Mathematics - 2012-09-27 (page 3 of 4)

@JasperLoy: thanks, that's what I was thinking it would be. In my context, A is a K-algebra, and then if $e\in A$ then $eAe\subset A$ is a K-vector subspace

Charlie

@JohnJunior Here when the girl grabs his hands and realizes it's him...and he gives her "the look"...it's really touching...makes mewanna cry every time.

hhh

@JayeshBadwaik I have a ball in the North dropped and now I try to understand the point $[w\sin(\theta),0,w\cos(\theta)]$ near the bottom

(I have corelios -effect, the centrapetal -thing, moving coordinate -term and accelerating -coordinate --thing)

(i am stuck to this geometric point...cannot see how the $\hat j$ is zero.)

user19161

8:25 PM

I don't know physics. =)

user19161

Post on the main site. It's hard to understand your question like this.

user19161

Chat is for trivial questions only, like 1+1=2.

hhh

This is trivial: $w$ is the angular velocity of planet and $\bar v$ is the velocity of the ball.

Jayesh Badwaik

@hhh I did not understand a word of what you said. Also, in that figure, I cannot figure out what you are talking baout. It will be probably better if you can draw a sketch representing just what you want to be asked. :-)

hhh

You have a falling ball.

The ball is twisted by the rotation of the Earth and its velocity.

Jayesh Badwaik

8:28 PM

Okay, a ball is falling. Where is it falling?

hhh

Let say Germany :)

Jayesh Badwaik

Okay, so you drop a ball over Germany? Hmm, okay.

hhh

I sticked the coordinates to the surface of the rotating earth, the reference frame of the ball.

Jayesh Badwaik

Now, you mean to say the motion of the ball is twisted by the earth's rotation?

Charlie

@hhh when you make that cartesian $W\times V$ you gotta take some derivatives, doncha?

Jayesh Badwaik

8:30 PM

So, basically, you want to calculate the motion of the as seen from surface of the earth (which is rotating)?

hhh

$$\bar a=\bar a'-2\bar w\times\bar v -\bar\alpha\times\bar r -\bar w\times(\bar w\times\bar r)$$ where $\bar a'=\bar g$ (gravitation), $\bar \alpha$ is the angular acceleration of the Earth, $\bar w$ is the angular velocity of the Earth, $\bar r$ is the position of the ball in Germany falling, --.

Jayesh Badwaik

And with this you lost me again. Wait a sec, let me get my notebook.

N3buchadnezzar

Giii

Charlie

@JohnJunior why do you like skull?

hhh

where $\bar \alpha\times\bar r=0$ because Earth is rotating with constant speed.

So

$$\bar a=\bar g-2\bar w\times\bar v -\bar w\times(\bar w\times\bar r)$$

where gravitation, corelios -effect and central-petal-thing -effect (keskipakoiskiihtyvyys -thing in Finnish).

Jayesh Badwaik

8:34 PM

@hhh What is $\bar{a}$? Acceleration of the ball wrt the point on the surface of earth in germany?

Charlie

@hhh ok

John Junior

@Charlie dontcha like skull?

N3buchadnezzar

Anyone mind helping with some linear algebra? =)

Jayesh Badwaik

@hhh okay.

Charlie

@JohnJunior :-D just a little

hhh

8:37 PM

@JayeshBadwaik $\bar a$ is the acceleration of the ball falling.

Jayesh Badwaik

@hhh Wrt a point on the earth right?

Wait, you have fixed the co-ordinates to earth, so okay.

hhh

@JayeshBadwaik Yes like the picture shows

Charlie

@hhh I didn't undertand what you didn't...

hhh

You can see there $\hat k$, $\hat j$ and $\hat i$ at the top. It is the point where the ball is falling in the direction of $\hat k$, pointing to the center of planet.

Charlie

no I cannot

Jayesh Badwaik

8:40 PM

@hhh okay, I am getting it now.

Now ask your question.

You want to know why the acceleration in direction $\hat{j} = 0$?

hhh

(fixed it with my wacom :D -- hopefully now clearer, the falling ball is the red point)

user19161

@charlie How was the test?

Jayesh Badwaik

Okay.

