Mathematics

Alias User

12:04 AM

You're the best. Thank you!

@anon - how can I prove that that is an anti symmetric bilinear form?

anon

that what is?

Alias User

$a^TMb$

anon

it's antisymmetric iff M is antisymmetric

so, not generally antisymmetric

Alias User

I have M is the inverse of a positive definite matrix. Is that sufficient?

anon

definitely not

for example if M is the identity matrix, it's symmetric not antisymmetric

Mike Miller

12:14 AM

as far as I know one only says 'positive definite' of symmetric matrices in the first place...

Alias User

I didn't think so. Hrm. I'm trying to think - if M is the inverse of a covariance matrix, it would be the inverse of a symmetric matrix. That's not anti symmetric, is it?

It looks like it is symmetric. I'll post my work on a SE question and see if there's any errors.

Thanks!

dREaM

1:15 AM

I need to prove if G is a group with exactly one element $a$ of order $2$ then $\prod_{g\in G}g=a$

I'm pretty sure that product isn't uniquely defined unless the group is abelian. Any feedback?

In the abelian case it is trivial.

Nevermind, I found the errata

it has to be abelian.

Kevin Driscoll

2:27 AM

@StanShunpike Hey, I hope you've found a good answer to your question by now, but if you haven't let me say something and then give a link. In the wave equation, $u(x,t)$ is generally the displacement of the string from flat at a particular point, AKA it's the amplitude. If you think about it like $\vec{F} = m \vec{a}$ then you can identify the term that is the 2nd derivative in time as the acceleration. The other term, then, must be a force that is proportional to the 2nd derivative in space.

Stan Shunpike

@KevinDriscoll But what is a second derivative in space?

Thanks for replying by the way

Ted Shifrin clarified this I think

He said it is just the usual curvature of a line

or like that is what is is referring to

Kevin Driscoll

@StanShunpike And in fact if we start with an ideal string, consider a very small segment of it, and apply Newton's 2nd Law to that segment we find exactly that it leads to the wave equation. You can see a full derivation from newton's Law here: math.ubc.ca/~feldman/m267/wave.pdf

Stan Shunpike

so a straight line would have a second derivative of zero

Kevin Driscoll

@StanShunpike Yes, that's right. The second derivative is related to the curvature. The more curved the string is, the bigger the force with which the edges of the string are pulling on the middle with

Stan Shunpike

Ah! That makes intuitive physical sense :)

Kevin Driscoll

2:30 AM

@StanShunpike You can see that directly in the derivation. The 2nd derivative term arises precisely as a way of rewriting the vertical component of the tension in the string.

Stan Shunpike

@KevinDriscoll I was looking at this explanation tutorial.math.lamar.edu/Classes/DE/TheWaveEquation.aspx

And I was a bit confused what the term

$Q(x,t)$ stood for

I know it says it

But I still didn't really quite get what it was

It seemed to be saying something about gravity affecting the string

But I didn't really understand that because I thought we usually ignore it

Kevin Driscoll

Yes, that's right. The $Q$ are some external forces that are not caused by the string itself

Often yea, they are negligible

Stan Shunpike

But that's my confusion. What does "other forces" mean?

Like is this besides

the force creating the plucking / wave in the first place?

Kevin Driscoll

The derivation I linked to goes through in more detail how the derivative of the tension shows up, just by thinking geometriccaly about a small segment of the string

Stan Shunpike

ooo very nice. I didn't see the link sorry. I will look at that in detail!

Kevin Driscoll

2:35 AM

Yea, so there are internal forces which are caused by segments of the string pulling on other segments. Those forces go in the $T$ terms, for tension. Then external forces like gravity, air resistance, if you're driving the string by constantly plucking it, etc go in the $Q$.

Stan Shunpike

But would a single pluck be included in Q?

Or do we assume the string starts out in a plucked state?

Kevin Driscoll

Usually not. There are 2 ways one could include the initial pluck.

Either you could say at time $t<0$ the string in flat. That at $t=0$ we come along and pluck the string. And so we get some kind of force like $\alpha \delta(t)$ and that would go in the $Q(x,t)$.

