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Let $\{U_{k, n}\}_{k=1}^n$, denote the order statistics of a sample of $n$ iid uniform $[0, 1]$ variates.
Note that, marginally $U_{k, n}$ is distributed $\mathrm{Beta}(k, n+1 -k)$.Therefore, let us denote the corresponding mean and variance parameters, respectively, by$$\mu_{k, n} = \frac{k}{n+1} \quad \mbox{and} \quad \sigma^2_{k,n} = \frac{k (n+1 - k...

I have a question about a very basic construction that seems to be absent from the standard literature.
Every differential geometry textbook introduces$$S^2 = \{ x \in \mathbb{R}^3 : \|x\| = 1 \}$$and develops the geometry of the sphere using the ambient metric of $\mathbb{R}^3$. This is convenient, but it regularly leaves students (and even some graduate...

Consider a vector bundle $E$ with compact structure group $G$ over $\mathbb{R}^n$, and a smooth connection $D$ in this bundle compatible with the structure group. Denoting the curvature of this connection as $F$, suppose this connection has finite Yang...

I had asked this question on MathStackExchange but did not get any response. It will be very helpful to have an answer:
I have some doubts from the book 'Homological algebra' by Cartan and Eilenberg. On page 253, the authors describe a resolution as follows:...