Mathematics - 2015-03-10 (page 3 of 4)

Chris's sis

21:00

@robjohn where?

teadawg1337

@Chris'ssis I've already solved this one, and what do you mean by natural?

Chris's sis

@teadawg1337 Ah, if you solved it then it OK ...

@teadawg1337 Did you ever hear of Catalan-Botez identity?

teadawg1337

Nope

Chris's sis

@teadawg1337 Use that at first. The rest is pretty boring usual stuff.

$$\sum_{k=1}^{2n} \frac{(-1)^{k-1}}{k}=H_{2n}-H_{n}$$

@teadawg1337 ^^^

Theorem

@robjohn What i have done is differentiation with respect to $s$ and i have applied the fundamental theorem of calculus of variation . What i am afraid is that i am not getting some nice equations . :(

Chris's sis

21:07

Partial fractions and a bit of use of the generating function of the harmonic function complete the job.

Vrouvrou

@iwriteonbananas ?

robjohn

@Chris'ssis In this answer

@Theorem When dealing with $L^p$ rather than $L^2$ sometimes the reverse Hölder inequality helps.

teadawg1337

@Chris'ssis ??? This arose in my work with polylogs and $\frac18\log^2(2)$... I know nothing about the harmonic function...

Well, identities thereof

robjohn

It often helps to linearize things similar to $L^2$

iwriteonbananas

@Vrouvrou let $x_0 \in E$ and let $(x_n)_n$ be the sequence described in the assignment. since you know $x_0$ is an adherent value of $(x_n)$, let $x_{n_k}$ be a subsquence such that $d(x_{n_k}, x_0) \to 0$. to prove that $x_0 \in \overline{f(E)}$, it suffices to show that any $B_\epsilon(x_0)$ has nonempty intersection with $f(E)$. suppose toward contradiction that $f(E) \subset B_\epsilon(x_0)^c$. Then $d(x_0, x_{n_k}) \geq \epsilon$ which contradicts convergence

teadawg1337

21:13

@Chris'ssis The answer I got is $\frac18\ln^2(2)-\frac14\ln(2)+\frac{\pi^2}{48}$, but I question its validity after seeing your method...

Theorem

@robjohn In the first step , to find the necessary conditions i just perturbed the minimizer $u^*$ to $u^*+s \phi$ , differentiated with respect to $s$ and equated it to $0$.$s$ is a scalar , $\phi$ is a function that belongs to $H^{1,2}$.

Chris's sis

@teadawg1337 I told you above what you should do. Just let me know where you're stuck.

robjohn

@Theorem yes, but you are trying to get $L^p$ estimates, right?

teadawg1337

@Chris'ssis As I've said, I have no experience with using identities of the harmonic function...

Theorem

@robjohn aah ok , you are talking about my second question , i.e. for which $q$ such minimiser actually exists .

Chris's sis

21:16

@teadawg1337 I think you know more than you say you know. I'm very good at noting that. ;)

robjohn

@Theorem Oh, I didn't realize the first part was a question. It sounded like a statement of things you knew.

Theorem

@robjohn my first question is if the necessary condition , i.e. the equations i have gotten are correct . :)

iwriteonpeoplewhowriteonbanana

@Ramanewbie irccloud.com/#!/ircs://irc.freenode.net:6697/%23le_room_42

Ramanewbie

I'm on @iwriteonpeoplewhowriteonbanana

Ted Shifrin

bonsoir, les frères Jacques

hi @robjohn, rehi @teadawg, bananas

Ramanewbie

21:23

@ted hi how is that ?? "les frères Jacques"

Ted Shifrin

Penses-y un peu :P

robjohn

@TedShifrin G'day

Ted Shifrin

You practicing to go down under, @robjohn? :P

teadawg1337

@Chris'ssis $$\frac18\ln^2(2)=\sum_{n=1}^{\infty}\frac{\sum_{k=1}^{2n}\frac{(-1)^{k-1}}{k}}{‌16n^2-8n}+\sum_{n=1}^{\infty}\frac{1-4n}{64n^4-64n^3+16n^2}+\sum_{n=1}^{\infty}\f‌rac{1}{16n^2-8n}$$

Ted Shifrin

ugh, @teadawg, what horrific things are you doing to that poor series?

iwriteonbananas

21:25

heyho mctedster

Ted Shifrin

I'm tired of grading differential geometry homeworks, so I'm having a martini :P

iwriteonbananas

that's a great idea

Ted Shifrin

want one, bananas?

iwriteonbananas

yes, but i'd rather smoke weed

Ted Shifrin

LOL, well, I don't do that, so I can't offer :)

iwriteonbananas

21:27

im just trying to be one of the kewl kids

Ted Shifrin

aren't you too old to be kewl?

