Mathematics - 2012-08-31 (page 1 of 2)

peoplepower

12:01 AM

@OldJohn Finished a game of chess. My psychosis allowed me to sacrifice a few pieces and checkmate.

Khromonkey

12:26 AM

what does community wiki mean?

Peter Tamaroff

12:55 AM

@Khromonkey Read the FAQ =)!

@anon Hello.

anon

within seconds of opening my laptop

Peter Tamaroff

@anon Uhuh?

anon

so my prof pitched a couple research questions to me and a couple undergrads. one is about Hopf orders and the others about Galois (scaffolding?) stuff...

skullpatrol

Which one are you going to take a swing at?

anon

Actually that latter was his research, not one being pitched. The other one has something to do with analogues of Wilson primes in finite fields.

Not sure. I'll read a bit about Galois and field theory, and Hopf algebras, and see which one I like best.

skullpatrol

1:02 AM

Well, I hope you hit a grand slam :-)

Peter Tamaroff

@anon I can't help with that :(

robjohn

@PeterTamaroff which FAQ would that be?

Peter Tamaroff

@anon What's the difference between using $S=\{x:P(x)\}$ and $S=\{x\mid P(x)\}$?? I mean the "$:$" versus "$\mid$". I thought that they meant the same, but that is wrong.

anon

It's wrong?

I use : personally, but I was under the impression they meant the same thing. The potential ambiguities occur (for me anyway) when working with group indices [G:H] and divisibility relations d|n.

peoplepower

The set theoretic arrow notation can be a (very unlikely) ambiguity.

Peter Tamaroff

1:09 AM

@anon I'm :confus:

@robjohn Dunno. I thought there was a FAQ for CW.

robjohn

@PeterTamaroff why do you say it is wrong?

peoplepower

259

Q: What are "Community Wiki" posts?

Some questions and answers are marked Community Wiki and are owned by a Community Wiki user. Why have Community Wiki posts? How do Community Wiki posts work? How does a post become a Community Wiki post? How can the Community Wiki status be removed from a post? Return to FAQ Index

robjohn

@PeterTamaroff There are a lot of questions and answers, but I don't think it is mentioned in a FAQ.

Peter Tamaroff

@robjohn Because they are not the exact same "operator"

anon

"operator"?

robjohn

1:11 AM

@PeterTamaroff what is the difference? what operators?

Peter Tamaroff

@robjohn logical operator

robjohn

@PeterTamaroff in that context, they both mean "such that"; they are not logical operators

Peter Tamaroff

@robjohn sorry, they are not logical operators, yes

anon

@PeterTamaroff Are you saying that since the symbols : and | have different meanings in different areas of math, they must mean different things in set constructions?

Peter Tamaroff

@anon in the context of set builder notation

@robjohn Rob, am I wrong or do you work in $\bf W\mid A$?

skullpatrol

1:14 AM

conditional?

robjohn

@PeterTamaroff I don't use W|A

Peter Tamaroff

@robjohn I thought you worked in Wolphram Research.

anon

@PeterTamaroff that is not a "yes" or a "no" response, nor is it technically a complete sentence. Are you saying they mean different things specifically in set-builder notation? If so, what are these two meanings, what is the difference between them?

robjohn

@PeterTamaroff No, I work at UCLA...

peoplepower

@PeterTamaroff The entire expression $\{x:P(x)\}$ is some set which can be constructed by the axiom of separation. Same goes for $\{x|P(X)\}$. They are just different notations.

Peter Tamaroff

1:16 AM

@robjohn =P Sorry.

@anon I'll ask my prof. tomorrow. The difference was delicate

anon

Did you read somewhere that there was a difference? You are being oddly silent...

robjohn

@PeterTamaroff Perhaps in some usages there is a difference, but in all that I've used them they are the same.

Peter Tamaroff

@anon I can't really recall what he said. Let's not wastemore time on this =)

skullpatrol

@Khromonkey When you mark a post community wiki on a Stack Exchange site, that means …

Peter Tamaroff

@MarianoSuárez-Alvarez Hola/Hello

Peter Tamaroff

1:41 AM

2

No offense intended, peoples

@skullpatrol Waittttttttttt

If $a\in \Bbb R$; then $a=0$ oor $a\neq 0$, right?

skullpatrol

right

Peter Tamaroff

@skullpatrol OK, so the theorem is "If $a \cdot b =0$, then $a=0$ or $b=0$"

skullpatrol

right

Peter Tamaroff

Note that $a=0$ or $b=0$ is true in three cases.

skullpatrol

three cases?

