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Orthogonal Matrices

Representation Theory

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Basic question about real-analytic functions summing this binomial series If $f: \mathbb{N} \to \mathbb{N}$ and $\forall x \in \mathbb{N},\,(f \circ f )(x) = x^{4}$, is $f$ polynomial? Need help with this proof via induction If $X\subset L^1$ is a closed vector space and $X\subset \bigcup_{1<p\leq\infty} L^p$ then $X\subset L^q$ for some $q>1$. Three externally touching circles have their centers on the same line and have radii $a$,$b$ and $c$ (where $a<b<c$).Prove that $b^2=ac$. Can a triangle ABC be made if $\frac{\cos A}{2}=\frac{\cos B}{3}=\frac{\cos C}{7}$ Given a self-map $h$ of a (closed?) manifold, is there a vector field $\xi$ with flow $\Phi_t$ such that $h = \Phi_1$? The relation between uniform integrability and dominated convergence theorem in the case of counting measure Write $\cos(5x)$ as a function of $\cos(x)$, answer with a polynomial. Limit related to $ f(x) = \prod_{i = 1}^x \left( \sin\left( i \frac{\pi}{n}\right) + \frac{5}{4}\right) $? Understand part of a proof to show that if K is normal in G, then K is the kernel of a homomorphism

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