OR you could say at time $t=0$ I tell you the initial amplitude at some point on the string and the velocity at that point. Then there is no $Q$ but our string started out in some non-flat state and we can see what happens from there

Stan Shunpike

is $\delta$ the dirac delta function?

Kevin Driscoll

either way is a valid way of modeling that initial 'pluck,' but usually people use the 2nd one because it's easier.

Yea

Stan Shunpike

huh, that's really interesting.

Kevin Driscoll

2:39 AM

It's a powerful and important thing that both methods work.

Because it gives you flexibility. Sometimes you care exactly what kind of 'pluck' is happening. For example, it could be a constant oscillating for like $\sin(x,t)$

But sometimes you DON'T care very much what kind of pluck it was, you just want to know how the string behaves in the long run. Then you can just use the initial conditions instead and not include any external forces.

Stan Shunpike

That's awesome.

Kevin Driscoll

It's actually exactly the same as with Newton's Laws. If you have a ball at rest, you can either described exactly what kind of force you use to throw it

Stan Shunpike

I got into all this when I was learning about music theory. Is this stuff used in QM?

Kevin Driscoll

OR you can saw, look after Ive thrown it, its at posiiton $x_0$ with velocity $\vec{v}_)$, now what happens?

Yes, the Schrodinger equation is another kind of wave equation and one goes about solving it in a very similar way. In that case, the $Q$ terms are like the potential that the system is moving in.

Stan Shunpike

Yes! That's what I thought. :)

Excellent

What's ur favorite QM book?

Kevin Driscoll

2:49 AM

Hmmmmm, good question. In my opinion, there is not a single best 'beginner'

quantum mechanics book, like there is for E&M (in that case its Griffiths)

Stan Shunpike

I have Griffiths and Ballentine

Griffiths E&M book is great

I also like Purcell

and I have Jackson

Kevin Driscoll

Jackson is really quite poor because its so old fashioned. It has a lot of good information but it isn't presented so well. I prefer Zangwill for graduate level E&M, which is sort of a modern Jackson

Stan Shunpike

mmm excellent

Kevin Driscoll

for Quantum, I think Griffiths is good for starting out, I havent read Ballentine

Stan Shunpike

I dislike Jackson for that reason

I just got it because it came up a bunch of times

but it seemed really oddly organized

Kevin Driscoll

2:51 AM

yea, many many schools use it

Stan Shunpike

I got a used version that clearly is reaaaaally old

Kevin Driscoll

Then I would say Shankar is pretty good for a slightly deeper level of QM

and my favorite graduate level book is Cohen-Tannoudji

Cohen-Tannoudji is excellent because it is organized in an unusual way for a physics textbook. The chapters are shorter, general theoretical outlines. Then, each chapter has 5+ appendices where each appendix is devoted to a specific problem using what was discussed in teh chapter.

It's less good when you want a sort of 'guided tour' broad overview. But its excellent when its time to get down to brass tacks and learn, say how to apply perturbation theory to a system with degenerate energy levels.

Stan Shunpike

3:06 AM

@KevinDriscoll Great! I have been unsure

I have heard good things about Shankar

But I had not heard of Cohen-Tannoudji

I will have to look at that one.

user147690

@robjohn Shouldn't blocked people not appear on the starboard by design?

Mike Miller

@AlexClark No, the starboard is universal. This isn't a bad idea, necessarily; for instance, what if you had ignored Clarinetist? Then you wouldn't see the chat guidelines.

Kevin Driscoll

@StanShunpike Yea the key on Cohen-Tannoudji is that he spend half or maybe more than half of the 2nd volume going over how to do perturbation theory. This is really the right thing to do because there are almost 0 exactly solvable problems and almost everything that we do is perturbation theory of some kind.

user147690

@MikeMiller Well you could set pinned posts to override to counteract that

Mike Miller

Sure, though one can make similar arguments (what if a room has no mods but still has chat guidelines? what if it's a message users unanimously agree should be seen?) In any case, this is a SE thing, not an MSE thing.