David Wheeler

@TedShifrin How do you take your martinis?

Ted Shifrin

LOL, that got @David's attention. Very dry, usually on a rock when I'm home, with multiple olives ... and gin, of course.

iwriteonbananas

i dunno :O

teadawg1337

@Chris'ssis $$\sum_{n=1}^{\infty}\frac{1-4n}{64n^4-64n^3+16n^2}+\sum_{n=1}^{\infty}\frac{1}{‌16n^2-8n}=\frac14\ln(2)-\frac{\pi^2}{48}\implies \sum_{n=1}^{\infty}\frac{\sum_{k=1}^{2n}\frac{(-1)^{k-1}}{k}}{16n^2-8n}=\frac18\‌ln^2(2)-\frac14\ln(2)+\frac{\pi^2}{48}$$

Ted Shifrin

21:28

* thinks he's going to ignore teadawg until this passes *

hi @Alizter

iwriteonbananas

leave that poor series alone teadawg

it's never done anybody no harm

David Wheeler

@iwriteonbananas I gave up pot decades ago, because everyone drug-tests nowadays, and that stuff (THC) sticks around in your system for ever...

Ted Shifrin

bananas is in a different country, @David

teadawg1337

Hey, I'd show the rest of my work but it'd fill up half the screen... I just put the bare minimum required to prove my result

iwriteonbananas

@DavidWheeler dont worry, i have gallons of old urine in the freezer

David Wheeler

21:29

@TedShifrin When I was a younger person, I had a neighbor who had brought some hashish back from some arabic country where it was legal

Alizter

@TedShifrin Are you good?

Ted Shifrin

somehow, bananas, I don't think that will pass, as it were

David Wheeler

It still had the brand logo stamped in it

Chris's sis

@teadawg1337 Great!

Ted Shifrin

my back is killing me, @Alizter, but I'm ok otherwise, and you?

Alizter

21:30

@TedShifrin Did you see my probability question? Well it's not mine but you get the idea.

Ted Shifrin

no, @Alizter ... I've been working

it's official: @teadawg has now gone over to the dark side.

Mike Miller

morning ted

iwriteonbananas

@DavidWheeler isn't weed very illegal in all those countries?

Alizter

40 mins ago, by Alizter

@TedShifrin One hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a random seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats at random otherwise. What is the probability that the last person to board gets to sit in his assigned seat?

David Wheeler

@iwriteonbananas might be now, this was long ago....

Ted Shifrin

21:31

oh, that's the letter-in-the-envelopes question, @Alizter

Mike Miller

I'm sure he's heard the problem before, @Alizter

Ted Shifrin

hmm, more or less

Mike Miller

mhm

Alizter

Yes, I get 1/2

Ted Shifrin

good night, @Mike

robjohn

21:31

@Theorem What is preventing us from simply adding a constant to $u$ to bring down the ratio?

teadawg1337

@Ted I was being considerate, I don't think a page and a half of work in my handwriting would translate very well to this chat :P

Alizter

However my reasoning is far from clear

iwriteonbananas

@DavidWheeler and smuggling it to the USA sounds stoopid

gee wilikers

Mike Miller

@Alizter He gets in his assigned seat iff it's the last one available. First peron has $(n-1)/n$ odds of not sitting in that seat; next has $(n-2)/(n-1)$...

evinda

@DanielFischer Could you take a look at my question?

-1

Q: Can linear execution time be achieved

The SELECT algorithm determines the $i$th smallest of an input array of $n>1$ distinct elements by executing the following steps. Divide the $n$ elements of the input array into $\lfloor \frac{n}{5} \rfloor $ groups of $5$ elements each and at most one group made up of the remaining $n \mod 5$ ...

Alizter

21:32

Yes @Mike

David Wheeler

@iwriteonbananas I cannot vouch for my former neighbor's intelligence

iwriteonbananas

:(

Mike Miller

multiply

Theorem

@robjohn I am wondering how thats going to help me .

Ted Shifrin

The famous question, @Alizter, is what's the probability that no one is in the right seat?

Yours is different.

Mike Miller

21:33

And that's got the cutest answer.

Ted Shifrin

Yup @Mike ... I even have that one in my algebra book.