@Charlie you had a test?

Charlie

@JasperLoy it's tomorrow.Thanks for asking.

@JayeshBadwaik i have.tomorrow.

Jayesh Badwaik

@Charlie Best luck.

user19161

8:47 PM

OMG I just got 40 points for aiming for some ground fruit @jay.

Jayesh Badwaik

@JasperLoy Awesome!

Charlie

@JayeshBadwaik Thanks!

Jayesh Badwaik

Where?

user19161

@JayeshBadwaik Here. math.stackexchange.com/questions/203626/…

hhh

@JayeshBadwaik Now I want to express $\bar w$ in the ball coordinates, how?

user19161

8:49 PM

Again, the result of algebra teachers not making students understanding their steps. It happens here on a large scale.

hhh

I know the solution i.e. $[\hat i,\hat j, \hat k]=w[\sin(\lambda),0,\cos(\lambda)]$ but cannot see it from the picture, some projections?

Jayesh Badwaik

@JasperLoy Or probably too much homework and fatigue? I know your explanation is probably the more likely one, but benefit of doubt. Because, else, it would be depressing.

user19161

@JayeshBadwaik Hmm, not likely. Too much homework yes, but it is useless as they are drilled without understanding...

Jayesh Badwaik

@JasperLoy Yup.

@hhh So, basically you cannot understand why the $j$ component should be zero?

hhh

@JayeshBadwaik precisely

Charlie

8:51 PM

@JohnJunior "freak like me" she doesn't know what freak is...

user19161

@Charlie Neither do I.=)

Jayesh Badwaik

@hhh Actually, that is the easy part. I actually expected you to ask why the $i$-component is non-zero!

Charlie

@JasperLoy Dontcha?

user19161

@Charlie I know very few words. I don't read fiction. I only read math books, and "freak" is not in math books.

Jayesh Badwaik

@Charlie who are we talking about?

Charlie

8:54 PM

@JayeshBadwaik I was talking about this

@JasperLoy Nice!

Jayesh Badwaik

@Charlie Ohh. Don't cha. Hmm. Nicole Scherzinger is kinda cool (if i have got her last name right).

hhh

@JayeshBadwaik Perhaps it is but simple things first, trying to do some new pictures to help here.

Charlie

@JayeshBadwaik scherzinger

Jayesh Badwaik

@hhh Try to picture the earth revolving underneath while the ball is falling from above towards the center.

Charlie

@JayeshBadwaik this reminds me vector calculus.

Jayesh Badwaik

8:57 PM

@Charlie It is vector calculus. "Physics is geometry --Einstein"

Charlie

@JayeshBadwaik that's why it reminds me

Jayesh Badwaik

@hhh Even better. Try to think of a train running on the ground in a straight line and a ball dropped above it. Now convert that train into a moving latitude and think of what should be the acceleration.

user19161

@jay You are a night owl too!

Charlie

Bye @JohnJunior!

user19161

@cha What is the test about tomorrow?

Charlie

9:00 PM

@JasperLoy analysis.do you wanna help figuring out one exercise?

Jayesh Badwaik

@JasperLoy Yup, I like the quiet in the night. I wrote a poem about diwali nights some months ago describing the beauty of the sodium vapor lamps, the "akaashdive" or the hanging lamps and the chill of the winter air.

@Charlie I do.

Charlie

@JayeshBadwaik yay,thnx

Jayesh Badwaik

@Charlie I need some practice too. :P

Charlie

"Let $f:\mathbb {R^n}\rightarrow \mathbb{R^n}$ a function that maps bounded sets to bounded sets. Prove that $f$ is continuous iff its graph is closed "

the part continuous implies closed is done.

hhh

@JayeshBadwaik xy -plane is a spinning circle. xy -plane and yz -plane have an oscillating point along their lines.

Charlie

9:04 PM

so if the graph is closed, then $f$ is continuous.

suppose it's not continuous.

Jayesh Badwaik

@Charlie Graph is closed means?

Charlie

@JayeshBadwaik diwali is kinda funny,isn't it?party of lights?isn't it?

@JayeshBadwaik $G'\subseteq G$

@JayeshBadwaik you write poems...

