Large Hardon Collider

[Kryssan]

Yeah, brain wasn't functioning this morning lol, sorry. It especially wasn't working for my Physics exam

Verify the On/Off switch is placed in the On position. It makes a world of difference. I promise.

[Kaylia]

Yeah, brain wasn't functioning this morning lol, sorry. It especially wasn't working for my Physics exam

It's all good, mine hasnt functionned properly since 2004. I feel your pain.

[Eliseos]

Spent 10 of the 50 minutes trying to figure out why I was getting a number larger than c for relativistic velocity transformations. I forgot to subtract 1 in the denominator

[Khamsin]

Spent 10 of the 50 minutes trying to figure out why I was getting a number larger than c for relativistic velocity transformations. I forgot to subtract 1 in the denominator

Were you momentarily excited that you discovered faster-than-light travel?

[Woozie]

Here's a real math question:

I'm trying to make an epsilon delta prove for lim x > c, x^2 = c^2 (limit as x tends to c of x squared is c squared). I'm just going computationally, so far...

0 < |x-c| < δ

<=> c-δ < x < c+δ Let δ ≤ c

=> (c-δ)^2 < x^2 < (c+δ)^2
= c^2 - 2cδ + δ^2 < x^2 < c^2 + 2cδ + δ^2
<=> -2cδ + δ^2 < x^2 - c^2 < 2cδ + δ^2

Here I'm getting close to the form |f(x) - L| < ε, but I'm not sure how to finish it.

Seems like you're starting from a different direction than I'm used to. Since I don't see where you're going with this, I'm going to just start the way I'd normally do this.

In order to prove this, I had to use the triangle inequality, which I hope you're very familiar with if you like to evaluate limits by hand like this.

To prove this limit, we need to show that for any given arbitrarily small ε, I can choose some δ such that |x-c| < δ => |x^2 - c^2| < ε

When we do limits in analysis, the professor makes us start with a discussion, then a proof. I'm going to do it the same way here.

Discussion:

We need to find a sufficiently small δ such that |x-c| < δ => |x^2 - c^2| < ε. But note that:

http://latex.codecogs.com/gif.latex?...2}|=|x+c||x-c|

Since we're making http://latex.codecogs.com/gif.latex?|x-c|%3C\delta, it follows that http://latex.codecogs.com/gif.latex?...%3C|x+c|\delta

So we have

http://latex.codecogs.com/gif.latex?...%3C|x+c|\delta

If we can show that http://latex.codecogs.com/gif.latex?...%3C\varepsilon, it will follow that

http://latex.codecogs.com/gif.latex?...%3C\varepsilon

or

http://latex.codecogs.com/gif.latex?...%3C\varepsilon

which is what we want.


So, to make http://latex.codecogs.com/gif.latex?...%3C\varepsilon, first note that:

http://latex.codecogs.com/gif.latex?...%3C\delta+2|c|

(note the use of the triangle inequality)

So http://latex.codecogs.com/gif.latex?|x+c|%3C\delta+2|c|

meaning

http://latex.codecogs.com/gif.latex?...(\delta+|c|)^2

which implies

http://latex.codecogs.com/gif.latex?...(\delta+|c|)^2

(do you see what I'm doing by now? Every step I take, the number either gets bigger or stays the same, yet it gets simpler. As long as it doesn't get smaller, any simplification is valid. If I need to show that p < r, it suffices to show find a q that is bigger than p, and then show that q < r instead. Because since q > p, then showing q < r will automatically imply that p < r. This last expression is my final q and our original expression is our p, and epsilon is our r in this analogy. The whole point in finding that middle man instead of working directly with what's given is that the middle man q may be simpler than p. In this case, we need an expression that is constant [i.e. has no "x" terms in it. c is fine since it's constant])

if you're following, then you understand that if I show that (δ+|c|)^2 < ε, then the desired result follows and we're done.

To make this inequality true (the inequality in bold print), we just solve for δ.

http://latex.codecogs.com/gif.latex?...arepsilon}-|c|

So we found our delta. The rest should be simple. To be honest, I feel like I made a mistake somewhere, but i can't be bothered to recheck this. In fact, I'm sure I made a mistake somewhere, because something about my answer doesn't make sense (look at my solution. There's a huge problem with it. Can you see what it is?)

[Kaylia]

While we are at it, I have a relatively simple quantum mechanics (mathematics) questions, but I fucking suck at linear algebra, and spent the last hours on this demonstration

S1 and S2 are ½ spin operator (they should have arrow, but ascii suck)

Show that projector on spin 1 and 0 are given by
P1 = ¾ + ℏ-²(S1· S2) and P0 = ¼ - ℏ-²(S1· S2)

[edit]
question 14.8
http://books.google.com/books?id=qHx...age&q=&f=false

[edit2]
nvm, that page isnt there. oh well..