Sabyasachi Mukherjee

This is a chat fest.

iwriteonpeoplewhowriteonbanana

what's the probability that ted smacks the passenger that took his seat ?

robjohn

@Theorem I'm just trying to understand the first part. It looks as if we can simply add a constant to $u$. That doesn't change $\nabla u$ but changes $u$

David Wheeler

Combinatorics is all about counting things. Unfortunately, if I can't use my fingers to count, it doesn't go well...

Mike Miller

21:33

0, @iwriteonpeoplewhowriteonbanana, unless he's you

teadawg1337

@Ted Hello, sorry for ignoring you during that

Ted Shifrin

If it's you, Hippa, 100%.

Mike Miller

We're not so different, you and I, @Ted

iwriteonpeoplewhowriteonbanana

@TedShifrin Hopefully, you don't know what I look like :D

Ted Shifrin

@teadawg: Did you figure it out by totally elementary means, knowing that $\sum_{k=1}^n 1/k \sim \log(n)$?

Alizter

21:34

(n-1)/n! as a wild guess

Ted Shifrin

I won't make any obvious comments, Hippa.

Way....yyyy too small, @Alizter :)

I'm not sure I've done your version before, @Alizter.

I need to ponder it.

Theorem

@robjohn Yes , but i i will still have a factor $\mu(u)$ , which really is giving me trouble .

Ted Shifrin

My not-so-educated guess, @Alizter, is that the answer to yours is $1/n$. I need to think harder to justify it.

Mike Miller

I said what it is and why above, @Ted :P

teadawg1337

@Ted Using asymptotic series? No

Alizter

21:39

@TedShifrin 1/n is too small. Thats as if the first pasenger guessed correctly. However there is also when he doesn't but possible the second and third swapped and rest were correct.

But the prob is definitely >1/n

Ted Shifrin

@teadawg: This is totally elementary, just using the sum to approximate the integral.

teadawg1337

Integral?? I never used any integration...

robjohn

@Theorem I guess I am asking if there is anything preventing $\varphi$ from being a constant.

Chris's sis

@teadawg1337 You have answers with integrals.

Ted Shifrin

You're using far more arcane things, @teadawg. You should understand that $\sum_1^n 1/k$ looks like $\int_1^n dx/x$.

@Mike: Where is said answer?

Mike Miller

21:43

@TedShifrin he gets in his last seat iff it's available. the first guy has probability (n-1)/n of not getting in his seat; the next guy (n-2)/(n-1), ... the second to last guy 1/2

take the (telescoping) product

teadawg1337

I actually know nothing about asymptotic expansion... This is all new to me

Mike Miller

what am I on about? that's 1/n, not 1/2

Ted Shifrin

Ah, good point. I was misapplying symmetry. I'm in geometry mode now, not probability :P

Ha ha ... My instincts were right, then :)

Indeed. remembers henceforth never to trust Mike

Mike Miller

did you trust me before?

Ted Shifrin

@teadawg: What I'm talking about is far more fundamental for you to learn/master/use.

Theorem

21:44

@robjohn I guess you are not considering $\varphi$ is as a constant .

Alizter

Yes, $H_n\sim \log n$ is used in NT quite a bit. The bit I have studied anyway.

Its a fun fact

Ted Shifrin

It's used everywhere, @Alizter.

Mike Miller

Another way of thinking about it: each assignment of seats is just a permutation of $[n]$. Precisely $(n-1)!$ of those preserve the last one - those are the permutations of $[n-1]$.

Alizter

@TedShifrin Well I have literally only studied Algebra and NT so thats my everywhere!

Ted Shifrin

@Mike: That's related to the group-theoretic interpretation of the envelope problem I put in my book ... counting fixed-point free permutations.

Mike Miller

21:46

mhm

I know the story behind the problems, though not from your book, @Ted

Alizter

I take it Homotopy doesn't make sense for graphs

Ted Shifrin

You're wise never to look at anything I write, @Mike, just as I'd be wise to ignore you.

Mike Miller

what, @Alizter?

Alizter

You either have rings, joined rings or a vertex

or cycles/rings

Mike Miller

I have no idea what you're talking about.

Ted Shifrin

21:47

@Alizter: I was reasoning it out that the problem should be symmetric in the passengers/seats. It should be the same for the first as for the last.

teadawg1337

I've gotta stop working on math for the day, there's too much stuff that I didn't know until now and it's overwhelming

Alizter

@MikeMiller As I understand homomorphic classes or whatever they are called can be morphed into eachother but classes are sepereated by holes and sticky-outy-bits. But in homotopy classes they are just holes.

Ted Shifrin

oy @Alizter

Mike Miller

I don't understand what you're saying, but I suspect you don't either.