Jayesh Badwaik

Ohh you mean the set $(x,f(x))$, okay.

Charlie

@JayeshBadwaik yeah yeah yeah(she loves you yeah yeah yeah :P)

Jayesh Badwaik

Sie Liebt dich, yeah yeah yeah.

Charlie

9:11 PM

@JayeshBadwaik yes!!!

Ich liebe dich

Beatles

hhh

@JayeshBadwaik Nothing?

Jayesh Badwaik

@hhh I could not understand what you wanted to say.

hhh

@JayeshBadwaik "--think of what should be the acceleration" -Nothing, zero -- correct?

I mean the accelaration of the ball, the ball is not affected by the moving train or latitude or?

Jayesh Badwaik

@hhh Correct.

hhh

roger that is why $\hat j$ is zero, now other terms...thinking.

Jayesh Badwaik

9:15 PM

@Charlie You have closed and bounded sets. So they are compact.

Charlie

@JayeshBadwaik the graph is closed,not the set.

@JasperLoy Good morning,Jas.

Jayesh Badwaik

@Charlie projections? (nvm, they are not absolutely necessary).

Old John

Good evening

Charlie

@JayeshBadwaik if the graph is closed,then there's a sequence that converges to a point of accumulation:$(x,y)\in G' \Rightarrow \exists (x_n,y_n)_n \subset G$ such that $(x_n,y_n)\rightarrow (x,y)$

Jayesh Badwaik

@Charlie Suppose $E$ is a bounded set in $R^{n}$ then assume $(E,f(E))$ is closed. That means both $E$ and $f(E)$ must be closed, right?

hhh

9:24 PM

@JayeshBadwaik I cannot see the $\hat i$, how is it different to $\hat j$? Longitudial?

Charlie

Suppose that $f$ is not continuous

hhh

Err it changes the longitude all the time! Unless corrected...with $r\cos(\lambda)$ or $r\sin(\lambda)$.

Charlie

@JayeshBadwaik yes

Jayesh Badwaik

So, the function pulls back closed sets to closed sets. Hence it is continuous. QED.

Charlie

@JayeshBadwaik But the exercise doesn't say that the sets are closed,only bounded.

N3buchadnezzar

9:27 PM

@OldJohn Heeeey sexy old man =)

Old John

@N3buchadnezzar whaaaat???

Jayesh Badwaik

@Charlie You are proving the reverse! You are assuming the graph is closed and want to prove that the function is continuous!

Charlie

@JayeshBadwaik yep

Jayesh Badwaik

@Charlie So?

N3buchadnezzar

@OldJohn I just wanted to grab your attention to help me with a few problems that have deadline for tomorow =)

robjohn

9:28 PM

@JohnJunior Google image search says that is Ruben Cortada

Old John

@N3buchadnezzar LOL - OK:)

Charlie

@robjohn damn it!I thought it was him... :-P

@JayeshBadwaik so...

N3buchadnezzar

Is a possible basis for the solution set of
$$\begin{align*}
x_1 - 2 x_2 + x_3 & = 0 \\
2x_1 - 3 x_2 + x_3 & = 0
\end{align*}$$
$$
\begin{align*}
\begin{bmatrix}
\phantom{-} 3 \\ -6 \\ \phantom{-}2
\end{bmatrix} \end{align*}$$ ? =)

robjohn

@Charlie LOL!! If I run my gravatar through Google Images, it matches it to Aquaman!

Jayesh Badwaik

@robjohn I would have preferred Rainman. :P

@Charlie so, the graph is compact (it is closed and bounded in $R^m$)

robjohn

9:32 PM

@N3buchadnezzar it isn't even a solution of the first equation

Old John

@N3buchadnezzar Hmm - that vector doesn't seem to satisfy the equations

N3buchadnezzar

Oh, sorry. Of course Gauss jordan... I simply added them

hhh

@JayeshBadwaik $\hat i$ is projected to the $z$ -axis. $\hat k$ is projected to the $x$ -axis -- I cannot yet see why this way.

Old John

@N3buchadnezzar you need to find a solution of the two equations simultaneously - and examine the dimension of the set of solutions

Charlie

@HenningMakholm Hallo!Wie geht's dir?