Peter Tamaroff

1:46 AM

Yes.

This or is not exclusive

skullpatrol

OK

Peter Tamaroff

@skullpatrol So, where is this you're reading?

skullpatrol

I'm not reading anything, I'm trying to work it out myself :)

Peter Tamaroff

@skullpatrol OK, OK.

So, suppose $a\cdot b=0$. We know that either $a=0$ or $a\neq 0$ must be always true.

So suppose $a=0$ is true. Then for any $b$, it is the case $a\cdot b=0$, yes?

skullpatrol

But you just supposed what you were to prove.

Peter Tamaroff

1:53 AM

Now suppose $a\neq 0$. Then we may use the axiom (can't remember the name) that tells $$a\cdot b=a\cdot c\iff b=c$$ when $a\neq 0$, yes? (This axiom is logically equivalent to what you're trying to prove)

@skullpatrol No. The statement "It is rainy". is not the same as "It is rainy or sunny."

Enunciate the theorem a few times
> " $\color{red}{\text{ If } a\cdot b=0 \,\,\,} \color{green}{\text{ then }} \color{blue}{a=0\text{ or } b=0}$

skullpatrol

If ab=0,
then a=0 or b=0.

Now you tell me suppose if a=0???

Michael Boratko

(He is working one of the possible cases)

Have you had a class in logic? (philosophy usually?)

Peter Tamaroff

@skullpatrol Re read up there. After "OK, OK"

Michael Boratko

You have a statement that goes "If p then q or s"

now p is true

it is always true that q or not-q

assume q

Peter Tamaroff

@MichaelBoratko duuuuuuuuuuuude

wait up

You're killing it.

Michael Boratko

1:58 AM

@PeterTamaroff Sorry, didn't mean to stomp all over

:)

Peter Tamaroff

@MichaelBoratko ;)

Michael Boratko

No worries - sorry for interrupting! :)

Peter Tamaroff

@skullpatrol Do you understand that "either $a=0$ or $a\neq 0$" is always true?

skullpatrol

@PeterTamaroff No, because "a=0 or b=0" is the conclusion that you are working towards.

Given only that ab=0

anon

"a=0 or b=0" is not the same proposition as "a=0 or a=/=0." the latter is just the law of excluded middle: either something is true, or its negation is true (here we speak of "something" as "a=0")

Peter Tamaroff

2:02 AM

@skullpatrol What is your native language?

@anon Is that for skull or for me?

anon

that is a silly question

Peter Tamaroff

@anon I like silly.

skullpatrol

@anon Hmm.. "a=0 or b=0" is not the same proposition as "a=0 or a=/=0." I think I'm getting confused by the overlapping of "a=0" in the two propositions :)

peoplepower

@skullpatrol Maybe consider the statement "a=0 or a$\neq$0 $\implies$ (ab=0 $\implies$ a=0 or b=0)"

If "a=0", then the conclusion of the second implication follows immediately, proof complete in that case.

The other thing we have to do is check the second implication if "a$\neq$0".

skullpatrol

If ab=0,
then a=0 or b=0.
Case 1: If a=0 the theorem is true; there is nothing to prove.
Well, here is where I get confused because you have just assumed the conclusion in this case, namely "a=0 or b=0."

peoplepower

2:18 AM

I have assumed "a=0", part of the hypothesis "a=0 or a$\neq$ 0".

anon

Case 1 assumes "a=0," not "a=0 or b=0," although the second follows directly from the first. At any rate, "a=0" is essentially the description of case 1. Of course you assume the description of cases...

Case 2 is "a=/=0," and together Case 1 or Case 2 exhaust all possibilities for a. What we have here is an example of a conclusion ("a=0 or b=0") following from hypothesis A as well as following from hypothesis B ("a=0" and "a=/=0" respectively), and we also know that necessarily one of the two hypotheses must be true. If the conclusion follows from both hypotheses, and at least of the two hypotheses is true, so too must the conclusion be true.

