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Boy Born on a Tuesday - is it just a language trick? The following probability question appeared in an earlier thread: I have two children One is a boy born on a Tuesday What is the probability I have two boys? The claim was that it is not actual
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?
What is the relationship between SL (n) and SO (n)? To add some intuition to this, for vectors in Rn R n, SL(n) S L (n) is the space of all the transformations with determinant 1 1, or in other words, all transformations that keep the volume constant This is because the determinant is what one multiplies within the integral to get the volume in the transformed space SO(n) S O (n) is the subset in which the transformation is orthogonal (RTR
problem solving - Diophantus Lifespan - Mathematics Stack Exchange "The son lived exactly half as long as his father" is I think unambiguous Almost nothing is known about Diophantus' life, and there is scholarly dispute about the approximate period in which he lived There is no reason to think that the problem has a historical basis
semi-simple and simple lie group,SO (n) for n even You should edit your question using MathJax More importantly, you should use SO(n) S O (n) instead of so(n) s o (n) (the latter would be the notation for a Lie algebra) Lastly, do you know the definition of a simple (semisimple) Lie group?