@TedShifrin I've heard of Backlund transformationss in the context of integrable systems. Could never understand what they were or why they were so interesting.
@TedShifrin MM wants me to learn Riemannian geometry, so sounds like I'll get back to your diff. geom. notes and further soon enough... but I have got like 6 months to figure out how to fit in that in my schedule
@TedShifrin the domain of $ f(x,y) = \dfrac {x+y}{x-y} $ is $ {(x,y) | x \neq y}$. How can I graph that ? Its all the numbers on the graph but the diagonal that goes through origin with slope 1
Like in a class, there's only so much to do, so it'd make sense for you to be selective @Ted. In the case of my physics TA, we'd be done with complex analysis whenever we needed to be done
If ${\sim}:=(x\sim-x)$, in general, $(S^n\times S^m)/{\sim}$ isn't the same as $(S^n/{\sim})\times(S^m/{\sim})$, right?
(I've never been sure the best what the best way to write the definition of equivalence relations is.)
(I mean the quotient maps obtained by identifying opposite points.)
The latter space is $\rm P\Bbb R^n\times P\Bbb R^m$.
They're the same in the case of $n=m=1$; they both become the torus. I don't quite know what $(S^n\times S^m)/{\sim}$ would be for general $n$ and $m$, though. (I mean, hopefully it's easily describable in terms of spaces I know.)
@AkivaWeinberger By ~ in S^n x S^m you mean identifying (x, y) with (-x, -y) as a subspace of R^(n+1) x R^(m+1), yeah? I think (S^2 x S^1)/~ should be a nontrivial RP^2-bundle on S^1.
@Arrow What is the context? A map between manifolds with boundaries?
@BalarkaSen the more general the better. For starters, I don't see why pulling back compacts should have anything to do with boundaries in the continuous case.
I mean, are you working with boundaries of subsets of metric spaces, or manifolds, or what? Those are not the same things. I can't tell you anything unless you give me a precise statement.
In general any map from a compact manifold with boundary to something with no boundary is proper, because, well, the domain is compact. So that's rubbish.
Precise statements are what I'm looking for. Say a set-function between spaces is proper if it pulls back compacts. What are some facts in any of the settings you mention which relate properness to boundaries
@Arrow Ok, suppose your map is $f: U \to V$. $x \in \partial U$ be a point in the boundary; pick $x_k$ converging to $x$. Then $f(x_k)$ converges to something in the codomain (could be not in $V$ - that's what we are supposed to verify). If it converges to something in $V$, the $f(x_k)$ along with it's limit point would be compact. Preimage would be compact by properness; rubbish, the limit point of $x_k$ lies outside of $U$
So $f(x_k)$ converges to something not in $V$; so it's in the boundary.
I think you worry about intuition more than the math. I didn't use much about the intuition up there, just the definition. The point is proper maps shouldn't sent "stuff at infinity" to finite areas
I think proper map between locally compact Hausdorff spaces extend to their one point compactifications
it's about Lagrangian/Hamiltonian mechanics. For the sake of concreteness let's say I have a system described by $2$ spatial coordinates, let's say $R$ and $\theta$ (motion of a point on a cone or something) and I get a Langrangian $L(t,R,\theta,\dot R,\dot\theta)$ such that $\frac{\partial L}{\partial \theta}=0$ so that the quantity $\frac{\partial L}{\partial \dot\theta}$ is conserved.
If I now want to write an Hamiltonian $H$ for the system I use the transformation $P_\theta=\frac{\partial L}{\partial \dot\theta}$ and similarly for $P_R$. Now in the equations for the motion of this point I have $\frac{d}{dt}P_\theta=0$ so it's kinda like my system "lost a dimension" since my Hamiltonian only depends on $t,R$ and $P_R$ now
I'm having trouble understanding why this "losing a dimension" means physically
@MikeMiller It's quickly getting harder to understand the symbolically written stuff in C-C, so let me verify. Suppose $M$ is a manifold with a foliation; let $x$ and $y$ be two points in a leaf; pick a path $\gamma$ from $x$ to $y$. Cover it by by a plaque chain; then it takes a small nbhd of the transversal at $x$ diffeomorphically to a small nbhd of the transversal at $y$ by moving along the plaques using the holonomy cocycles.
Germ of this local self-diffeom (R^q, 0) --> (R^q, 0) is the holonomy of the leaf associated with $\gamma$, yes?
I'll probably check an example. For trivial bundles it's... trivial. For the foliation associated to the Klein bottle bundle it spits out germ of an orientation-reversing diffeomorphism of R at some point, doesn't it?
I'm not good at accepting exceptions, @Balarka. Hmm, maybe I am, though. Just not you.
@Alessandro: I don't see why you say you lost a dimension. The constant of motion is still a parameter (like an initial condition).
But I did have to giggle at Balarka's telling you to avoid someone. You certainly know whom he means from months in chat here. I'm surprised he's in physics and not back here.
I see a better way to write my bijection between it (unit singular matrices) and $S^1\times S^1$, @TedShifrin. Map $(u,v)\in S^1\times S^1/{\sim}$ to $uv^{\top}$, where $(u,v)\sim(-u,-v)$.
The science volunteering I'm doing with 4th graders is fun, but it's literally killing my body.
yeah, DogAteMy, I'm fine with it.
So the (unit) singular $3\times 3$ ones should not be a smooth manifold, DogAteMy. This is something I worked with a lot in research for many years. The rank 2 guys form a smooth hypersurface, but the rank 1 guys form a (smooth) 4-dimensional submanifold inside it. The question is whether those are singular points of the top stratum. I believe so.
I like the mathematical aspects of the Lagrangian and Hamiltonian formulation and I see some of the beauty in them, but then the exercises are just meh at best, a lot of calculations and not much more
@TedShifrin I suppose it's equivalent to how $\int_0^{2\pi}\cos(n\theta)\operatorname d\!\theta$ equals $0$ unless $n=0$, at which point it equals $1$. EDIT: $2\pi$.
I am trying to calculute this tripple integral over the region x^2+y^2-z^2 = 1 , 0<z<1 , i did solve it but am curious how can it be solved piece by piece , ie setting up the integral dx dy dz or any permutation of that
@TedShifrin I'm supposed to write something about Baire spaces for a small project thingy, I wonder how much descriptive set theory I can fit in there :P
@TedShifrin the way i solved it was by first taking double integral of the circle with radius sqrt ( z^2 +1) then last took integral wrt dz from 0-1 , how cylindrical help us here?
Aren't those synonymous of meagre and nonmeagre? Anyway I'm going to sleep now since I have this famous physics exam tomorrow in the morning, good night everyone!
