Recent content by InvisibleBlue

aahhhh! I completely forgot about the whole independence issue! Thanks a lot. This really helped!

Hi, I'm trying to prove that the projective n-space is homeomorphic to identification space B^n / ~ where for x, x' \in B^n: x~x'~\Leftrightarrow~x=x' or x'=\pm x \in S^{n-1}, The way I have tried to solve this is, I introduced: {H_{+}}^{n}=\{x\in S^n | x_n \geq 0\} Then...

Actually, I'm now suddenly very confused. We say that B_{t} - B_{s} = B_{t-s} for B = (B_{t})_{t\geq0} brownian motion. So this must mean that they have the same distribution. But if they are standard brownian motion (i.e B_{t} ~ N(0,t)) then we get that B_{t} - B_{s} ~ N(0, t+s) and B_{t-s}...

So You think it's enough to show that those 2 have the same distribution? Problem is that the notion of Gaussian Process is not introduced or used in this course. I guess I just have to use the properties without saying where it comes from.

Hi, I'm trying to prove that X=(X_{t})_{t\geq0} is a Brownian Motion, where X_{t} = tB_{1/t} for t\neq0 and X_{0} = 0. I don't want to use the fact that it's a Gaussian process. So far I am stuck in proving: \[ X_{t}-X_{s}=X_{t-s} \quad \forall \quad 0\leq s<t \] Anyone has any ideas?

Yes there is nothing illegal about it. The empty set is also an object (as is any other set). But you should remember that in the context of above {} is being treated as an object not a set, otherwise there is room for confusion: The range of f as defined above is not empty, it has one element...

-Regarding your problem with ({}(0), {}(1),...,{}(n),...) - {}(0), {}(1), etc are not different empty sets. They are all {}. Th integers assigned to them only determine the position. It's ok to do this with sets, in the same way that it is ok to have n-tuple (0,0,0,...) in a ring where 0 in a...