits the Iverson notation

Srivatsan

The series is not well-defined, of course.

leo

I dare to ask: why? :)

Srivatsan

02:57

I have to leave now. See you all.

leo

See you Srivatsan

Zhen Lin

More evidence for my suspicion...

t.b.

@ZhenLin Qing Liu?

Zhen Lin

Indeed.

J. M.

Coincidentally, Georges used to use "elgeorges", so they're somewhat in the same boat... :)

t.b.

03:01

@leo Do you know the law of the unconscious statistician?

Zhen Lin

@tb I think there are more people who know it than there are people who know there's something non-trivial to be proven...

leo

no. go to see

t.b.

@ZhenLin I'm quite sure of that. By the way, donkey kong seems to share your suspicion

Zhen Lin

Also, the edited question seems to have invalidated Georges's nice answer.

t.b.

@ZhenLin well, but now I'm confused

Zhen Lin

03:14

@tb I'm confused about your confusion.

leo

$\int f$

t.b.

Does it work?

leo

$\int \sum xyzI^123$

:)

yes

t.b.

wonderful!

robjohn

hello all!

leo

03:22

hello

t.b.

hi robjohn!

J. M.

Hey rob. Your bookmarklet has another satisfied customer!

robjohn

@leo I see you have installed the MathJax bookmark.

@JM cool!

leo

yes

Zhen Lin

$$(1 - \textstyle\int)^{-1} 1 = \exp$$

t.b.

03:23

@robjohn It's intriguing that almost all people use \int f for testing purposes :)

robjohn

@tb how are things?

t.b.

Oh, well, I'm in hibernation mode...

leo

I haven't reasons to that choose

J. M.

@leo It's still a bit peculiar that a lot of people test with \int, though. ;)

leo

yep, I think that

perhaps because $\int$ is nice

robjohn

03:25

@tb tired?

t.b.

@robjohn tired of winter mostly. But yes, I haven't caught too much sleep, lately.

robjohn

@tb I like the snowman ping icon :-)

t.b.

What do you mean?

Zhen Lin

Before long we'll have snowflakes falling across the screen...

leo

@tb why the series in my comment are not defined rigorously?

robjohn

03:27

@ZhenLin Ack! don't say that... someone might hear.

leo

I mean, a rigorous explanation

just curiosity

t.b.

Well, how do you define today, monday, friday?

For those who joined later, leo is talking about his comment to Didier's answer here

leo

just as we expect

t.b.

@robjohn I still get the same old ping notification

Zhen Lin

@tb You probably need to refresh.

@leo: While your series makes sense pointwise (i.e. when evaluated termwise on a particular day), it's not clear that it makes any sense uniformly...

t.b.

03:31

Can someone ping me now?

J. M.

@tb Like this?

leo

@tb

t.b.

@JM oooh, thanks! Now I get that, too

leo

but is the constant sequence $0$ in the other days

J. M.

@leo Well, since Iverson brackets are zero when false, your "series" converges except on Fridays.

Still, I don't quite see why one would use the brackets in that way.

robjohn

03:33

Does $[\text{today is monday}]$ evaluate to $1$ on Monday?

leo

yes, that's my point

and I agree with J.M.

robjohn

Okay, I didn't get that exactly. Now I do.

J. M.

@robjohn According to the strictest interpretation of Iverson, yes.

t.b.

I'm somewhat tempted to upvote four answers by mixedmath and downvote two of them

robjohn

@JM I haven't ever used Iverson brackets before.

t.b.

03:35

@JM So Iverson's not case sensitive?

leo

:)

J. M.

@robjohn Once you've gotten used to 'em, you'll wonder how you got along without 'em...

@tb IIRC, no.

leo

but, what's Iverson brakets, it is a function?

robjohn

@tb Is truth case sensitive?

J. M.

@leo It's a construct, used to build piecewise functions (among other things).

leo

03:36

otherwise allows things like that

t.b.

@robjohn See here for a convincing argument

leo

yes, but that allows thing like my series

J. M.

For the Iverson virgins among you: see this.

@leo e.g. I'd say there's no difference between $[p\text{is prime}]$ and $[p\in\mathbb P]$.

or $\mathrm{sign}(x)$ and $[x > 0]-[x < 0]$.

leo

the J.M.'s link says, "$[S]$, were $S$ is a mathematical expression"

J. M.

@leo Yes, what about it? Did you also see the Knuth paper I linked to?

t.b.

03:41

I for one would argue that today is Monday is not a mathematical expression, so the bracket isn't defined. I guess that was Srivatsan's point.

leo

is "today is Monday" a mathematical expression

why not it is

J. M.

Sidestepping that argument: I haven't seen anybody use brackets like that, so far. Except you.

leo

i'm reading @JM

robjohn

@tb But it has a truth value...

leo

that's my point

t.b.

03:43

@robjohn well, I'm feeling comfortable with that abus de notation

Zhen Lin

We could define a domain { Monday, Tuesday, ... } and make these into honest functions. The domain is even finite, so we can make it into a measure space...

robjohn

@tb According to the Wikipedia trail, it doesn't appear to be an abuse.

