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Maths and Theoretical Physics Course Thread TR031 TR035

Maybe_Memories

There's a load of people popping up saying they're doing maths this year, and there's a load who already do maths, so we should have a thread, especially since the engineers have one. :P

So yeah, first years, ask questions, etc, etc... :cool:

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Aoibheann

\o/ Finally we get a thread. How about including the TPs in the title though? There's a good few of 'em lurking around, even if most of us do eventually transfer to Maths. >_>

Oh, and I'll be happy to answer any questions too.

Maybe_Memories

Good idea! I've no idea how to change the title though

Use your Modly powers!

Aoibheann

Maybe_Memories wrote: »

Good idea! I've no idea how to change the title though

Use your Modly powers!

My modly powers are useless in this forum.. :pac: You might be able to do it by editing your post, or perhaps one of the TCD mods would do us a favour?

Jonathan

Aoibheann wrote: »

My modly powers are useless in this forum.. :pac: You might be able to do it by editing your post, or perhaps one of the TCD mods would do us a favour?

Done.

PurpleFistMixer

Hey, not most TPs switch to maths!

SS TP here, in case anyone has any questions... (or questions about doing schols in TP I guess...)

Maybe_Memories

I'd say during the year this thread will become full of "can someone explain how to do such and such a problem." :pac:

Aoibheann

PurpleFistMixer wrote: »

Hey, not most TPs switch to maths!

SS TP here, in case anyone has any questions... (or questions about doing schols in TP I guess...)

I could have been talking about the lurkers! I'm sure plenty of 'em have switched.. Or will switch!

Actually, surprisingly only 5 people have transferred from my year, so you are of course correct. A lot more have dropped out, and a few (3/4 perhaps) have wandered into physics.

Someone should just tell the first years how to do the Jordan Normal questions now tbh.. >.> Also, if you guys don't like Pete, you're dead to us. :P

Tears in Rain

Aoibheann wrote: »

I could have been talking about the lurkers! I'm sure plenty of 'em have switched.. Or will switch!

Actually, surprisingly only 5 people have transferred from my year, so you are of course correct. A lot more have dropped out, and a few (3/4 perhaps) have wandered into physics.

Someone should just tell the first years how to do the Jordan Normal questions now tbh.. >.> Also, if you guys don't like Pete, you're dead to us. :P

I'm not convinced anyone knows how to do the Jordan Normal questions :S

Maybe_Memories

Tears in Rain wrote: »

I'm not convinced anyone knows how to do the Jordan Normal questions :S

I do! :P

And posting a method for JNF is actually a very good idea, which I might do later on today.

Fringe

I will never switch!

Maybe_Memories

Since JNF is one of the things most people find difficult and the supplemential exams are next week I thought some people might find this useful..

Suppose we have a linear operator A:V->V where A is the matrix of the operator relative to some basis, presumable the standard orthonormal basis of R^3.

The purpose of finding the Jordan Normal Form is because it makes the matrix appear simpler in a different basis so we can more easily understand what the operator does.

Right, so take the matrix A. Firstly, find its characteristic equation found by
det(A-tI)=0, where I is the identity matrix.

The solutions to the equation are the eigenvalues of our matrix.
Define a new matrix B=A-tI. There will usually only be one or two eigenvalues, I haven't seen a question of Vlad's yet where there was 3 distinct eigenvalues.

For each eigenvalue find rk(B), rk(B^2), ..., rk(B^k-1), rk(B^k)
until rk(B^k-1) = rk(B^k).
Obviously then dimKer(B^k-1) = dimKer(B^k) because of the formula
rk(A) + dimKer(A) = dim(V)

NOTE: CASE 1 AND 2 ASSUME TWO EIGENVALUES

CASE 1: rk(B) = rk(B^2)
Find a basis of ker(B) given by the vector v, Bv=0.
v is our basis vector. Since the kernels of powers stabilise instantly we get a thread of length one, and we have our one basis vector, and we are done.

CASE 2: rk(B) =/= rk(B^2) = rk(B^3) =/= 0
Reduce the vectors which span Ker(B^2) using the vector which spans Ker(B) to get a relative basis vector f, which is the non-zero vector obtained after reducing. Since the kernels of powers don't stabilize for two steps, find Bf and reverse the order to get the basis vectors Bf, f.

CASE 3: If B is nilpotent, that is to say (B^k)=0 for some k. There are certain intricacies to this case.
Firstly, assume two distinct eigenvalues.
Find the vectors that span ker(B^k-1), and our first basis vector is the vector which "makes up for the missing leading 1". If k=2, find Bf and we are done.
Reverse the order to have a basis; Bf, f.

Now assume only one eigenvalue. If k=3, do the same as above, but find f, Bf and (B^2)f. Reverse the order for the basis. (B^2)f, Bf, f.