[Woozie]

Might want to wait for woozie seal of approval before writting this in your homeworks. There is probably easier way to do it too.

Well, my answer was a lot longer and more complicated, and has to be wrong because my delta could become negative.

Edit: I'll work on your problem, Kaylia, but now that House is on, it will probably take a while.

[Altariel]

I thought its been theorized that neutron stars might collapse into strange stars? Has that been refuted now or just unconfirmed?

[Cadsuane]

Hmmm, let me try


Definition

Lim(x) = C ⇔ ∀ε> 0 ∃δ>0, for all x ∈ f domain and 0 < | x - x0 | < δ => |f(x) - L | < ε
x->x0

Proof
For f(x) = x²
We are trying to show that

∀ε> 0 ∃δ>0, 0 < | x - C| < δ => |x² - C²| < ε
x->C

Since we can write
|x² - C²| = |x - C|*|x + C| ≤ (|x|-|C|) * (|x -C|)

Let's assume
(|x -C|) ≤ 1, then we have
|x|-|C| ≤ ||x|-|C|| ≤ |x -C| ≤ 1

Going back to our original statement, we can say that
|x² - C²| ≤ (2*|C| +1) (|x -C|) < ε
=> (|x -C|) < ε/(2*|C| +1)

As long δ < Minimum[ ε/(2*|C| +1) , 1 ], the statement ∃δ>0, making the lim x² = C² true


[edit]

Might want to wait for woozie seal of approval before writting this in your homeworks. There is probably easier way to do it too.

First of all, thanks for taking the time to solve it. Secondly, I'm afraid you lost me.

Can you explain how the first statement (|x² - C²| = |x - C|*|x + C| ≤ (|x|-|C|) * (|x -C|)) is true? particularly how |x² - C²| ≤ (|x|-|C|) * (|x -C|). I don't see anything really obvious like the triangle inequality.

And (|x² - C²| ≤ (2*|C| +1) (|x -C|) < ε => (|x -C|) < ε/(2*|C| +1)) I can't fathom at all how we got there.

The structure of the proof is a bit confusing to me since what I've been doing for these is choosing a delta instead of putting bounds on it, which I still don't get for anything besides first degree functions. Do you mind going through it a bit more verbose?

[kuronosan]

What did the photon say to the neutron?

Nothing, it just waved!

Oh lord I laughed.

[Kaylia]

I thought its been theorized that neutron stars might collapse into strange stars? Has that been refuted now or just unconfirmed?

I'm not the most qualified person to give you an answer, but I think it's more along the line of "we have never seen anything weird about neutron star that seem to indicate they collapsed into strange star". The emission and maths probably correspond to what we expect from our normal matter model.

First of all, thanks for taking the time to solve it. Secondly, I'm afraid you lost me.

Can you explain how the first statement (|x² - C²| = |x - C|*|x + C| ≤ (|x|-|C|) * (|x -C|)) is true? particularly how |x² - C²| ≤ (|x|-|C|) * (|x -C|). I don't see anything really obvious like the triangle inequality.

And (|x² - C²| ≤ (2*|C| +1) (|x -C|) < ε => (|x -C|) < ε/(2*|C| +1)) I can't fathom at all how we got there.

The structure of the proof is a bit confusing to me since what I've been doing for these is choosing a delta instead of putting bounds on it, which I still don't get for anything besides first degree functions. Do you mind going through it a bit more verbose?

Only reason why I didnt use more words is because I never done maths in english, and I'm lacking the correct vocabulary (or rather, it's a pain to find the correct wording).


|x² - C²| = |x - C|*|x + C|
We simply decomposed it into 2 term |x+c||x-c| = |x² +xc-cx+c²|

|x-C| |x + C| ≤ (|x|-|C|) (|x -C|))
Both side have |X-C|, so it's the equivalent of |x + C| ≤ |x|-|C|, which is one of the numerous variation of triangle inequality. If you can't see it, the right term is always a "difference" between 2 positive number, while the left term is at worst a difference between two positive number (positive + negitive number), but can also be a sum.


From the first and last term of this line |x|-|C| ≤ ||x|-|C|| ≤ |x -C| ≤ 1, we can say that |x| ≤ C +1 (add |c| on both side). Going back to the original statement |x² - C²| ≤ (|x|-|C|)·|x + C| < ε, we replace the |x| with C +1 , sum both X for simplicity and get the final line.

The structure of the proof is a bit confusing to me since what I've been doing for these is choosing a delta instead of putting bounds on it, which I still don't get for anything besides first degree functions. Do you mind going through it a bit more verbose?

Can you give us your definition of "limit"? To make this kind of demonstration, it's important to use the correct definition, and depending of the one you use in your class, it might be slightly different.


I'm pretty sure it's doable the other way around, but I've no clue at the moment what it would look like.

