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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
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
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
Why do $SO(n,\\mathbb{R})$ and $O(n,\\mathbb{R})$ have the same Lie . . . If one is willing to avoid all of the details regarding special types of matrices, there is a simple abstract reason that O(n;R) O (n; R) and SO(n;R) S O (n; R) should have the same Lie algebra, namely that the Lie algebra is an invariant of the local group structure of a Lie group, and the Lie groups O(n;R) O (n; R) and SO(n;R) S O (n; R) have isomorphic local group structures In more detail
Distinguishing $SU(n)$ from $SO(n)$ - Mathematics Stack Exchange Suppose that I have a group G G that is either SU(n) S U (n) (special unitary group) or SO(n) S O (n) (special orthogonal group) for some n n that I don't know Which "questions" should I ask to determine which one it is? e g which structural differences are there between these groups? Of course a question like "Is G G isomorphic to SU(n) S U (n) for some n n?" doesn't count I'm hoping for
group theory - Generators Of $SO (n)$ - Mathematics Stack Exchange As a starting point, consider the fact that g1,g2 ∈SO(n) g 1, g 2 ∈ S O (n) are conjugate if and only if they rotate through the same oriented angle So for example, gxgyg−1x =gz ∈SO(3) ⊆SO(n) g x g y g x 1 = g z ∈ S O (3) ⊆ S O (n), where gz g z is the rotation by π 2 π 2 about the z z -axis, etc So we can certainly get away with fewer generators for n ≥ 3 n ≥ 3
orthogonal matrices - Irreducible representations of $SO (N . . . @Jahan: 2) This is also not a problem You can check that if a connected Lie group G G acts on a finite-dimensional vector space V V then V V is irreducible as a representation of G G iff it's irreducible as a representation of g g We aren't classifying all representations here, just checking whether particular representations are irreducible, so the existence of the spin representations
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