J. M.

In any event: I'm not sure I'd like to see a mathematical expression whose values depend on when you saw them...

That's my source of discomfort there, even if the validity is defensible.

leo

what about things like [the door is open]

...

J. M.

That's a bit vague. Which door? (I haven't seen Iverson extended to three-valued logic.)

leo

03:47

you can construct a series that converges if the Riemann' Hypothesis is trow and diverges otherwise

robjohn

@JM people felt uncomfortable with $\sqrt{-1}$, too

J. M.

@leo I think you don't need Iverson to build those sorts of series.

leo

that's a result that depend directly of RH

yes, i know, change the truthness of RH by anyother conjecture

J. M.

@robjohn Yeah, I guess I'm biased. :) (On that note, I was introduced to the Argand plane early on as a kid... :) so there was no discomfort for me.)

robjohn

@JM now you get to experience discomfort ;-)

J. M.

03:49

@leo No I mean there are indeed series whose convergence is true if the hypothesis is true.

No need to invoke Iverson there.

(I think it was mentioned in MO somewhere...)

leo

yes, i know. My point is you can force, for example, the convergence of a series to depends of the truth value of any conjecture

I don't know how to say that I'm thinking. I'm not native speaker

robjohn

replace Riemann Hypothesis with Continuum Hypothesis.

t.b.

@JM Do you mean this?

J. M.

@tb That's it! Thanks!

leo

yes like that @robjohn

robjohn

03:57

@JM: you have had the same torus for quite a while.

J. M.

@robjohn Yeah, I still haven't decided on my "Christmas torus"...

...but I'll think of something.

robjohn

I have a candy cane torus already rendered :-)

Srivatsan

@JM But it'll be torus, eh?

J. M.

@rob: The LifeSaver you showed me a few weeks ago?

@Srivatsan Or a variation thereof. ;)

robjohn

@JM Here: i.sstatic.net/Le0Ou.png

t.b.

04:01

@JM: Segue: do you know how to find a stable link to this?

robjohn

@JM That's the thinner one. The thicker one looks like a life saver.

J. M.

@tb Yeesh, the Japanese sites are pretty notorious for not giving them. Let me check...

leo

well Knuth says things like "arbitrary entities", "any relation", so it seems that there isn't problem with [today is monday]

oh, today is monday! :)

t.b.

Not for me :)

J. M.

@leo Not where I am, so... :)

leo

04:06

well 0 for you, guys

Srivatsan

And I thought mathematical propositions are meant to be eternal. =)

J. M.

@Srivatsan Precisely my discomfort with leo's statement...

robjohn

@Srivatsan unlike romantic proposals...

leo

you mean, invariant by physical time

robjohn

off to pick up some dinner. bbl

Srivatsan

04:08

Hey, hi robj. // bye robj.

robjohn

@Srivatsan Hello. Nice to see you again.

t.b.

bye robjohn

robjohn

It's been since this morning :-)

leo

@Srivatsan that's make sense

robjohn

be back in a bit

t.b.

04:09

bbiab?

Zhen Lin

Pre-war Japanese orthography looks rather more scary... 「或函數方程式ニ就テ」

@tb I think the problem is that this article has not been allocated a JOI.

t.b.

JOI = Japanese DOI?

Zhen Lin

Presumably.

Srivatsan

That's rather strange. It's as if the "D" in DOI stands for Denmark.

J. M.

@tb Nothing. I tried most of the Japanese alternates I know. :(

@Srivatsan The "Digital" world is certainly different... ;)

Zhen Lin

04:15

If you inspect the source code there's a URL: joi.jlc.jst.go.jp/JST.Journalarchive/tmj1911/…

It doesn't work, of course.

J. M.

It's too bad only the "Second Series" is in Project Euclid...

Zhen Lin

Hmmm... it seems that the website itself is a little confused. The print ISSN given is for the second series.

leo

@JM Adriano Garsia says: $$\chi(\mathcal{A})=\begin{cases} 1 &\text{if } \mathcal{A}\text{is true}\\ 0&\text{if not}\end{cases}$$"were $\mathcal{A}$ is any statement whatever"

t.b.

leo you miss an \end{cases}

leo

@tb yep, thanks :)

Srivatsan

04:23

@leo Well, we disagree not in the usage of $\chi$ or the Iverson notation. We disagree whether [this is Monday] is a valid statement or not.

t.b.

@JM Thanks, I tried tinkering with the openurl instructions they provide, but it doesn't seem to work. So I'll try again in a few months.

Thanks everyone for looking into this!

leo

@Srivatsan I see

J. M.

Characteristic functions (what Garsia was referring to) show up a lot in measures, no?

Srivatsan

@JM - Who is Garsia? [I don't get what you're alluding to? Are you referring to some post?]

J. M.

@Sri summarized it well I think.

t.b.

04:26

@JM Well, they're the basic ingredient for defining the Lebesgue integral.