****** Now suppose only one eigenvalue and k=2. Find f, and Bf, and then reduce ker(B) using Bf to get a vector g.
Basis given by g,Bf,f.

Right, you have your jordan basis. More than likely you will be using two of the cases above, one for each eigenvalue.
So you get a transition matrix C, which transforms A into it's Jordan Normal Form, J.

J = (C^-1)AC

In general, J is given be each of your eigenvalues along the diagonal.
The multiplicity of each factor of eigenvalue in your characteristic equation determines the size of your jordan block. Best way to illustrate is with an example. Suppose the characteristic equation is t(t-3)^2
Eigenvalues are 0 and 3, and 3 has a multiplicity of 2, so the jordan normal form is given by

.... 0 0 0
J = 0 3 1
.... 0 0 3

A one is placed over the intersection of two same eigenvalues, so if your characteristic equation was (t-3)^3, J would be

.... 3 1 0
J = 0 3 1
.... 0 0 3

This is true in all cases except the one marked with ******. In this case you essentially have two jordan blocks of different size, but each has the same eigenvalue, so you'd get

.... 3 0 0
J = 0 3 1
.... 0 0 3

This is a result of the Cayley-Hamilton theorem and minimal polynomials etc.

One last thing, you put the basis vectors in corresponding order of the eigenvalues of your JNF. Example, if your eigenvalues are 2 and 3, and for 2 you get a basis of f, and for 3 you get a basis of g and Bg, and you arrange your J as

.... 2 0 0
J = 0 3 1
.... 0 0 3

then the columns of your transition matrix C is f, Bg, g

So there you have it, Jordan Normal form explained.

Lisandro

Fantastic, we have our own thread (with the mathematics students, of course, I haven't forgotten you). I've noticed that there are a lot of fellow theoretical physics students knocking around here, it should mean this thread will be quite active over the next year.

Maybe Memories, I can tell you're going to be getting a lot of thanks for that post, Jordan Normal Form was a big problem area for our year, most people seemed to just take the bitter pill and learn the algorithm and how to apply it. I would, however, like to add an adjoint (groan) to your explanation, because my understanding of the Jordan Normal Form comes from having read Lectures on Linear Algebra by I.M. Gel'fand rather than from Vladimir (this is a section where the two of them very peculiarly differ in their approaches.

To preface this note, I will assume the reader is familiar with the properties of linear operators, changing bases and how basis vectors transform under said operators. The elements of a canonical matrix will be the co-ordinates of the images of the new basis vectors with respect to that basis; hence a diagonal matrix of eigenvalues multiplies every basis eigenvector by a factor lambda. However, when an eigenvalue repeats itself, we seek canonical basis vectors of the form:

Ae1=λ1e1
Ae2=e1+λ1e2
Ae3=e2+λ1e3
...
Aen=e(n-1)+λ1en

Af1=λ2f1
Af2=f1+λ2f2
...
Afm=f(m-1)+λ2fm

...etc., etc., the numbers are subscripts and the total number of basis vectors will equal the dimension of the matrix. Most of those sets of vectors (eg. e1 to en) will consist of just one vector; often it is just one of these sets that contains an extra eigenvector. The canonical matrix, therefore, will take the form:

λ1...1..0.
.0..λ1..0.
.0...0..λ2

The eigenvector corresponding to λ2 can easily be computed, as can one of the two basis vectors (the eigenvector) corresponding to λ1. The problem is finding that third basis vector - remarkably, this process is quite simple. Note that:

Ae1=λe1
(A-λ1E)e1=0

and that:

Ae2=e1+λ1e2
(A-λ1E)e2=e1

Now:

(A-λ1E)^2(e2)=(A-λ1E)e1=0

Our inference is that the missing basis vector maps onto an eigenvector, then onto the zero vector. It is always possible to select such a vector and there are infinitely many of these vectors, it is simply necessary that that vector yield an eigenvector as its image. Once this condition is satisfied, we now have a set of basis vectors that satisfy the conditions laid out several paragraphs earlier.

I wanted to mention this because finding the basic vectors can be done rather intuitively by re-examining the form of the vectors we seek - by going back to basics, we can unjumble the algebraic spaghetti that forms the end product of any Jordan Normal Form algorithm (incuding this one). Much as Vladimir is fantastic, the way he taught the Jordan Canonical Form was very difficult to digest, and I'm surprised he didn't refer more to Gel'fand when he was explaining it, as he occasionally drew on methodologies from Gel'fand's book during other sections of the course.

PurpleFistMixer

While it is a good idea to provide explanations etc. for common problems, I would personally advise doing up nice LaTeX documents and putting them on your personal maths webpages - it's more permanent, easier to refer to, and easier to read than a forum post! (not that I object with forum posts, it just makes more sense to LaTeX stuff like that, to me.)