@JayeshBadwaik jay...I'm confused now...

Old John

9:35 PM

each of those equations defines a plane in $\mathbb{R}^3$, so the intersection could be a line or a plane

Henning Makholm

@Charlie Um, do I know you? (Are you Skullpatrol?)

Charlie

@HenningMakholm I think so.The other day you said that I should speak german or dansk,I know a little german...

Jayesh Badwaik

No, she is meandmath. JohnJunior is skullpatrol.

N3buchadnezzar

@OldJohn Seems like i end up with

$\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 & 1 & -1 & 0
\end{bmatrix}$

Henning Makholm

Actually what I said translates to "Speak Danish, you German dogs!"

(which is the first half of a joke that continues "... said the farmer to the Frenchman")

Old John

9:38 PM

@N3buchadnezzar How do you get get a matrix???

Jayesh Badwaik

@HenningMakholm I have a album by Maria Montella. (And so the story goes...)

Charlie

@HenningMakholm It happens that I'm not german...and that was portuguese...you could have explained the joke...

N3buchadnezzar

@OldJohn Oh, it is just easier for me to solve the equations in that way

x_1 = x_3
x_2 = x_3

Old John

Ah - I like that better

Charlie

@N3buchadnezzar LOL

Old John

9:40 PM

so - for example, taking then all to be 1, we have the vector (1,1,1) as a solution of both equations, yes?

Henning Makholm

@Charlie I strongly suspect the joke was no more opaque to you than the Portugese was to most of the rest of us.

Charlie

@Jay

N3buchadnezzar

This seems very one dimensional to me

Old John

(I'm writing my vectors as a row to make it easier for me)

Charlie

@HenningMakholm Portuguese.and I doubt of that...

N3buchadnezzar

9:41 PM

Since the solution is $x_1=x_2=x_3$

Old John

Yes - the solution set of both equations is a line - the intersection of the 2 planes

Jayesh Badwaik

@Charlie I made a mistake of assuming that the graph is compact.

Charlie

@JayeshBadwaik I was trying to say it...

Jayesh Badwaik

I was trying to simply extend the proof.

I have a proof from Rudin, where I have proved that if the graph is compact, then it is continuous.

Problem 6 from the chapter of continuity

hhh

@JayeshBadwaik moved it here.

N3buchadnezzar

9:42 PM

But I wanted a basis for the solution

so I can use the basis t(1,1,1) ? =)

Jayesh Badwaik

@Charlie Yup, you were.

Old John

@N3buchadnezzar so, a basis of a 1-dimensional space (the line) would be the set consisting of the single vector (1,1,1)

Jayesh Badwaik

@Charlie I found this

Charlie

@JayeshBadwaik brilliant Jayesh!

Jayesh Badwaik

@Charlie what it just took a google search.

robjohn

9:44 PM

@OldJohn: you put dark glasses on God?

Old John

@robjohn I like my blue glasses :)

robjohn

@OldJohn I just discovered search on Google images... pretty cool!

Old John

@robjohn ah - you found where I got the original image from? (Man khoda hastam)

Jayesh Badwaik

@Charlie I guess the limit approach is necessary, instead of the closed set approach I was talking as the compactness is not given and boundedness is a metric property, not a topological one.

robjohn

@OldJohn I found it mainly on a YouTube Video

Charlie

9:47 PM

@JayeshBadwaik hmmm. the given answer does what I already did...

Old John

It is a Persian YouTube cartoon - with the main character being "God" - talking abut various things - there was a great episode about the Higgs Boson :)

robjohn

@OldJohn I added a link to my previous comment

N3buchadnezzar

@OldJohn indeed

Old John

@robjohn My Persian is not that great, but I an still amused by what I can understand of the episodes

robjohn

@OldJohn My persian is limited to cats and rugs.

4

Jayesh Badwaik

9:50 PM

@Charlie hmm. Let me prove it. I will get back to you with it.

