Mathematics - 2013-02-22 (page 5 of 5)

NBP

22:01

@OrangeHarvester, you all seem to know Did by name

@OrangeHarvester, this suggests he doesn't mind

Orange Harvester

@NBP No, I don't.

Dominic Michaelis

@orange maybe you did :D

Orange Harvester

@DominicMichaelis Did not. :P

Charlie

@DominicMichaelis BADUMTSS

user19161

@NBP I don't think anyone here wants to help over the phone.

user19161

22:07

Of course, if it is my bro like Jonas, that is a different story. But Jonas doesn't need my help.

NBP

Why won't you guys help me?

user19161

@NBP I think you need to hang around on this site for a few days to see how it works. I think it is not what you think it is.

user19161

@NBP Because we may not have the knowledge, the time, the skill or the mood.

NBP

@JaconBlack, I don't have time. My exam is due in 2 days and I don't get anywhere near close to solving this issue.

user19161

@NBP Ah, that explains it. Exams.

user19161

22:10

Even worse, if you post a question saying URGENT EXAM, nobody will help you.

user19161

So what? Just fail your exam! What is the big deal? People are suffering in the world...

NBP

@JaconBlack, I didn't. But I didn't expect it to be ignored for so much time either.

Git Gud

22:27

@DominicMichaelis It's a picture of Zermelo: http://upload.wikimedia.org/wikipedia/commons/7/7d/Ernst_Zermelo.jpeg
Then it says ZFC.

Dominic Michaelis

zermelio franko ?

shall it look like KFC ?

NBP

@DominicMichaelis, It's Zermelo Fraenkel..

Git Gud

It's supposed to be reminiscent of KFC, yes.

NBP

Fraenkel was a great jewish mathematician

Matt

quick notational question: for a set X and a group G, we have X^G which is the set of fixed points. Isn't this the same as the stabilizer? Why do we have both kinds of notation? I guess the only thing i can think of is that X^H would be useful if we wanted to look at H, a subgroup of G

its described in the first section of orbits and stabilizers : en.wikipedia.org/wiki/Group_action

Charlie

22:34

@NBP being jewish does not affect anything...

Dominic Michaelis

after 2000 reputation you don't get reputation from editing ?

Orange Harvester

@DominicMichaelis MWUHAHAHA. And you thought you can become an edit moderator eh?

NBP

@Charlie, I'm not saying it does, but Franekel was jewish, I don't appreciate someone calling him Franko which is obviously not a jewish name.

Peter Tamaroff

@NBP Bleh.

Dominic Michaelis

what is an edit moderator ?

NBP

22:36

@Peter Tamaroff, excuse me?

Orange Harvester

@DominicMichaelis a person who acquires enough reputation points and popularity to become moderators only on the basis of edits.

Peter Tamaroff

@NBP I think you're making a fuss out of nothing.

NBP

@Peter Tamaroff, I don't mind that much, just wanted to point it out.

user19161

@NBP So is Paul Cohn and Walter Rudin, authors of my holy books.

Peter Tamaroff

My surname is Tamaroff, and some people call me Tama. Is "Tama" Russian? Who knows?

Well, actually they call my brother Tama, but it is just for illustrative purposes.

user19161

22:38

@PeterTamaroff Tama is a vulgarity in Mandarin lol.

Orange Harvester

For a moment I thought, oh no, where is my Pedro? ;-)

Peter Tamaroff

@JacobBlack What is it?

user19161

@PeterTamaroff It means "his mother", lol.

Peter Tamaroff

@JacobBlack I don't understand. "His mother"?

user19161

@PeterTamaroff Yes, maybe I don't understand why it is vulgar myself.

Orange Harvester

22:39

@PeterTamaroff if you do not understand that, then you obviously need some more practice.

Peter Tamaroff

@OrangeHarvester I don't know mandarin.

Orange Harvester

@PeterTamaroff I mean, if you do not understand why "his mother" can be vulgar, you need some practice in swear words.

It is short for "his mother" and something obvious in front of it.

Peter Tamaroff

@OrangeHarvester OK, give me an example.

user19161

@PeterTamaroff Haha, are you trolling?

Peter Tamaroff

Nope.

user19161

22:41

Maybe it means something like **** his mother.

NBP

His mother sounds perfectly fine to me

Orange Harvester

@PeterTamaroff okay, quickly read it, I will erase very very quickly.

user19161

@OrangeHarvester I think I have said enough.

Orange Harvester

@JacobBlack okay.