... And if you don't know LaTeX, it's a good reason to start learning.

Now my own question: does anyone have any idea of who will be teaching general relativity this year?

edit: Also some exciting news which TPs may be interested in: this year's Hamilton lecture will be given by Edward Witten! It will be sometime in October on something like "The Quantum Theory of Knots".

Tears in Rain

Thanks for the two explanations of JNF guys

PurpleFistMixer wrote: »

blah blah blah words...your personal maths webpages...

What's the deal with those anyway? How do we get/use them?

PurpleFistMixer

Tears in Rain wrote: »

What's the deal with those anyway? How do we get/use them?

Do they not do 061 in first year any more? The incredibly basic computer course where you get your maths account and are encouraged to make a website? (there are probably other elements to the course but I don't remember them) In practice I just scp to my maths account and do everything from there, but if you don't have a maths account I can see a problem.

Tears in Rain

PurpleFistMixer wrote: »

Do they not do 061 in first year any more? The incredibly basic computer course where you get your maths account and are encouraged to make a website? (there are probably other elements to the course but I don't remember them) In practice I just scp to my maths account and do everything from there, but if you don't have a maths account I can see a problem.

I went to the first few but they were painfully slow so I gave up on them. That said, there doesn't seem to be anything on Timoney's 061 page explaining how to do it...either way it's a moot point since I just figured it out

dabh

PurpleFistMixer wrote: »

Now my own question: does anyone have any idea of who will be teaching general relativity this year?

It has come up in conversation. The School of Mathematics were interviewing a number of candidates earlier this month for temporary lectureships to meet essential teaching needs...

PurpleFistMixer

dabh wrote: »

It has come up in conversation. The School of Mathematics were interviewing a number of candidates earlier this month for temporary lectureships to meet essential teaching needs...

Just so long as they don't cut the course... that would be very uncool.

Maybe_Memories

What's the story like with the maths schols?
Are the papers generally the same standard as final exam papers or way more hardcore including stuff you've never done in class?

Bottleopener

Will be doing Economics/Maths so guess this thread partly applies to me

. Looking forward to this big time, and it's awesome to see there's a few others from here doing the same course =D

Fringe

Maybe_Memories wrote: »

What's the story like with the maths schols?
Are the papers generally the same standard as final exam papers or way more hardcore including stuff you've never done in class?

They're fine. Maybe a bit harder than the summer ones but everything on the paper will be covered in class. Similar questions come up every year so it's something that you can prepare for.

fh041205

I remember last year I think we were asked by Donal to fill out a sheet giving advice to people coming in to Maths. I wrote a decent paragraph or two. But now I wish to retract it.

My advice; wiki everything.................EVERYTHING

Didn't Stalker do GR last year? Why wouldn't he do it again this year? Actually has anyone left the maths dept since last year lecturer-wise?

Maybe_Memories

fh041205 wrote: »

Didn't Stalker do GR last year? Why wouldn't he do it again this year? Actually has anyone left the maths dept since last year lecturer-wise?

Yeah Vlad left this year. As in he was here for all of this year but wont be here from September on. Pete is taking over Linear Algebra but apparently they don't have someone to do analysis yet. I'd imagine Stalker could do it.

fh041205

Oh no Vlad was so nice. Stalker, first year analysis. Didn't that go not-so-well last year.....? Good luck freshers!

Lisandro

It's a pity Vladimir's leaving, he was such a fantastic guy. He'd always begin a lecture with, "Uh, I'd like to begin this lecture with a note on linear operators..." in his deep Russian voice that would inevitably take at least ten minutes. When he did proofs (which was often), he would get really excited as he approached the end of one, and once finished would turn to face us and announce the bottom line with an elated smile on his face. The Maths Department will miss Vladimir.

Stalker (fantastic name) took Analysis of the Real Line in 2010, but Pete took it in 2011. It was half and half between the two of them (Pete Analysis I, Stalker Analysis II) but last year, it was just him. I've never had him, but from what I've heard, his lecturing style was very dry, which caused people to stop coming to lectures and not get very good results in the end of year exams. Himself and Pete came over together from MIT a number of years ago, and Stalker himself taught Pete; I've heard that Pete was one of his best students.

If indeed Pete teaches Linear Algebra, it'll be a surprise, because he's always struck me as being more inclined towards calculus, and algebra of vector spaces is a distinctly different domain. Indeed, it makes me curious to see how different he'll be to Vladimir; Pete's style is very clear and structured, whereas it was very easy to fall off Vladimir's intricate train of thought if you're not functioning at peak capacity. I'll have to interrogate a couple of first years on that one.

As first years will learn soon, his real name isn't Pete; it's Paschalis Karageorgis. Pete is simply a nickname he likes to use with English speakers.

mr_brightside

Starting Theoretical Physics now in a few weeks, just wondering how many people are actually in the course? And when do we actually start?