Old John

@robjohn I like the rugs better :)))

robjohn

@OldJohn we all need prostheses as we get older ;-D

Old John

@robjohn tell me about it :)

@robjohn I really like my gravatar, but feel a bit guilty that I have probably breached some guys copyright - and I did upset Gigili with it

Jayesh Badwaik

@Charlie You can use the fact that the graph is bounded and closed, and hence, the graph is compact.

hence it is sequentially compact.

Old John

If I could draw, I would create my own version (I also liked the image of muppet drummer "animal")

Jayesh Badwaik

9:54 PM

@Charlie Hence, every sequence has a convergent subsequence.

Charlie

@JayeshBadwaik yes

robjohn

@OldJohn she recognized the character?

Old John

@robjohn No - but got upset when I mentioned that it came from "Man Khoda Hastam"

robjohn

@JayeshBadwaik every sequence is subconvergent...

Old John

she is a native Persian speaker

Jayesh Badwaik

9:55 PM

@robjohn shorter way to write it?

robjohn

@OldJohn ah, I can see that

@JayeshBadwaik no, just playing with words.

Jayesh Badwaik

@robjohn okay.

robjohn

@OldJohn I have found that out.

Charlie

@Jay I don't if this is right...

Old John

@robjohn I got into a silly argument about the meaning of the Persian words

Jayesh Badwaik

9:57 PM

@Charlie If this is the right approach?

Hmm, me too.

Charlie

@JayeshBadwaik exactly

robjohn

@OldJohn I wouldn't have been able to do that.

Charlie

I'm starting to have pre test anxiety...palpitation...fast breath...

Jayesh Badwaik

Oh dear. That is bad.

Old John

@robjohn The Youtube videos are actually the creation of a sort of group of Persian atheists trying to promote rational thinking, I believe - I was pointed to them by an Iranian friend/teacher

Charlie

10:00 PM

@JayeshBadwaik once I made a real analysis test with so much nausea...I couldn't even move my head an epsilon without losing my balance...

or throwing up

was a bad day...

Jayesh Badwaik

@Charlie hmm. You can do the problem easily with projections. If you want to use the projectiosn that is.

Charlie

@JayeshBadwaik hmm

N3buchadnezzar

Sigh. Any quick tips to showing $\left\{ f(x,y,z) \in \mathbb{R}^3 \; \mid \: x = 3y \ \text{and} \ z = -y\right\}$

Jayesh Badwaik

@Charlie no, just a little hungry.

Will raid my fridge to see what I have. Thinking of making french toast for breakfast.

N3buchadnezzar

Is a subspace of $\mathbb{R}^3$ under the operations of addition and scalar multiplication defined on $\mathbb{R}^3$?

Charlie

10:04 PM

@JayeshBadwaik tasty!

Jayesh Badwaik

@Charlie Yeah, the thing is there are eggs and milk in the fridge, if I don't finish it, it would go stale.

Old John

@N3buchadnezzar That seems to be a subspace spanned by (3,1,-1)

Jayesh Badwaik

@Charlie The projection thing is simple. it is a simple map $\pi_X : X \times Y \mapsto X$ and it is continuous.

Charlie

@JayeshBadwaik then :FINISH THEM!

Old John

in other words, any vector in that set is of the form (3k, k, -k) for some real number k

N3buchadnezzar

10:06 PM

math.ntnu.no/emner/TMA4145/2012h/exercises/oving5.pdf It is the first one here. Usually these things are easy to prove. One just multiplies it by k or some other konstant and shws it still lies in the space.

Jayesh Badwaik

So, using that, you can pull back closed sets from closed sets.

Charlie

@JayeshBadwaik YES

Jayesh Badwaik

you can do the same to $\pi_Y: X \times Y \mapsto Y$

N3buchadnezzar

@OldJohn I see thanks =)

Old John

so you just need to show that the set of all vectors (3k,k,-k) is a subspace - i.e. closed under addition and scalar mult

Jayesh Badwaik

10:07 PM

Also, you use the fact that $Y$ is compact.

To make $\pi_X$ a closed map.

And then you are done.

Old John

@N3buchadnezzar I can't always remember the precise words to use in Linear algebra, but I still remember the basic ideas :)

Charlie

I still don't know...

N3buchadnezzar

=)

Yeah, I just need a small push in order to complete these things.