Stan Shunpike

3:15 AM

@KevinDriscoll omg thats great!

I need to be better at perturbation theory

user147690

@MikeMiller Sure

Stan Shunpike

@KevinDriscoll I didn't know where to look

Kevin Driscoll

@StanShunpike Yea its all specific to the problem one is working on. There are about 1,000,000 different types of perturbative approaches, depending on what you're doing

Stan Shunpike

Thats why its confusing lol

Too many books to choose from

Kevin Driscoll

@StanShunpike The best book I know of is Bender and Orsazg, Advanced Mathematical methods for scientists and engineers

Stan Shunpike

3:24 AM

Nice! I will check on that too! This has been very productive lol

Kevin Driscoll

@StanShunpike If you want apreview, Carl bender has a series of 15 or so lectures on youtube. It's fascinating stuff. Convergence acceleration, resummation of divergent series, local and global analysis of differential equations, etc.

Stan Shunpike

Nice! I will definitely do that. I love youtube lectures

Mike Miller

4:31 AM

@PVAL: The question never got answered on MO so if you get a chance to ask Dan about his thoughts I'd definitely still be interested.

robjohn

@AlexClark do you mean blocked or ignored?

Anthony

@MikeMiller Ayo mike

do you know why the existence of sylow p-subgroups implies the existence of p-groups of all lesser powers?

Mike Miller

@Anthony: At that point all you need to do is convince yourself every $p$-group has a subgroup of order $p^k$ for all $k$.

You can induct this upwards.

Anthony

Do you mean starting from Cauchy's theorem?

Mike Miller

4:47 AM

Yeah.

Anthony

I see. I'll think about it!

Thanks.

Mike Miller

I remember this being a little tricky and i don't remember the details.

Anthony

It's cool. Thanks!

Stan Shunpike

5:19 AM

@KevinDriscoll So when did you first start liking physics?

Kevin Driscoll

@StanShunpike Middle school sometime. I read The Universe in a Nutshell and had a series of like binders where they sent you a pack of inserts every month. A good bit was focused on astronomy but there was particle physics in it too.

Stan Shunpike

We were taught some in middle school and I thought it was stupid

Kevin Driscoll

My perception radically changed when I took my firs tphysics class in high school

Stan Shunpike

Exactly! my physics teacher in high school taught me Newtonian mechanics and I was hooked. I think it was the difference between shitty teachers in middle school and having him in high school.

I felt appalled that I almost didn't like it

before and realized how much the teaching influenced me

Kevin Driscoll

because I had never really considered how physics really worked at a fundamental level. I thought of it more like being a mechanic or engineering before I took a class. But once I got some experience I was really impressed with the power of even simple things like predicting trajectories of thrown objects

Stan Shunpike

5:25 AM

Yeah! It was neat. I remember covering SR and was floored by it. I just couldn't believe how much physics could shape how I think about the world.

Kevin Driscoll

When I think about Quantum mechanics its funny becuase there are certain questions I don't ask now that I found incredibly strange when I first started

For example, when one considers a particle in an infinite square well, theres a nonzero probability of finding it in the left or the right half of the well

But 0 probability for the exact center. So, how does it get from one side to the other?

Huy

3

Kevin Driscoll

@Huy ....What book is that from?

Stan Shunpike

@KevinDriscoll I remember being super confused in chemistry class because my teacher

Huy

Apparently Glencoe Algebra II 2014 edition. @KevinDriscoll

Stan Shunpike

5:35 AM

tried to explain about wave functions in high school

and I just wasn't really read to grasp it because I didn't even know what a vector space was yet

Kevin Driscoll

@Huy Whoops!

Stan Shunpike

@TedShifrin evening professor :)

Ted Shifrin

Hi @Stan et al

Stan Shunpike

lolol

et al

Huy

morning @Ted

Kevin Driscoll

5:38 AM

@StanShunpike Yes, unfortunatley trying to explain what wavefunctions ARE is something that takes a physics degree, a philosophy degree, and several years of arguing to decide

We're like the end of the Gilligan's Island theme song......" And the rest..."