Peter Tamaroff

@anon Goodbyes fellow stranger!

skullpatrol

@anon Hypothesis A is ab=0 ?

anon

@skullpatrol No, I said what the two hypotheses were. Do you know what "respectively" means? Hyp A is "a=0" and Hyp B is "a=/=0" (I'm simply using the term hypotheses instead of the term cases to make it easier to understand).

skullpatrol

2:36 AM

@anon OK so what does "Case 1: If a=0 the theorem is true; there is nothing to prove. " mean?

If ab=0,
then a=0 or b=0.
Case 1: If a=0 the theorem is true; there is nothing to prove.

It looks like we are assuming the conclusion...

because "there is nothing to prove" :)

anon

He means that a=0 implies "a=0 or b=0" so directly that nothing more needs to be said.

that's the translation of the phrase "there is nothing to prove"

from lazy mathspeak

skullpatrol

Hmm..

Case 1: a=0 directly implies "a=0 or b=0."
Case 2: a=/=0 ....

...b=0.

therefore "a=0 or b=0"

together Case 1 and Case 2 exhaust all possibilities for a

and we also know that necessarily one of the two cases must be true. If the conclusion follows from both cases, and at least of the two cases is true, so too must the conclusion be true.

and at least one * of the two cases is true,

@anon Is that the idea?

anon

3:09 AM

yes

I get a strange sense of deja vu reading that

skullpatrol

copy paste ;-)

@anon It still sort of looks like we are assuming part of the conclusion in Case 1 because it follows so directly that I don't really think about the difference between "a=0" and "a=0 or b=0"

anon

that's because we are assuming part of the conclusion in case 1, and we are assuming the negation of that part in case 2

skullpatrol

@anon Yes, I think that is the source of my confusion. We are not assuming the entire conclusion; only a part of it :)

anon

In 1859, 24 rabbits were released in Australia. Within 6 years, the population grew to 2 million. - flipside of my snapple bottle cap

skullpatrol

3:27 AM

That gives more meaning to the phrase "breeding like rabbits." ;-)

experimentX

hello

skullpatrol

hi

phku

4:54 AM

Hi

taylor

ola

phku

having some trouble proving that if n is an odd integer and is not a multiple of 3, then 24|(n^2+23)

taylor

im definitely not the person to ask lolz

anon

write n=2m+1 and this is 24|(4m^2+4m+24), which you can reduce to 6|(m^2+m)

3|n entails 2m+1=0(mod3), or m=1(mod3). can you put these together?

I suppose I should instead say 3not|n implies m=0 or 2 mod 3...

skullpatrol

6:01 AM

hi @WillHunting

user19161

@skullpatrol It's not good to use Jonas as your avatar, please change it...

Jayesh Badwaik

spikedmath.com/521.html

skullpatrol

@WillHunting Why so serious?

user19161

@skullpatrol Well, OK. It's not a big deal. But I would not use it unless Jonas is my boyfriend.

3

Jayesh Badwaik

@WillHunting But you want to get into a shower together!!

user19161

6:05 AM

@JayeshBadwaik That's different!

Jayesh Badwaik

@WillHunting :P

user19161

Does Jonas know about this already?

Jayesh Badwaik

@WillHunting about shower or about avatar?

user19161

@JayeshBadwaik The pic of course.

user19161

Well, at least tell him first.

Jayesh Badwaik

6:08 AM

@WillHunting okay, I will leave @skull to answer it.

skullpatrol

It's not a big deal, relax...

user19161

I am very relaxed...

user19161

I have many things to worry about.

skullpatrol

Worried is not relaxed :)

user19161

Many other things, not this!

user19161

6:10 AM

And thanks for the star guys, I love stars!

3

skullpatrol

Worry has a way of affecting everything...

Ilya

8:03 AM

taylor

wonder how many answers

Morning.

good morning

morning

hi

8:15 AM

"good morning" for some reason sounds like something that should be said greeting people at funeral

Matt

It is. This is my funeral.

skullpatrol

The death of old ideas gives birth to new ones :)

taylor

well then

i should have dressed better

skullpatrol

J. M.

@Matt ...

Matt

8:29 AM

@J.M. Silent expression of your consolations?

taylor

i plan to live on as a computer program when my bodily functions fail

skullpatrol

@taylor In what language are you going to write that program?