Peter Tamaroff

I don't get it

Dominic Michaelis

22:42

@peter like "his mother is so fat, she contradicts whitneys embedding theorem"

Orange Harvester

@PeterTamaroff I don't want to get flagged. I can mail you if you want.

Peter Tamaroff

@OrangeHarvester Don't be such a cat's offspring.

Orange Harvester

Okay!

Good!

user19161

@orange Cat is not vulgar.

Orange Harvester

@JacobBlack Transform it now. That is a job upto you.

user19161

22:44

Pussy is also not vulgar, for pussy is cat.

user19161

Pussy is listed in reputable dictionaries to mean cat.

Peter Tamaroff

@OrangeHarvester I wasn't looking!

user19161

OK, I think I have said enough.

Peter Tamaroff

Come again?

user19161

Geezis Pedro.

Orange Harvester

22:45

looked enough?

Peter Tamaroff

Fuck this shit. I'm gonna go and do some algebra.

Like, come on, guys.

Dominic Michaelis

LOL

Orange Harvester

LOL TROLL

I am having a good time tonight trolling.

user19161

@PeterTamaroff Good! Algebra is more important than cats!

Orange Harvester

Really?

Dominic Michaelis

22:46

you get flagged for "his mothers pussy" ?

Orange Harvester

@JacobBlack I thought algebra had cats everywhere. The joy of cats.

@DominicMichaelis I am trolling today.

user19161

The Joy of Sets and The Joy of Cats.

Dominic Michaelis

is cat not short for category ?

user19161

Yes, it is.

user19161

Once again, pussy is not vulgar.

Orange Harvester

22:47

@JacobBlack In your dreams.

Peter Tamaroff

Dominic Michaelis

a typo

Orange Harvester

@JacobBlack By the way, you can have even better choice there.

user19161

@PeterTamaroff I reserve that curse word only for people who have severely abused me.

Peter Tamaroff

@JacobBlack You know nothing, Jacob Black.

user19161

22:50

@PeterTamaroff Trust me, I have been through some deep shit bro.

Peter Tamaroff

@JacobBlack I was quoting Ygritte.

user19161

@PeterTamaroff WTF is Ygritte? That guy in the pic?

Peter Tamaroff

@JacobBlack But seriously, there are much better insults than motherfucker, like: "[THIS IS SOMETHING REAL ABOUT YOU, NOT SOME 12-YEAR OLD CURSE WORD]"

user19161

@PeterTamaroff OK OK anyway we better not say more and attract the flaggers.

Orange Harvester

I agree with Pedro.

Peter Tamaroff

22:52

@JacobBlack Flaggers gonna flag.

Orange Harvester

They are gonna flag I am out of this world.

Peter Tamaroff

Really, why would you be around all like "Uh, I won't participate in this chat room and get to know the "house tenants" but decide wether something is flaggable or not."

user19161

@OrangeHarvester Oh well, that is a misunderstanding, we are on good terms now.

Orange Harvester

@PeterTamaroff I asked that question on meta.

Peter Tamaroff

I call dibs on this one, man

Dominic Michaelis

23:22

mpfh i didn't cap today

damn exam

Orange Harvester

What was the exam on?

Dominic Michaelis

theoretical physics

Orange Harvester

specifically?

Dominic Michaelis

lagrangian hamiltonian, oscillators ...

Orange Harvester

nice.

NullOverNull

23:27

anyone familiar with summation shortcuts and totient function phi (and gcd)?

I am trying to find a way to express this in terms of the totient function: $\sum\limits_{{\Large 1 \leq b<a} \atop {\Large \gcd(a,b)=1}}{bk}$ for some constant k

It seems like it should be $k\frac{a\varphi{(a)}}{2}$ but then further simplifications go awry because wolfram says it's an "odd parity function"

Ethan

@NullOverNull

k is just a constant, why not factor it out?

and the latter sum should be $a\phi(a)/2$

NullOverNull

isn't what what I have?

the problem is that when I do some further simplifications the numbers stop working

it's driving me nuts

Ethan

well its correct

NullOverNull

well when I do that, my main equation after simplifying is $(N+1-bk)((N+1)\varphi(b)-kb\varphi(b)/2)$ which seems to work okay as long as I treat the last division as a floor operation. Something just feels very much off to me

Ethan

what main equation?

NullOverNull

23:34

4

Q: A way to simplify $\gcd(a,b)$ condition in a double sum?