Alizter

Probably

Ted Shifrin

21:49

Go back to understanding the airplane problem :P

Alizter

@TedShifrin I get it now :P

A and R are part of the same homeomorphism class

so are O and D

Ted Shifrin

Then figure out the no-one-gets-the-right-seat version, @Alizter.

Alizter

@TedShifrin ok, let me incoherently say stuff first

Mike Miller

Depends on how you write your R's, but yes.

teadawg1337

@Ted Do you think I should shelve my work in analysis until I take a college course to understand the material better? :/

Alizter

21:51

But then if considering homotopy classes

R and O are in the same class

Ted Shifrin

@teadawg: I think you should know fundamentals before you specialize in arcane things.

Alizter

As far as I ~~understand~~ misunderstand

Ted Shifrin

To be working on these polylogs and not know what I told you is criminal, IMHO.

Alizter

@TedShifrin I thought polylog was something WolframAlpha made up

Ted Shifrin

LOL

Alizter

21:52

;)

Ted Shifrin

they're cousins of polliwogs, @Alizter

Mike Miller

You're correct, @Alizter, those are homotopy equivalent.

teadawg1337

They could be made up as far as I know, since I did learn about polylogs from WolframAlpha...

Mike Miller

But you should try to understand the definition - and use that to see why - and not just the intuiting idea that you can collapse lines to points.

Ted Shifrin

And I'm telling you about monologs, @teadawg.

Mike Miller

21:53

Mathematics isn't made up, it's discovered. Except polylogs, those are made up.

5

Ted Shifrin

Draw the picture from the integral test and work out why $\lim\limits_{n\to\infty}\sum_{k=1}^n \dfrac 1k - \log(n) \to \gamma$.

Ramanewbie

are you gone

@iwriteonpeoplewhowriteonbanana

Ted Shifrin

His skeleton is still here, @Ramanewb.

Ramanewbie

@ted hum... I would prefer him alive, I don't like speaking to a skeleton !

Ted Shifrin

Sometimes it's better, @Ramanewb. Skeleta don't talk back as much as live ones.

teadawg1337

21:55

I just feel like such a fool for jumping into something without knowing the underlying fundamentals beforehand...

evinda

3

Q: Execution time of function

The following pseudocodes are given. quicksort(A,p,r) if p<r then q<-partition(A,p,r) quicksort(A,p,q-1) quicksort(A,q+1,r) partition(A,p,r){ x<-A[r] i<-p-1 for j<-p to r-1 if A[j]<=x then i<-i+1 swap(A[i],A[j]) swap(A[i+1],A[r]) return i+1 Wh...

algorithms computer-science

Could anyone help me?

Ramanewbie

@ted but you can't get cleverer talking to a skeleton

Ted Shifrin

I'm not belittling you, @teadawg. I'm giving you (perhaps unsolicited, but then solicited) advice.

You can't get cleverer talking to Hippa, either, @Ramanewb. :D

Ramanewbie

@ted talking to him alive, yes. Not to his skeleton !

Ted Shifrin

no, @Ramanewb, talking to him alive just makes everyone exasperated.

Ramanewbie

21:56

@ted you're not wanna tell me what he's done to you uh ?

Ted Shifrin

Not I :P

teadawg1337

@TedShifrin I mean, it explains why I've been having so much trouble... Now I just don't know what the fundamentals are that I should learn before diving back in to where I was :/

Ramanewbie

@ted "when I get dementia I may forget all the evil things you've done to me, Hippa, but not yet !"

Ted Shifrin

It takes time, @teadawg. Just chill and learn slowly.

You found that on a crypt wall, @Ramanewb?

iwriteonbananas

@Ramanewbie lol....pure gold

Mike Miller

21:58

OK, it's time for a nap.

Ted Shifrin

No, it's time for dinner soon, @Mike.

When is your talk?

Mike Miller

I have to clean my room tonight before the prospectives see it... :P

Ramanewbie

it's purely real, he send that @iwriteonbananas

iwriteonbananas

@teadawg1337 dont listen to ted, study all day until the wheels fall off

Ted Shifrin

oh god ... don't let them near all that infestation.

Mike Miller

21:59

Friday. Ciprian wants me to make some modifications but I'll be done with those tonight.

iwriteonbananas

@Ramanewbie oh i believe you

Mike Miller

Only one is nontrivial to do.

Ted Shifrin

I still remember my first grad school seminar talk, @Mike. Make us proud.

Ramanewbie

@ted absolutely not, you posted that a few month ago (the message was even starred)

Transcript for

$Mathematics$ Mathematics