Ted Shifrin

Goodnight @Huy

Stan Shunpike

@KevinDriscoll Yes, exactly. In retrospect, I feel like my teacher probably should not have been so ambitious.

But I liked the enthusiasm for the subject matter

a lot of people don't have that

Ted Shifrin

You can be Maryanne, Kevin.

Mike Miller

Hi @Ted.

Ted Shifrin

Goodnight, Mike

Kevin Driscoll

5:40 AM

@Ted I guess that makes you the professor

as long as I don't have to be the Millionaire's wife

Agawa001

@TedShifrin i perceived a little change in your identicon, does that imply some reputational growth ?

Ted Shifrin

I may well have no cell/wifi for a few days, Mike, but I'll try to let you know when I get to Santa Barbara or thereabouts.

Stan Shunpike

I didn't have service while in canada. It was a traveshamockery

Ted Shifrin

I Have no idea, Agawa.

Mike Miller

@Ted: I'm sending you something on facebook.

Ted Shifrin

5:43 AM

oh.

Mike Miller

lol

Kevin Driscoll

Its an invitation to help Mike pick varieties in FarmVille

Mike Miller

I have no idea why my thing from earlier has three stars.

13

Agawa001

me too

just take first initiative and wait for a torrent of stars

Stan Shunpike

LOL

is that how it works

?

Agawa001

5:53 AM

wait and see

Mike Miller

-_-

Stan Shunpike

people just click on stuff if it looks mildly interesting and clickable.

doesn't really matter what it is.

Agawa001

theres nothing easier than a mouseclick

the lower button is rather less arduous to move up the cursor to

Ted Shifrin

BTW, @Stan @Anthony, I strongly support the endorsement of Bender/Orszag. Also,I like Jim Keener's books.

Stan Shunpike

6:08 AM

@TedShifrin what in particular do you like about Bender/Orszag?

Agawa001

7:31 AM

someone good at financial mathematics ?

Stan Shunpike

7:42 AM

@Clarinetist is quite good in my experience

I would recommend asking him

@Agawa001

Lucas

What you choose between Faculty of Accounting and Faculty of Mathematics

Agawa001

@StanShunpike problem: my questions are in french

i ll try translate some technical terms when he logs in

Stan Shunpike

Yes that could be an issue but Clarinetist might at least get u started

Lucas

@Agawa001 what you advise me to choose between Faculty of Accounting and Faculty of Mathematics

I saw a question from you about financial mathematics

Agawa001

@Lucas i cant listen your inner voice :)

Stan Shunpike

7:50 AM

Lol

Lucas

Nice answer

Agawa001

anyways, i hate statistics and finance. it was the most hateful 1 hour session per week i ever spent

Lucas

@agawa where are you from?

Agawa001

north algeria

Lucas

You say that .."it was the most hateful 1 hour session.." I suppose that you're beginning?

Agawa001

8:05 AM

no, its the last year i study such stuff

Lucas

What college are you?

Agawa001

do you know my region ?

what a wonderful piece of heaven on earth-face ruled by idiots

thats where i live :)

Owatch

8:22 AM

Has anyone done Derivation of compound statements?

It's basically given x,y = A, B, find statements that realize (x) post condition.

user147690

thetruesize.com - Type a country in and see it's size in comparison with other countries

Owatch

This is a bit confusing to me: X == A, Y == B, to X*Y < 0, X+Y==A+B

user147690

@robjohn Well neither hide nor ignore work, but I meant ignore

user147690

thetruesize.com - If you are going to use this someone, type in New Zealand and drag it up to the UK

Owatch

8:43 AM

Well, I'm stuck.

If I make x = A-B, and y = B-A, then x*y is < 0

Because unless A and B are equal, then x*y will be negative.

However, this also makes x+y = 0

IDK what I'm doing.

robjohn

9:17 AM

@Owatch If $(x+y)^2=0$, then $xy=-\frac{x^2+y^2}2$

don't know if that helps

Transcript for

$Mathematics$ Mathematics