ah thanks for that.
Things You've Always Wanted To Know
- Thread starter Jonesy
- Start date
ah thanks for that.
Mr. Tea
Let's Talk About Ceps
I love that quote of his about doing algebra, whereby he got the 'important bits' done on the first pass and then went back and adjusted all the fiddly details like factors of 2, pi and minus one. Or possibly even just left them to be filled in by some lackey, a bit like how renaissance masters did the interesting bits of a painting and had anonymous apprentices fill in the background.
IdleRich
IdleRich
Think that this is pretty much true. To some extent you can almost get through a maths degree with this approach. The problem is that if you don't have the understanding you can't deal with something very well when it varies from the versions you've seen in the past. I've been playing a lot of squash recently and I would draw an analogy with technique in this sport - if you haven't got a good and natural feeling technique you may well be able to deal with a certain number of shots but then you get tougher ones where you need to improvise and if you haven't got the basics instintively you will struggle.
Not to say that I have good squash technique or that can remember anything from my degree for that matter.
I'm interested in the minds whose workings you do have an idea about.
nomadthethird
more issues than Time mag
Not that I've taken a lot of advanced math but I remember I usually took someone else's answers and worked backwards to figure out algebra/trig.
High school geometry would have been easier if I had taken it after college level symbolic logic, which I had to pass to get a philosophy degree for some reason.
High school geometry would have been easier if I had taken it after college level symbolic logic, which I had to pass to get a philosophy degree for some reason.
don_quixote
Trent End
unfortunately the maths teaching at university is usually so poor that you have to rely on that level of understanding. after three years (i've been out of it a year and a half) i still don't have an understanding of what an isomorphism is and it came up every fucking year.
Mr. Tea
Let's Talk About Ceps
Ah, I seem to have confuzzled isomorphism with homomorphism:
"The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that iσj's are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one homomorphism from SU(2) to SO(3)."
http://en.wikipedia.org/wiki/Pauli_matrices#SO.283.29
So it depends whether you're talking about groups or the Lie algebras describing the groups. Bit out of my depth here, to be honest.
"The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that iσj's are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one homomorphism from SU(2) to SO(3)."
http://en.wikipedia.org/wiki/Pauli_matrices#SO.283.29
So it depends whether you're talking about groups or the Lie algebras describing the groups. Bit out of my depth here, to be honest.
Slothrop
Tight but Polite
It's one to one and 'structure preserving'. If there's an isomorphism between two things (an isomorphism from A to B automatically gives an isomorphism from B to A because they're invertible) it means that "in terms of the sort of structure that we're looking at, these are identical."
The 'structure preserving' is intuitively pretty sensible once you've seen a couple of examples, but I'm not sure I could write down a general definition without resorting to category theory. But you get things like the map
f: integers -> integers
given by f(x) = x + 1
Which is an isomorphism of sets (since it's 1 to 1 and sets have no further structure) but isn't an isomorphism of additive groups, since f(0) =/= 0 and f(x+y) =/= f(x) + f
But essentially, "A is isomorphic to B" means "A and B are the same thing described in different ways."
Mr. Tea
Let's Talk About Ceps
OK, cool. I remember the thing about SU(2) and SO(3) because in quantum mechanics SU(2) describes rotations in spinor space that correspond to the SO(3) rotations in configuration space - with the 2-to-1 homomorphism (as groups rather than algebras) being related to the way spin-1/2 particles have to be rotated through 720 degrees before the wave-function looks the same as it did originally, which I find fascinating.
Kinda confusing because as far as I can see, 'iso' and 'homo' are both ways of saying 'the same'. Edit: apparently 'iso' means 'equal', not 'same'. Isotope, isobar etc. etc...
Kinda confusing because as far as I can see, 'iso' and 'homo' are both ways of saying 'the same'. Edit: apparently 'iso' means 'equal', not 'same'. Isotope, isobar etc. etc...
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Mr. Tea
Let's Talk About Ceps
Yeah sure, like I said, I got isomorphism mixed up with homomorphism. JUST GIVE ME A BREAK OK, MISTER MATHS FACE!
baboon2004
Darned cockwombles.
Who changed the channel to Open University when I was away?
Mr. Tea
Let's Talk About Ceps
I love the way there are hundreds of threads full of ultra-abstruse cultural theory but the moment someone mentions maths it's like 'OMG nerdfest!'.
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Slothrop
Tight but Polite
Yeah, pretty much. A lot of the terminology is legacy stuff afaict.
If you get bored / have a chance, you might enjoy reading up on some category theory, which starts from the observation that whether you're dealing with Lie algebras, rings, function fields, topological spaces, sets or whatever, a lot of what you're doing can be talked about purely in terms of basic objects - and the group (or whatever) is a basic object in this case, rather than an element of the group - and the morphisms between them, and thus you can look at stuff in a much more clear and general way than always having to worry about the precise properties of a homomorphism of whatever baroque object you're looking at at the time.
don_quixote
Trent End
well i think you've gathered my problem. loved groups rings and modules and then suddenly they start throwing in iso and homos and my brain goes numb.
Mr. Tea
Let's Talk About Ceps
No need to be isophobic.
Cheers for the tip Slothrop, I may well get round to brushing up on some of this stuff, or learning it in the first place even, when I've got my thesis corrections done and sorted out a proper job.
don_quixote
Trent End
haha
ok, here's one: what's the best way to cook rice? i have my own methods but want to see other people's first.
ok, here's one: what's the best way to cook rice? i have my own methods but want to see other people's first.