J. M.

@Srivatsan this

Srivatsan

@JM Oh =)

J. M.

@tb One of these days I really should sit down and understand the Lebesgue way of doing things... :)

Srivatsan

This sounds like it should be rather obvious.

Zhen Lin

@Srivatsan: I imagine that the main ingredient would be to show that $H$ acts properly-discontinuously on $G$.

t.b.

04:32

@ZhenLin That's overkill but perfectly obvious -- $H$ is a closed subgroup after all.

Zhen Lin

@tb: I can't say; I learned algebraic topology but not topological groups!

Actually, come to think of it, I've never studied topological X for X anything other than spaces...

t.b.

@ZhenLin set X = "set"? :) Anyway, the problem with the question is that OP doesn't specify the definitions used. As soon as those are written down, it must be a straightforward exercise in plugging in the definitions.

Ilya

hi, the sleepless Swiss )

hi, Zhen

Zhen Lin

@tb: Why yes, a topological space is a model of the empty theory in the category of topological spaces. But so is any object in any category... :p

Srivatsan

@Ilya hi, vacationers in America =)

t.b.

04:39

@Ilya sleepless in Zurich, yes...

Ilya

@Srivatsan hi ) how are you?

Zhen Lin

It's nearly 1pm here...

Srivatsan

Yes, I leave the chatroom, tb is here. I enter, and tb is still here. =)

@Ilya I am good. How do you like US?

leo

@ZhenLin Where are you?

Zhen Lin

@leo I'm at home. :p

Ilya

04:41

@Srivatsan maybe there are two on the other side of the Internet )

J. M.

For context: I'm eating lunch...

Ilya

@Srivatsan I guess I had too high expectations - but people from US I meet here are not happy with this place as well

@JM hi, enjoy

Srivatsan

@Ilya this place refers to the country? Or Florida?

J. M.

@Ilya Why "not happy"?

Ilya

@Srivatsan Hilton Bonnet Creek

@JM 1. prices are incredible 2. food is bad 3. service is odd 4. $\dots$

J. M.

04:43

@Ilya Ah, snooty hotel staff? Yecch...

Zhen Lin

@Ilya: It's a Hilton hotel. How could it ever be cheap?

J. M.

(I am however surprised at the "food is bad" part.)

What Zhen said.

Srivatsan

@JM Really? I think the Americans are wonderful in general when it comes to treating others.

Ilya

@ZhenLin the room price is like a very simple room in the Netherlands. everything else is higher

J. M.

@Srivatsan In most places, yes. I have however seen snooty hotel staff.

Ilya

04:45

@Srivatsan as I've said, apparently I had too high expectations

t.b.

@JM I first read "snotty", but I guess I got the correct meaning right away :)

Ilya

the people I met here are mostly from CA and NY so they don't like it here very much

J. M.

@tb "snotty" actually works, too. :D

Ilya

but the nature is cool and I like it )

also, loads of cars and that big-country-which-lives-the-full-life kind of stuff

there are things which I like here

Srivatsan

@Ilya I can believe that. Pittsburgh $\gg$ Cal, NY > other places.

Ilya

04:47

and the difference with Europe is unbelievably huge

@Srivatsan we talked about Pittsburgh today in fact )

t.b.

@Srivatsan In terms of $-$degrees Celsius?

Srivatsan

@tb Did you know that Pittsburgh is the most livable city [or some such nonsense]?

I mean, in this country.

t.b.

No I didn't know that. Never been to the US.

Srivatsan

Yes. You should come to this city. Unless I moved out, of course. Then you should go wherever I am.

@Ilya So what did you do? This is your second day at the conference?

t.b.

I'm more drawn to the southern hemisphere, these days... But should I happen to go to Pittsburgh, I'll make sure to let you know. I'm sure you know the most decent coffee shops around :)

Ilya

04:53

@Srivatsan yes. I did my presentation in the very morning, then attended some others and talked to some people about the collaboration. Then napped for an hour, took a shower and meet exactly those people, had a dinner, gone for a walk, drank some beers and started talking to you guys

t.b.

@DylanMoreland: Did you see this?

Ilya

I'm planning to go for a sleep now to wake up in 7 and swim a bit

t.b.

@Ilya I hope your roommate lets you sleep :)

Good night!

Srivatsan

@Ilya Sounds great.

Good night.

Ilya

@tb actually he snores like a devil again, so I will use my ear plugs

J. M.

04:55

Good night, Ilya.

Ilya

thanks, guys - have fun and see you later

t.b.

@Ilya I feared that...

Dylan Moreland

@tb No, I didn't. I don't usually click on MO questions involving model categories :)

Ilya

@tb oh really? me too ( I tried to sneeze to wake him up - didn't help

J. M.

(Moral lesson: I'm not taking him as a roommate the next time...)

Ilya

04:57

I better pour a bottle of Icelandic Glacier water (\$5 per bottle) on him

@JM one assistant professor from our department haven't arrange a room for the first night, so he was sleeping on our floor. In the morning I've found him with a pillow on his head but didn't tell anything. Today (after 2 days) he finally asked me if I slept that night ))

the first part of my message maybe a bit shocking I guess )

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