Recent content by eddo

Incase anyone doesn't understand the notation, GL(2;C) is the group of linear transformations on C^2 which are invertible. Another way of looking at it is all complex 2x2 matrices with non-zero determinant. It is fairly easy to show that GL(2;C) is not simply connected (just define a...

Thank you. Intuitively that's what I expected to be the case, because of the relationship between O(n) and SO(n). It makes sense that antipodal points get identified if you mod out by O(n) because the reflections that O(n) has over SO(n) would identify these points. But you can't always count on...

For an assignment, my prof asked that we show that S0(3)/SO(2) is isomorphic to the projective plane (ie the 2-sphere with antipodal points identified). Here's my problem. I checked in a textbook for some help, and it claimed that SO(3)/SO(2) is isomorphic to the 2-sphere. So which one is right...

That sounds very interesting. I'll have to take a good look at that when I have time. I actually noticed some very serious mistakes in the statement of the theorem on Wikipedia. First of all a minor type: they keep saying deviance when they mean deviation. Much more importantly, in the...

Alas, my good friend Wikipedia has answered my question for me. Here's the link for those who are interested: http://en.wikipedia.org/wiki/Curvature_vector It generalizes to n-dimensions pretty much how you would expect. I found the part about how the curvature vectors are simply found by...

I just finished learning the fundamental theorem of curves in 3 dimensions. As a reminder, this is the theorem that states that a continuous, C infinity, unit speed curve in 3d is uniquely determined by its curvature and torsion (up to actions by SE(3), that is rotations and translations)...

Thank you, you could use that to prove by contradiction that if x<=y<=z, then d(x,y)<=d(x,z) and d(y,z)<=d(x,z). This was the only missing link in my proof, so assuming the rest was correct, i guess the squeeze theorem is true for an arbitrary metric on R.

It is relatively simple to prove the squeeze theorem on the reals, using the usual metric. My question is, can you prove the squeeze theorem on R for an arbitrary metric (on R)? Does this even hold for an arbitrary metric on R? It seems to me that part way through the proof, you would need to...

Given nxn matrices A and B, and an n-vector x, are there any conditions that can guarantee Ax=Bx implies A=B? I started thinking about this well working on an assignment. It is clearly not always true, since you can easily think up 2x2 examples where it is not. You can also think up examples...

I have two questions. How do you find the equation of a cone given data points? I've found lots of info on the equation of a cone, but can't find anything on one that is rotated and not centered at the origin. What is the equation for a rotated translated cone? Second, given the equation of a...

KataKoniK, the way you solved it only applies to 2 people, his question was referring to a class with n people. Clearly the answer is going to involve n, since the more people there are, the more likely two of them share a birthday. You can think of it as each person "chosing" a birthday...

As part of my summer job working with one of my professors, I've written a C++ program to do Runge-Kutta of order 4 on a system of 3 equations. I've been playing around with it using the Lorenz equations and using maple to graph the results. I was wondering if anyone could tell me some initial...

I have limited knowledge of this, but since no one else is answering, I will offer what I have. Incase you don't already know, the first moment is the expectation, which can be interpreted as a sort of center of mass. The second central (about the mean/expectation) moment is the variance, which...

Nevermind I got it. You just have to turn P(AnB) into 2P(AnB)-P(AnB), then group one of each of the positive terms with your other two terms. Axiom 3 of version 1 can than be used in reverse to turn each of these into a single probability, where the events in question simplify, using boolean...

I tried this but wasn't able to work it out. How do you go from P(AnB) to -P(AnB)? Once I get to the second step I'm not sure how to write the two terms other than P(AnB) in any other way, since they involve intersections, and the axioms version 1 don't say anything about intersections. You can...