Thanks a bunch for helping me otherwise, I am not sure I would have been able to.

Jayesh Badwaik

@Charlie I am hungry now. Will come back in some time. Will try to finish it off completely.

Old John

@N3buchadnezzar No problem - it is actually quite fun trying to recall stuff I did so many years ago :)

Charlie

10:10 PM

@JayeshBadwaik Thanks Jay.Bon appetit!

Old John

I think it is important to get a feel for what is going on with these things, as well as being able to work with the precise definitons

Jayesh Badwaik

@Charlie Think of the simple function $f(\bar{x}) = ||x||$

It maps bounded sets to bounded sets right?

It is also continuous.

N3buchadnezzar

10:25 PM

@OldJohn Any tops on this one?

$\{ (x,y,z)\in\mathbb{R}^3\mid \, 5x^2 - 3y^2 + 6z^2 \} $

Old John

@N3buchadnezzar erm - something missing at the end there, I think?

N3buchadnezzar

=0

Old John

ah!

Hmm - I don't see that one being a linear subspace ...

N3buchadnezzar

I gave a suprised face, and I answered your question!

Charlie

@JayeshBadwaik Oh, Jayesh...

Jayesh Badwaik

10:28 PM

@Charlie what?

Charlie

@JayeshBadwaik I'm freakin' nervous!

Old John

@N3buchadnezzar It is definitely not a subspace :)

user19161

@Charlie Well, drink some beer then.

user19161

More beer, less nervousness!

Charlie

@JasperLoy You know I don't drink...

Jayesh Badwaik

10:36 PM

@Charlie Even @JasperLoy doesn't

user19161

@Charlie No, I don't know that. You are not Muslim or Jewish are you?

user19161

@JayeshBadwaik I mostly drink tea, coffee, pepsi, coca cola, and water. QED.

Charlie

@JasperLoy No,I already said that.

I'm catholic...

@Jas

user19161

Ah OK.

N3buchadnezzar

@Charlie Catholics drink too

Charlie

10:44 PM

@N3buchadnezzar I'm different.Priests should not be pedophiles,some are...

N3buchadnezzar

@Charlie So you like children huh? Well thats cool bro

Charlie

@N3buchadnezzar I don't like children.You are misunderstanding me...

N3buchadnezzar

@Charlie It was a joke! Take life easy, I understood what you meant.

Charlie

@N3buchadnezzar don't joke about these things...

N3buchadnezzar

Well if you do not do anything crazy and live a boring live. Then what have you done in your life, spent it reading books? I mean I love reading books,but heck I also need to have fun. Chat with people, drink have fun.

Old John

10:47 PM

@N3buchadnezzar "do you like children? - yes, but I couldn't eat a whole one" (quote from some comedy film)

N3buchadnezzar

Great song

Oingo Boingo - Little Girls

Charlie

@N3buchadnezzar I'm talking to people right now.And I don't like drinking...Yes I Spend my life studying and reading and listening some good music.I never got hurt because of that.I'm not a type of girl who likes "parties", so what,I'm like that...

N3buchadnezzar

I hate parties too ;)

Charlie

I like my liver...

N3buchadnezzar

Your argument is the same as saying you are a vegetarian, because you do not like the places they serve meat.

Charlie

10:54 PM

@N3buchadnezzar no it's not

N3buchadnezzar

The point of life is to live, not merely exist.

Old John

@N3buchadnezzar Is snus legal in Norway? - and if so, is it commonly used?

Charlie

@N3buchadnezzar You think living is doing those things you said?

N3buchadnezzar

@OldJohn Yes, and it deppends where you are. Where i live maybe 1/10 does it. But smoking is much rarer between younger people.

Old John

Ihteresting - I used to smoke years ago, and managed to quit by using snus

N3buchadnezzar

10:58 PM

I think you do not drink, not because of your liver, but because you are afraid of letting go. You hurdle up inside a shell and is afraid of what is outside, therefore you coat yourself in "reasons" and "believes" so you can continue doing what you have always been doing. Never daring, never doing anything out of the ordinary.

Charlie

@N3buchadnezzar I see no problem in living like that...

Transcript for

$Mathematics$ Mathematics