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{displaystyleqquadqquadqquadqquadqquadqquadquadtextbf{Homotopy
Dimension of SO (n) and its generators
The generators of $SO(n)$ are pure imaginary antisymmetric $n times n$ matrices. How can this fact be used to show that the dimension of $SO(n)$ is $frac{n(n-1
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Mathematical Fallacy
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The Tuesday Birthday Problem
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
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Fundamental group of the special orthogonal group SO(n)
Also, if I''m not mistaken, Steenrod gives a more direct argument in "Topology of Fibre Bundles," but he might be using the long exact sequence of a fibration (which you mentioned).
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lie groups
Welcome to the language barrier between physicists and mathematicians. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators.
Boy Born on a Tuesday
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
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Prove that the manifold $SO (n)$ is connected
The question really is that simple: Prove that the manifold $SO (n) subset GL (n, mathbb {R})$ is connected. it is very easy to see that the elements of $SO (n
Find Bayesian Nash Equilibria
The father then randomly handed each son one of the two envelopes with a probability of $0.5$. After both sons opened their envelopes, his father privately asked each son whether he
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