I have $$\sum_{{\Large 1 \leq a,b \leq L} \atop {\Large \gcd(a,b)=1}}(L+1-a)(L+1-b)$$ Which means iterating $1 \le a \le L$ and $1 \le b \le L$ and only adding $(L+1-a)(L+1-b)$ to the sum if $a$ and $b$ are coprime. Is there a faster way that is not $O(N^2)$? I know how to count how many copr...

number-theory

I actually forgot to include the constant in the question

should be -ka and -kb

Ethan

well if gcd(a,b)=1

gcd(a,b+a)=1

so the sum can be broken into periods of a

which can be given explictly

NullOverNull

so how would you simplify that expression? the (L+1-ka)(L+1-kb) for 1<=a,b<=L for gcd(a,b)=1

Ethan

what do you mean by a,b

all the pairs ab less then l

NullOverNull

yes

Ethan

such that gcd(a,b)=1?

NullOverNull

23:36

right

correct

Ethan

are we counting ab=ba as two different pairs?

NullOverNull

yes, it's like if you ran two for-loops

for a=1 to L
for b=1 to L
if gcd(a,b)=1 then add to sum etc

the person who answered the above question though pulled out the a=1 b=1 condition to make the rest symmetrical though

Ethan

Did someone give you this problem as an exercise

NullOverNull

it's not homework if that's what you mean (I've long since graduated)

Ethan

where did you encounter the problem?

NullOverNull

23:40

you'll find similar constructs in the following paper: survo.fi/papers/PointsInGrid.pdf

i am trying to find a way to express a subproblem of this in terms of totients

page 2 shows an example (page 4 too)

Ethan

im kinda busy, whats wrong with the answer you got?

NullOverNull

it's wrong

because of this parity thing

Ethan

which part

NullOverNull

when you use his answer as given

i don't know which part is contributing the problem; i think it's the division-by-2

for some reason when I reduce the summation, the end result is now incorrect

even though it was correct before the totient translation

Ethan

ok well id multiply out the sum

because parts of it can be given

explictly

clearly

NullOverNull

23:43

that is what i did

Ethan

can you write that out 4 me

here

user19161

I am very upset, because someone told me a lie yesterday.

user19161

I think I can no longer trust people I meet on the internet.

Ethan

@NullOverNull im still having trouble understanding your summation notation, the sum is over the pairs a,b, with a<L and b<L, such that gcd(a,b)=1?

And were not counting ab=ba?

NullOverNull

tinyurl.com/aowtvch

Ethan

23:46

?

NullOverNull

@Ethan ab is different from ba, so for example if you look at a,b=3,2, you add the inner part to the sum. if a,b=2,3, you add the inner part to the sum too

the link I just gave shows what it "Expands" the totient stuff to

Peter Tamaroff

@JacobBlack I am very upset because I am having a hard time with an exercise.

Ethan

a,b

user19161

@PeterTamaroff But I trust Pedro. =)

Ethan

@NullOverNull

1 sec

NullOverNull

23:49

@Ethan It's like a dual for-loop

Ethan

ok

@NullOverNull

$$\sum_{{\Large 1 \leq a,b \leq L}}ab$$

where gcd(a,b)=1

not counting ab=ba

is the sum over the square free integers less then L^2

NullOverNull

the equation is not $ab$ though

Ethan

I know, but if you multiply it out

you should get a factor of ab

and some other stuff

multiply it out, and try to tackle each thing individually

NullOverNull

yeah that's what i have done

a factor already pulled out

when a,b=1,1

Ethan

wait

my bad

NullOverNull

23:55

so for example after I reduce my summation, I get the sum of 2<=b<=L of $b^2k^2\varphi(b)/2-3bkN\varphi(b)/2-3bk\varphi(b)/2+N^2\varphi(b)+2N\varphi(b)+\varphi(b)$$

Khromonkey

hi guys

Ethan

@NullOverNull

your sum should be equal to

$$\sum_{{\Large 1 \leq a,b \leq L} \atop {\Large \gcd(a,b)=1}}(L+1-a)(L+1-b)=(L^2+2L+1)\sum_{{\Large 1 \leq a,b \leq L} \atop {\Large \gcd(a,b)=1}}1-(2L+2)\sum_{{\Large 1 \leq a,b \leq L} \atop {\Large \gcd(a,b)=1}}a+\sum_{{\Large 1 \leq a,b \leq L} \atop {\Large \gcd(a,b)=1}}ab$$

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$Mathematics$ Mathematics