[Woozie]

While we are at it, I have a relatively simple quantum mechanics (mathematics) questions, but I fucking suck at linear algebra, and spent the last hours on this demonstration

S1 and S2 are ½ spin operator (they should have arrow, but ascii suck)

Show that projector on spin 1 and 0 are given by
P1 = ¾ + ℏ-²(S1· S2) and P0 = ¼ - ℏ-²(S1· S2)

[edit]
question 14.8
Quantum mechanics: from basic ... - Google Books

[edit2]
nvm, that page isnt there. oh well..

Sorry, I've been trying to solve this and couldn't get an answer. I'm going to have to start working on my own homework soon. I'll keep trying until House goes off, but I'm really screwing up here, even worse than in the limit problem above. Today is just not my day (you should have seen how bad I screwed up on my Computer Science assignment).

Edit: Now that I think about it Julian asked me a question a week ago and I couldn't solve it. I've been having problems in Probability, Abstract Algebra, and Mechanics I for the past few weeks (all three of which I've done before so it should be really easy). There's something else wrong with me here.

[Kaylia]

Sorry, I've been trying to solve this and couldn't get an answer. I'm going to have to start working on my own homework soon. I'll keep trying until House goes off, but I'm really screwing up here, even worse than in the limit problem above. Today is just not my day (you should have seen how bad I screwed up on my Computer Science assignment).

Edit: Now that I think about it Julian asked me a question a week ago and I couldn't solve it. I've been having problems in Probability, Abstract Algebra, and Mechanics I for the past few weeks (all three of which I've done before so it should be really easy). There's something else wrong with me here.

It's alright, thanks for trying haha.


I'm having so much trouble in quantum mechanics lately. I never had linear algebra class...and it's starting to show. I know it's simple, but when there is a ton of unclear concept, it's getting harder to figure out a solution.

[Khamsin]

I thought its been theorized that neutron stars might collapse into strange stars? Has that been refuted now or just unconfirmed?

I think it's more of a "it hasn't happened yet, and you'd think it would have by now" thing than a straightforward refutation.

But, the universe is still young. We're barely into our teenage (billion) years, and while there was a finite beginning there's no finite end, so as the universe ages indefinitely we'll finally begin to see things that took a trillion years to happen. Unless "ultimate fate of the universe" events constitute an end to time, as well.

[Mizango]

Were you momentarily excited that you discovered faster-than-light travel?

It's all about VSL baby! V-S-L!!

[Woozie]

I thought its been theorized that neutron stars might collapse into strange stars? Has that been refuted now or just unconfirmed?

I don't know about that, but I do know that there are certain conditions under which strange matter can exist and it probably can exist inside of neutron stars. But I've never heard of an entire neutron star becoming a strange star.

The whole strange matter Ice9/greygoo doomsday thing relies on the idea that strange matter is actually more stable than normal matter. If strange matter is sometimes stable, that's fine, as long as it isn't the most stable state under normal conditions.

If you go to youtube and type in "supercooled water" or "superheated" water. These are examples of what are called "metastable states". This means the state is relatively stable, but there is a more stable state. Water under 0 degrees C (under our atmospheric pressure) is most stable as ice. When you supercool water, you bring water below the freezing point without it actually freezing. As I mentioned earlier, everything in nature likes to be in its most stable state, so the water wants to go into the ice state. It just needs something that can bring it out of the metastable state. In the case of water, pretty much any little disturbance can do so, and the chain reaction is the entire bottle freezing over (or exploding if it's superheated. If you've never seen it before, watch the videos, it's pretty cool).

The situation with strange matter is the same. If strange matter is the most stable state, then normal matter is just a metastable state. In that case, bringing strange matter into contact with normal matter will cause normal matter to turn into strange matter for the same reason hitting a bottle of supercooled water turns it into ice.

[Woozie]

http://www.youtube.com/watch?v=DpiUZI_3o8s

http://www.youtube.com/watch?v=1_OXM4mr_i0

Now imagine that happening to the entire Earth, but instead of ice or steam, it all turns into a huge hot blog of degenerate matter. Well, not huge, it would actually be extremely small because it's degenerate matter, which is basically the smallest (densest, most compact) way possible to arrange fermions. That's why a small neutron star smaller than the city you live in would be heavier than the sun.

Edit: But as I've mentioned a billion times, the strange matter thing wont happen.

[Eliseos]

I had read about that, but never saw any video of it happening. Filing this under "The Cool Science Experiments to Show Your Friends" category. In the same category is burning steel wool:

[Woozie]

Wow @_@ I've never seen that before. How exactly does that work?

[Eliseos]

Not sure exactly on the science behind it, google doesn't turn up much. All I know is that it burns extremely well, and twirling it around like that accelerates the burning. You can also start them on fire with a 9V battery. I used to detail cars at a dealership, burning that shit was so much fun lol. The videos don't do it justice, it's so bright in person.