J. M.

@Matt I usually just take off my hat and hang my head at such...

taylor

javascript

J. M.

(playing "Taps" in the background is optional.)

Matt

8:33 AM

@J.M. I don't know what I'd do. I've not had the chance. Although I have quite a long list of people whose funeral I'd like to attend...

3

J. M.

@Matt Commiserative, or celebratory?

Matt

@J.M. Celebratory.

J. M.

*facepalm*

taylor

what is being celebrated at funeral time

J. M.

I see the appeal of Schadenfreude, though.

Matt

8:35 AM

No.

I'd celebrate the improvement of this world after ridding it of one of these individuals.

2

taylor

sounds kinda evil

Matt

8:50 AM

Sure. I never claimed I wasn't evil.

2

I have to go now, I'll see you all later!

taylor

rest in peace

skullpatrol

hi @JayeshBadwaik

Jayesh Badwaik

@skullpatrol hi

@taylor javascript? genius answer ;-)

I half expected you to say LISP, but oh well!

skullpatrol

@JayeshBadwaik What would you choose?

Jayesh Badwaik

@skullpatrol I am planning to design my own language for that. ;-)

peoplepower

9:04 AM

mixal

Jayesh Badwaik

@peoplepower yeah, but interfaces/bindings!! It would be more work writing the bindings than developing your own language.

Anyway, I have to go now, See you all later.

skullpatrol

later

@taylor Why "javascript"?

taylor

so then firefox would be my home

i wonder what i should do once i become a computer program

skullpatrol

compute?

taylor

well i guess, but to what end

oh no

what if im already a computer program

skullpatrol

9:16 AM

"we are programmed to receive..."

I have become comfortably numb.

or in my case dumb :)

taylor

would you rather be super dumb and happy or ultra smart but miserable

Jonas Teuwen

How about dumb and miserable?

taylor

i guess that could be an option too

skullpatrol

9:38 AM

Jonas Teuwen

Bullshit. That is for the "misunderstood geniuses" 8-).

taylor

aka the "tortured genius"

i believe that i would be a happier person if i were smarter

Jayesh Badwaik

It was true for galileo though.

"tortured genius"

skullpatrol

"ignorance is bliss"

: This proverb resembles “What you don't know cannot hurt you.” It figures in a passage from “On a Distant Prospect of Eton College,” by the eighteenth-century English poet Thomas Gray: “Where ignorance is bliss, / ‘Tis folly to be wise.’”

user19161

"Tortured genius" sounds like Will Hunting...

taylor

9:46 AM

or maybe its propaganda

user19161

Perhaps one day I will write my life story into a book...

user19161

@jonas What do you think of skull using your pic?

Jonas Teuwen

He does...?

taylor

i wanted to write my life story back when i was in highschool

user19161

Don't you see it?

Jonas Teuwen

9:51 AM

No.

Oh now I do.

skullpatrol

I don't even see it. I think there is something wrong with my gravatar account, it will not show any new changes of my avatar.

user19161

I thought I was seeing things...

user19161

Only those who believe can see, it's true.

Jonas Teuwen

Oh well.

skullpatrol

@WillHunting See he is relaxed about it, so why aren't you?

Jonas Teuwen

10:06 AM

Well, because he wants to make sure I am fine, I think. It is nice, but not needed here :-).

@WillHunting So thanks.

experimentX

hi

can anyone help me with with inequality>

$$ \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq {(a + b +c)(a^2 + b^2 + c^2)\over ab + bc +ac} $$

taylor

konnichi wa

experimentX

what property it uses ... hmm ... who's japanese here?

anyone any idea??

Jayesh Badwaik

@experimentX I am thinking inverting it would do something. working on it.

after inverting you can separate out the right hand side into two different easier to deal with partial fractions

left hand side I am not so sure

skullpatrol

The equality holds when a=b=c=1.

Jayesh Badwaik

10:20 AM

yesh shir

Jayesh Badwaik

10:30 AM

@experimentX
let $K = a + b + c $ and $L = a^{2} + b^{2} + c^{2}$
Then right hand side is
\begin{align}
RHS = \frac{1}{2}\frac{KL}{K^{2} - L }
\end{align}

hence
\begin{align}
\frac{1}{RHS} &= 2 \frac{K^{2} - L }{KL}\\
&= \frac{K}{L} - \frac{1}{K}
& \geq \frac{K}{L}
\end{align}

hence
\begin{align}
RHS \leq L/K
\end{align}

hence

\begin{align}
RHS < \frac{a^{2} + b^{2} + c^{2}}{a + b + c }
\end{align}

Now may be we can prove something from this

skullpatrol

@JonasTeuwen Has my avatar changed?