Oh and also, how many hours are the lectures, and all in all how much time does it take up? Many thanks.

Tears in Rain

I think there's 44 people in the course, it's roughly that number anyway. You start lectures the week after Fresher's Week, which starts Monday 19th September (so classes start Monday 26th).

As for number of hours, I don't know off-hand, but as courses go it's heavy enough hours wise.

Oh yes, one thing you should know about TP (well you probably know already) is that the course is jointly taught by the Maths and Physics department. There's not really much communication between the two, so for example, you'll be getting two timetables, one for the Maths modules from the Maths Department, and one for the Physics modules from the Physics department. Here's the Maths timetable for 1st year last year as an idea of the hours you'll have (although I know that your modules will be slightly different to ours). Unfortunately, I couldn't find the Physics timetables online, though I can say that it's less hours than Maths, and there's a big 3 hour lab in the middle of the week.

Also, you ask how much time it takes up...well, it'll take a lot more time than the lecture hours anyway. TP is a course that you need to put a lot of time and effort into to get anything out of it, and while 1st year is relatively forgiving, I still got bitten by making the mistake of not putting as much effort into it as I should have done, and I'm having to sit a repeat next week. Apparently, it only gets worse in 2nd year, so I won't be making that mistake again...

Lisandro

Mr. Brightside,

As Mark (Tears in the Rain) has already said, it's around 44 (our particular year was 41). Most lectures will have 80-100 people, as you mix with Mathematics and Science students. The course has about 24-hours of lectures per week, weighted in a ratio of approximately 2:1 in favour of mathematics (albeit it can be argued that it's really half and half, as the Mechanics course taught by the school of mathematics is very much physics taught from a slightly more abstract standpoint than the equivalent course that the science students get).

There'll be plenty of time for social stuff too, you'll get to be around your coursemates a lot; however, I would also encourage you to socialise with some of the Maths students and the Science students; it'll take you all over the east end of the college, but it'll give you a rather well-rounded Hamiltonian experience.

fh041205

I feel like Pete will be wasted in linear algebra. Its fairly self-explanatory, very leaving cert-esque. Having said that, I don't know what I'd rather he be teaching, or who I'd put in Linear ahead of him, but I feel he could teach a more complex course very effectively.

Also, is it just me, or are maths students the craziest bunch of people you've ever met for going out and social stuff? Honestly the first semester last year nearly killed me, and bankrupted me in the process.

Tears in Rain

fh041205 wrote: »

I feel like Pete will be wasted in linear algebra. Its fairly self-explanatory, very leaving cert-esque. Having said that, I don't know what I'd rather he be teaching, or who I'd put in Linear ahead of him, but I feel he could teach a more complex course very effectively.

Also, is it just me, or are maths students the craziest bunch of people you've ever met for going out and social stuff? Honestly the first semester last year nearly killed me, and bankrupted me in the process.

I had that impression too, that the course, at the start at least, was 'Leaving Cert-esque', but it's something I really only felt for the first half of the year. After Christmas, when we had moved on from computing numerical results to actually reasoning about vector spaces algebraically, proving theorems, and incorporating more abstract ideas into the fold, the course seemed to be a lot more in line with what I felt university level Maths was about. I found the first half of the course incredibly dull, but the second half really interesting. (That said, I couldn't do the second half, but that's a different story).

As for Pete being wasted on it, to be honest, I think to a certain extent, all first year Maths modules are 'beneath' all of the lecturers in the sense that it's so trivial to them, but on the other hand, it's the mark of a good teacher to be passionate and bring new insight into a subject that would ordinarily be considered dull and trivially easy. Who for example, would consider Feynman, a Nobel prize winning physicist, wasted in teaching the introductory Physics classes that went on to be compiled as his Lectures in Physics?

Furthermore, obviously as a student I'm no expert in pedagogy, but I'd imagine that teaching introductory Mathematics modules has additional difficulties and responsibilities, since in some sense you're responsible for stewarding the transition from the blind rule-following of Leaving Cert Maths (and let's face it, no matter how much they try to deny it, or change it, this is what LC Maths is and will be for the foreseeable future) to the world of university Maths which involves developing a sophisticated mathematical intuition where previously there was simple pattern matching from a small palette of possible problems, mathematical rigour where previously arm-waving arguments were acceptable, and working with abstract mathematical structures where previously there was a tacit assumption of working in the Euclidean plane over the field of Reals. This transition can be jarring, and being responsible for introducing it to students is a task I don't feel is 'beneath' anyone.

Also, nights out yeah, can't speak for Maths but got nicely comatose on some TP nights out

edit: Noticed you're going into SS while I've just finished JF, so if anything I said there is a load of bollocks, please correct me