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Fundamental group of the special orthogonal group SO(n) Question: What is the fundamental group of the special orthogonal group SO(n) S O (n), n> 2 n> 2? Clarification: The answer usually given is: Z2 Z 2 But I would like to see a proof of that and an isomorphism π1(SO(n),En) → Z2 π 1 (S O (n), E n) → Z 2 that is as explicit as possible I require a neat criterion to check, if a path in SO(n) S O (n) is null-homotopic or not Idea 1: Maybe
Dimension of SO (n) and its generators - Mathematics Stack Exchange The generators of SO(n) S O (n) are pure imaginary antisymmetric n × n n × n matrices How can this fact be used to show that the dimension of SO(n) S O (n) is n(n−1) 2 n (n 1) 2? I know that an antisymmetric matrix has n(n−1) 2 n (n 1) 2 degrees of freedom, but I can't take this idea any further in the demonstration of the proof Thoughts?
Improper integral of sin(x) x from zero to infinity I was having trouble with the following integral: ∫∞ 0 sin(x) x dx ∫ 0 ∞ sin (x) x d x My question is, how does one go about evaluating this, since its existence seems fairly intuitive, while its solution, at least to me, does not seem particularly obvious
Boy Born on a Tuesday - is it just a language trick? The only way to get the 13 27 answer is to make the unjustified unreasonable assumption that Dave is boy-centric Tuesday-centric: if he has two sons born on Tue and Sun he will mention Tue; if he has a son daughter both born on Tue he will mention the son, etc
Prove that the manifold $SO(n)$ is connected The question really is that simple: Prove that the manifold SO(n) ⊂ GL(n,R) S O (n) ⊂ G L (n, R) is connected it is very easy to see that the elements of SO(n) S O (n) are in one-to-one correspondence with the set of orthonormal basis of Rn R n (the set of rows of the matrix of an element of SO(n) S O (n) is such a basis) My idea was to show that given any orthonormal basis (ai)n1 (a i
The Tuesday Birthday Problem - Mathematics Stack Exchange In case this is the correct solution: Why does the probability change when the father specifies the birthday of a son? (does it actually change? A lot of answers posts stated that the statement does matter) What I mean is: It is clear that (in case he has a son) his son is born on some day of the week
Lie Algebra of U(N) and SO(N) - Mathematics Stack Exchange U(N) and SO(N) are quite important groups in physics I thought I would find this with an easy google search Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?