I see no change...

Jayesh Badwaik

10:47 AM

@experimentX solved it
Now once this is done
we show that
\begin{equation}
\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq a + b + c
\end{equation}

This can be easily shown using AM-GM inequalities
namely
\begin{equation}
\frac{a^{2}}{b} + b \geq 2\sqrt{\frac{a^{2} b}{b}} = 2a
\end{equation}

adding up the stuff, you get your answer.

Since
\begin{equation}
\frac{a^{2} + b^{2} + c^{2}}{a + b + c } < a+ b +c - \frac{ab + bc +ca}{a+b+c}
\end{equation}

Matt

11:00 AM

Me gusta:

</summer>

BenjaLim

@anon My calculations tell me that the real Lie algebras $\mathfrak{sl}_2(\Bbb{R})$ and $\mathfrak{su}(2)$ are simple

@anon I'm not sure if that's correct though because I try searching for a result and nothing comes up

taylor

11:41 AM

shouldnt it be called 'differential manifold' and not 'differentiable manifold'

robjohn

12:14 PM

@skullpatrol It looks very peaceful

Jonas Teuwen

@robjohn Does it...

robjohn

@JonasTeuwen It does

Jonas Teuwen

@robjohn Oh, I still see my photo there...

Son of a crack.

Jayesh Badwaik

Oddly enough, fans of oakland raiders defensive line are called skullpatrols

urbandictionary.com/…

robjohn

Matt

12:23 PM

@robjohn shudder : )

Wow. That was weird. I think a fighter plane just flew by, very low above ground.

Never heard that before.

Quite loud.

Jonas Teuwen

Yeah.

robjohn

@Matt Last week we had 3 days above 40°C

@JayeshBadwaik why is that odd... skullpatrol's avatar has often been an Oakland Raiders emblem

Yo guys

@OldJohn Hey guy

8-).

12:30 PM

@robjohn did not know that.. only today did a seach on skullpatrol.

@Matt more shudders for you
And these are average highs and low
last year absolute lows were around 6C (in january) and absolute highs at around 48C (in may)
http://en.wikipedia.org/wiki/Nagpur#Climate

Matt

@OldJohn Hello there.

@JayeshBadwaik Awful : (

Jayesh Badwaik

@Matt :-)

robjohn

@JayeshBadwaik 48°C ! That's hot! I think we've come within about 2.5°C of that here

Jayesh Badwaik

@robjohn yeah, those are the days when the streets are almost deserted. Those who are there have scarves around their heads.

Old John

@JayeshBadwaik Living in the UK makes that almost unimaginable

robjohn

12:44 PM

@JayeshBadwaik The day that we got to 116°F we were at an outside art fair and we would scurry from one tent to another.

Jayesh Badwaik

@OldJohn and we cannot imagine that you cannot have clear skies :P

I remember one day when there was not a single cloud over UK, a satellite had photographed it and it was a news on the internet

Old John

@JayeshBadwaik we had clear skies here - about a week ago - and for a whole hour !!

robjohn

@OldJohn amazing!

Jayesh Badwaik

@OldJohn Awesome!

Old John

Our "summer" here has just been declared the wettest in the last 100 years

Jayesh Badwaik

12:48 PM

@OldJohn Hmm. Our winter was the coldest and summer was the hottest of all time, and the monsoon hasn't changed much.

@robjohn but having grown up in Nagpur, I have become quiet accustomed to it and actually look forward to the hot air washing over my face while driving (especially when the fading winter is giving me the chills)

robjohn

@JayeshBadwaik 80's and 90's (F) are quite warm enough to thaw me out :-)

Matt

We chased a mosquito last night. -> tired.

I think we need one of those laser thingies that shoot mosquitoes.

Jonas Teuwen

@Matt I got hammered by some elephant pills -> tired. But I slept 8-).

Matt

@JonasTeuwen : )

Matt

1:30 PM

What a bad day.

