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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
- geometry - Find the coordinates of a point on a circle - Mathematics . . .
I have a circle like so Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees For example, I have a radiu
- 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?
- lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange
Welcome to the language barrier between physicists and mathematicians Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators So for instance, while for mathematicians, the Lie algebra so(n) consists of skew-adjoint matrices (with respect to the Euclidean inner product on Rn), physicists prefer to multiply them by I think − i (or maybe
- In a family with two children, what are the chances, if one of the . . .
For example, suppose there is a social science study on 2 child families with at least 1 daughter-- in this situation, about 1 3 of the families will be daughter-daughter, 1 3 will be daughter-son, and 1 3 will be son-daughter You have to consider the full probability space of two trials (d-d,d-s,s-d,s-s) and eliminate the s-s possibility
- Universal covering group and fundamental group of $SO(n)$
If H H is a topological group which is both path-connected and locally path-connected (i e a connected Lie group such as SO(n) S O (n)), then any path-connected cover of H H inherits a unique group structure making the covering map a group homomorphism In fact for any such cover p: G → H p: G → H,we have ker(p) ≅π1(H) p∗(π1(G)) k e r (p) ≅ π 1 (H) p ∗ (π 1 (G)) This
- Homotopy groups O(N) and SO(N): $\\pi_m(O(N))$ v. s. $\\pi_m(SO(N))$
I have known the data of $\\pi_m(SO(N))$ from this Table: $$\\overset{\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\textbf{Homotopy groups of
- 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
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