Hola

Hola

Hi Peter

challah

@taylor Looks tasty!

1:39 PM

later folks

Matt

Later!

@JonasTeuwen Do you have any elephant pills spare?

Jonas Teuwen

@Matt What... why?

Matt

I can't work. But I have no time to not work.

Jonas Teuwen

@Matt Right, and so you plan on not caring about it anymore?

Matt

No. I don't know. Seems to be the end.

Jonas Teuwen

1:49 PM

Oh.

skullpatrol

@JonasTeuwen hi

I think there is something wrong with my gravatar account, it will not show any new changes of my avatar. I apologize for the appearance of my avatar :)

Oh, I still see my photo there...
Son of a crack.

Unfortunately, I can't find away to delete my gravatar account.

skullpatrol

2:15 PM

Even removing my email address from my MSE account doesn't help :(

It removes the gravatar from the user profile but not the one in chat...

experimentX

@JayeshBadwaik thank you so much for your effort ... i had to be offline due to some technical reason

anon

@BenjaLim have you consulted this table on Wikipedia? (Particularly the second and third entries.)

experimentX

hi @anon

anon

hello

Steenrod

2:41 PM

it seems reference requests about real analysis texts are common.

:)

taylor

math textbooks in general are pretty easy to find though

i mean good ones

not the ones where half the proofs are "left as an exercise" lolz

Steenrod

I am studying

Understanding

Analysis.Being someone who plans a bit, i am wondering whether I should follow

it up with Rudin or Pugh.

taylor

yeah ive never look at rudins book

what book is Pughs?

Steenrod

well, i do not remember

the name

googling pugh analysis may help

taylor

oh, Real Mathematical Analysis

yeah

i usually prefer books from springer

they seem pretty reliable

Steenrod

2:47 PM

like?

Pugh i believe is from Springer

taylor

yup :)

Linear Algebra Done Right, from Axler

dumb title but very good book

Axler is on the editorial board of Springer

maybe this one too is worth a mention amazon.com/Introduction-Manifolds-Universitext-Loring-W/dp/…

An Introduction to Manifolds, Loring

Steenrod

:)

To be precise, I have just started doing higher math.So I pretty

ignorant

taylor

well the yellow covers i think are pretty reliable

but ive been a little underwhelmed by a few of them

Zhen Lin

... someone bumped a question to change "learnt" to "learned" ....

taylor

sounds like a pedant

Steenrod

2:55 PM

@taylor, looks like I need to get going.

good day

taylor

good day sir

Mats Granvik

Hi guys.

I am having problems with my calculations.

taylor

does not compute?

Mats Granvik

I get two different answers for the same expression. Or almost the same expression. One is a number and the other is the same expression used in an equation.

anon

I have now decided to use portals (orange & blue markers) in my whiteboard commutative diagrams

3

Mats Granvik

3:16 PM

I found it. It was a division of parentheses involving too many factors.

robjohn

3:27 PM

@anon my whiteboard is pleasantly free of commutative diagrams :-)

@ZhenLin they corrected a typo. If there are a lot that they didn't correct, then I might worry, but if that is the only error, then I wouldn't.

evidently we lost another user. I lost 60 points.

anon

I only lost 30

they must have liked you better :)

Zhen Lin

@robjohn "learnt" is not a typo.

robjohn

@ZhenLin It looks like it to me...

Zhen Lin

It's a standard form in British English.

robjohn

@ZhenLin It is non-existent in American English. here is what I found.

Zhen Lin

3:41 PM

shrug

robjohn

@anon Oh, joy -_-

Zhen Lin

I wouldn't like it if someone went about changing "colour" to "color" either.

robjohn

@ZhenLin before correcting spelling, I look things up to see if there is a variant spelling. I would have found the variant and not changed 'learnt' even though it looks wrong to me.

skullpatrol

North American English is rather "liberal" in it's usage of vocabulary and grammar, many of which may be considered wrong according to Queen's English.

robjohn

the British just don't know English! :-D

Zhen Lin

3:45 PM

It's funny how some other parts of the world divide their languages over less...

skullpatrol

This is probably why most PhD programs require their mathematicians to be able to read math in more than one language, no?

taylor

but not just any other langauge

skullpatrol

Usually English and German.

robjohn